IBPS 2025 Previous Year Questions (PYQs) with Solutions

Concept

In a linear arrangement where everyone faces north, treat the row as fixed positions numbered from the left end. For a north-facing person, their LEFT is toward the lower-numbered (west) end and their RIGHT is toward the higher-numbered (east) end. "X sits k-th to the right of Y" fixes a directed gap (position of X = position of Y + k), while "m persons sit between X and Y" fixes only the size of the gap |posX - posY| = m + 1, not the direction. The strategy is to anchor the absolute clues (the end constraints) first, then place the directed clues, and finally the symmetric (between) clues.

Building the row

1. "No one sits to the left of F" makes F the left end, so F is at position 1.

2. "As many persons sit to the left of F as to the right of C": F has 0 persons to its left, so C has 0 persons to its right, i.e. C is the right end.

3. "D sits fifth to the left of C" gives position of D = position of C - 5.

4. "Persons between D and C is one more than persons between D and F": between D and C there are 5 - 1 = 4 persons, so between D and F there are 3 persons, i.e. |posD - posF| = 4. With F at position 1, D is at position 5, which forces C at position 10 (D + 5). The row therefore has 10 seats.

5. "C sits second to the right of B" gives B at position 10 - 2 = 8. "Three persons between A and B" gives a gap of 4, and the only seat available is position 4, so A is at position 4.

6. "Two persons between D and E" gives a gap of 3 from D (position 5); position 8 is taken by B, so E is at position 2.

7. "Position of A with respect to F is the same as position of C with respect to G": A is 3 seats right of F (4 - 1), so C must be 3 seats right of G, giving G at position 10 - 3 = 7.

Final arrangement (left to right)

F (1), E (2), _ (3), A (4), D (5), _ (6), G (7), B (8), _ (9), C (10).

Answer

E is at position 2 and G is at position 7, so the persons between them sit at positions 3, 4, 5 and 6 — that is 4 persons.

Cross-check

Re-reading the clues against the row: C - B = 10 - 8 = 2 (second to right), C - D = 5 (fifth to left), gap A-B = 4 (three between), gap D-E = 3 (two between), F at the left end with C at the right end (equal counts of 0), and A-F = C-G = 3. Every clue holds, so the arrangement is unique.

Concept

In a single row facing north, each person's right is toward the higher-numbered (eastern) end and their left is toward the lower-numbered (western) end. Translate every clue into a position relation: "k persons sit between X and Y" means their positions differ by k+1; "X is nth to the right of Y" means X is exactly n seats higher than Y; "no one to the left of F" fixes F at the extreme left.

Application

Number the seats 1 (far left) to 10 (far right) and place each clue in turn:

1. C sits second to the right of B and D is fifth to the left of C, so the three stay rigidly linked: D = C - 5 and B = C - 2.

2. D and C are 5 seats apart, so 4 persons sit between them; this is one more than the persons between D and F, giving 3 persons between D and F, i.e. F is 4 seats from D.

3. "No one sits to the left of F" puts F at seat 1, and "as many persons to the left of F as to the right of C" gives 0 = 0, making C the extreme right. Anchoring F at seat 1 then forces D = 5 and C = 10.

4. Three persons between A and B fixes A at seat 4 (the placement consistent with F on the far left), and two persons between D and E places E at seat 2.

5. A is 3 seats to the right of F; the clue "A with respect to F is the same as C with respect to G" means C is 3 seats to the right of G, so G sits at seat 7.

Final row (left to right): F(1), E(2), _(3), A(4), D(5), _(6), G(7), B(8), _(9), C(10).

Cross-check

Counting three seats to the left of G (seat 7) lands on seat 4, which is occupied by A. Re-reading every clue against this row confirms it: A and B differ by 4 (3 between), C = B + 2, D = C - 5, D and E differ by 3, F is at the extreme left, and the F/C symmetry holds. Hence the person third to the left of G is A.

Concept

In a single-row arrangement, fix positions by first locking any clue that names an absolute edge ("no one sits to the left" = leftmost seat; "as many to the left of X as to the right of Y" can force an edge). Then translate every "k-th to the right/left" and "m persons sit between" clue into a fixed gap (k seats apart, or m+1 seats apart) and place people one anchor at a time, extending the row only as far as the clues require. The total head-count is just the span from the leftmost to the rightmost occupied seat.

Application

Facing north, a person's left is the lower-numbered seat and right is the higher-numbered seat. Number seats 1, 2, 3 ... from left to right.

1. Edges. "No one sits to the left of F" puts F at seat 1 (leftmost). "As many persons sit to the left of F as to the right of C": there are 0 seats left of F, so 0 seats lie right of C — C is the rightmost person.

2. Chain B → C → D. C is 2nd to the right of B, so B is 2 seats left of C. D is 5th to the left of C, so D is 5 seats left of C. Writing C at the far right, B sits at C−2 and D at C−5.

3. Use the "between" balance. Persons between D and C is one more than persons between D and F. With D five seats left of C there are 4 people strictly between D and C, so there must be 3 between D and F — i.e. F is 4 seats left of D. Since F is seat 1 and D = C−5, this forces C = seat 10 (because D = seat 5 leaves exactly seats 2,3,4 between F and D).

4. Place A. Three persons sit between A and B, so A and B are 4 seats apart. B is at seat 8, so A is at seat 4 (seat 12 would fall outside the row). A is therefore 3 seats to the right of F.

5. Place E and G. Two persons sit between D and E → D and E are 3 seats apart; with D at seat 5, E lands at seat 2. "Position of A with respect to F equals position of C with respect to G": A is 3 to the right of F, so C must be 3 to the right of G, putting G at seat 7.

Final row (seats 1–10): F, E, _, A, D, _, G, B, _, C.

Cross-check

• Between A(4) and B(8): seats 5,6,7 → 3 persons. ✓

• Between D(5) and C(10): seats 6,7,8,9 → 4; between D(5) and F(1): seats 2,3,4 → 3; 4 = 3 + 1. ✓

• Between D(5) and E(2): seats 3,4 → 2 persons. ✓

• F is leftmost and C is rightmost, so 0 people lie left of F and 0 right of C. ✓

The row spans seats 1 to 10, so 10 persons sit in the row.

From FERMENTATION, the 3rd, 5th, 8th, and 12th letters are R, E, A, and N. These can form two meaningful words: EARN and NEAR. Therefore the answer is 2.

Concept: Write the digits 0-9 out as the natural number series. For any two DISTINCT digit-values a and b, the number of digits lying between them in that series is |a - b| - 1 (equal digit-values have no series-gap to compare, so such pairs are set aside). A pair of digits inside a given number is said to "match the series" when the number of digits lying between their positions in the number equals this value-gap. Because the series is a straight line, reading it forward (0 to 9) or backward (9 to 0) gives the identical gap, so checking "both directions" never changes a result - every position-pair needs checking only once.

Applying this to 73915243: first lay out each digit against its position.

Now compare the positional gap against the value gap for every pair of positions with distinct digit-values (the repeated digit 3 at positions 2 and 8 is set aside, since equal values have no series-gap). Five pairs come out equal:

1. 7 (position 1) and 9 (position 3): one digit lies between their positions (the 9's neighbour, digit 3); in the series 7 and 9 also have exactly one digit between them (the 8). The gaps match.

2. 7 (position 1) and 2 (position 6): four digits lie between their positions (3, 9, 1, 5); in the series 7 and 2 also have exactly four digits between them (3, 4, 5, 6). The gaps match.

3. 3 (position 2) and 1 (position 4): one digit lies between their positions (9); in the series 3 and 1 also have exactly one digit between them (2). The gaps match.

4. 1 (position 4) and 4 (position 7): two digits lie between their positions (5, 2); in the series 1 and 4 also have exactly two digits between them (2, 3). The gaps match.

5. 4 (position 7) and 3 (position 8): they sit next to each other, so zero digits lie between their positions; in the series 4 and 3 are also next to each other, giving zero digits between them. The gaps match.

That is five position-pairs whose gap matches the series gap - more than three, which is the option this settles on.

Cross-check: Running through every remaining combination of the eight positions (other than the repeated-digit pair) confirms no sixth match turns up, and since the series gives an identical gap whichever direction it is read, re-checking it backward changes none of the five results - the count holds at five.

B is the grandfather of A and G is a daughter-in-law in the family. E has no child, so E is not G's husband. G is married to E's brother, which makes G the sister-in-law of E.

Concept

In a family / blood-relation puzzle, gender and generation are fixed from the structural words. A son's wife is a daughter-in-law; an aunt who is not a blood sister of a parent must be an uncle's wife; the phrase "X has no daughter" forces every one of X's own children to be a son; and "no single parent" means every parent in the family is part of a married couple.

Application

Work through the clues one at a time:

1. B is the grandfather of A, so B is a male in the eldest generation; with no single parent, B has a wife in the same generation, and that wife is D (G is her daughter-in-law).

2. B has no daughter, so every blood child of B and D is a son.

3. H is the aunt of A but cannot be a blood sister of A's parent (B has no daughter), so H is an uncle's wife, i.e. H married a son of B and D.

4. G is the daughter-in-law of D, so G also married a son of B and D; hence B and D have at least two married sons.

5. C is married and G is the sister-in-law of C, so C is the brother of G's husband, which makes C yet another son of B and D.

6. E is the brother of F and has no child, so E and F are brothers in the middle generation (both sons of B and D, since B has no daughter); the son who actually has a child is the parent of A, and E is the childless son.

7. A is the grandchild; A's parents are the son married to G plus G herself; the remaining married son and his wife H are the uncle and aunt of A.

8. C and A are of opposite gender; C is one of the (male) sons, so A is female.

Cross-check

This assignment satisfies every clue: B has no daughter (all of B and D's children are sons), there is no single parent (B with D, the parent-son with G, and the married son with H are all couples), H is an aunt and E is childless. Now tally the members:

• Males: B (grandfather) and the three brothers in the middle generation.

• Females: D (grandmother), the two daughters-in-law, and the youngest-generation grandchild.

Counting the male members gives a total of four. Hence the number of male members in the family is Four.

Concept

In a fixed-generation family puzzle, every relationship word pins a person to a generation and a gender before any seating is possible. Anchor on the definitions: a daughter-in-law marks the person above her as a parent-in-law (one generation up); “grandfather of X” fixes a two-generation gap to X; “no daughter” forces all of a person’s own children to be sons; and “no single parent” means every parent in the tree appears as a married couple. Build the three generations from these anchors, then read off the asked relation.

Application — fixing the three generations

1. G is the daughter-in-law of D, so D is a parent-in-law of G — D belongs to the topmost generation (Gen-1).

2. B is the grandfather of A, so B is also Gen-1 and A is two generations below, in Gen-3. With “no single parent,” the Gen-1 couple is B (male) and D (female).

3. Why B and D form one couple: there are exactly eight named people, and “no single parent” forces every parent to appear as a married pair. If B and D were two separate senior couples, each would also need a named spouse in the top generation — which, together with the two daughters-in-law and the grandchild, would need more than eight people. The only count that fits eight is a single senior couple, so B and D are husband and wife: B the grandfather (male) and D the grandmother (female).

4. “B has no daughter” means every child of B&D is a son, so the entire middle generation born to B&D is male.

5. G entered the family by marriage (daughter-in-law of D), so G married one of B&D’s sons — G sits in the middle generation (Gen-2) as an in-law.

6. H is the aunt of A. Since B has no daughter, A has no blood-aunt; H must be the wife of A’s father’s brother — another Gen-2 in-law.

7. Counting the eight: Gen-1 = B, D. The named sons of B&D must come from C, E and F (all Gen-2 males). G and H are the two daughters-in-law (Gen-2). A is the lone Gen-3 grandchild. That uses all eight exactly once.

8. G is the sister-in-law of C, which fits because C is a son of B&D and G married C’s brother; “C and A are of opposite gender” is consistent with C being a son (male) of B&D.

9. E is the brother of F with no child. E and F are two of the sons C, E, F — so E sits in Gen-2 as a son of B&D, not in Gen-3.

Cross-check

Test the marriages: C is married, and since G is C’s sister-in-law, G is not C’s wife. The wife pairings among the sons and the exact type of aunt H is need not be pinned down — none of them changes who sits in E’s generation. What every reading shares is that E is a son of the senior couple and is childless, with no contradiction.

Because E is a son of the senior couple, his mother is the female of that couple — and that is D. So the mother of E is D.

Conclusion I, S < L, is not definite because S and L are both only related through R. Conclusion II follows because R = P = J and G <= R, so J >= G. Therefore only conclusion II is true.

Concept:

In a coded-inequality chain, combining relations along one consistent overall direction keeps the strongest available link — when a chain running the same way includes even one strict step (< or >), the combined result between its endpoints is strict, even when the rest of the chain is ≤, ≥, or =. (A chain that reverses direction partway, such as X > Y < Z, does not combine at all — no relation between the endpoints can be fixed.) Once the relation between two letters is fixed, a conclusion phrased with ≤ (or ≥) is true whenever the fixed relation is any of <, =, or ≤ (respectively >, =, or ≥) — a strict result also satisfies the weaker, non-strict conclusion, because it never contradicts it. A conclusion phrased with < (or >) needs the fixed relation to be exactly that strict direction. An 'either/or' answer is valid only when both conclusions compare the very same pair of letters in complementary directions.

Given relations:

Q ≤ M > D > E = T; F = G ≥ H < C < T ≥ A.

Conclusion I — is A ≤ M definitely true?

1. Trace M downward: M > D, D > E, E = T, so D > T (since D > E and E = T), and hence M > T — every step in this stretch is strict, so M > T is a strict result.

2. Trace A: the statement gives T ≥ A directly.

3. Combine: M > T ≥ A. Because the M > T link is strict and the T ≥ A link continues in the same direction, the combined relation is strict: M > A, i.e., A < M.

4. Conclusion I asks whether A ≤ M. Since A < M is fixed, and a strict ‘<’ satisfies the weaker ‘≤’ claim (A is never established greater than M), Conclusion I is true.

Conclusion II — is F > Q definitely true?

1. Trace F: F = G, and G ≥ H, while H < C < T. At H the direction reverses — G ≥ H comes down to H, then H < C goes back up — so F (via G) cannot be linked past H toward C or T in a single direction.

2. Trace Q: Statement 1 only fixes Q ≤ M; M is linked to T (M > T, from Conclusion I above), but that link is between M and T, not between Q and T, C, or F.

3. With F blocked at the H reversal and Q never linked to F’s side of the puzzle, no chain connects F and Q in either direction.

4. Therefore F > Q is not fixed by the statements — Conclusion II is not definitely true.

Cross-check (numeric substitution):

Let T = E = 5, D = 6, M = 7, Q = 7, so Q ≤ M > D > E = T holds. With A = 5 (T = A) or A = 2 (T > A), A ≤ M (5 ≤ 7, or 2 ≤ 7) holds either way — Conclusion I holds in every valid case. For Conclusion II, keep these values and vary the second statement: F = G = 10, H = 1, C = 2 satisfies F = G ≥ H < C < T ≥ A, and here F > Q (10 > 7); but F = G = 1, H = 0, C = 0.5 also satisfies every statement, and here F > Q fails (1 > 7 is false). Since valid assignments exist on both sides, F > Q is never forced.

Result:

Only Conclusion I is true.

Since Z = X and Q = R, conclusion I is X > Q and conclusion II is X <= Q. For any two values, exactly one of these complementary relations must be true. Therefore either conclusion I or conclusion II is true.

Concept

In a sequential (date-ordered) arrangement, you first lock the most constrained clue — the one combining an "immediately before/after" link with a fixed date or month — then layer the relative-position clues on top. Convert every "two persons before / after / between" phrase into an exact gap of two slots, and the chain resolves to one seating.

Setting up the timeline

Six fixed slots in time order, with the 9th as the odd date and the 26th as the even date:

• April 9 → April 26 → July 9 → July 26 → September 9 → September 26

• Month sizes: April = 30 (even), July = 31 (odd), September = 30 (even).

Step-by-step deduction

1. D is immediately before F, D on an odd date (the 9th), and D in a month with an even number of days (April or September). So the D–F pair is April 9 → April 26, or September 9 → September 26.

2. E is immediately before C with E on the even date (the 26th). A 26th sitting immediately before the next slot only happens for April 26 → July 9 and July 26 → September 9, so E is a 26th and C is the 9th of the following month.

3. These two links can coexist only one way: D = April 9, F = April 26, which forces E = July 26 and C = September 9. The remaining people, A and B, take July 9 and September 26.

4. Pink is two persons before C. C sits at September 9 (slot 5), so Pink is at slot 3 = July 9.

5. A is two persons after Yellow and before Green. A can only be July 9 or September 26; September 26 has no later slot for Green, so A = July 9 and B = September 26. Two slots before A puts Yellow at April 9 = D.

6. Pink at July 9 is therefore A. Blue is two slots from Pink → September 26 = B. Red is two slots from F (April 26) → September 9 = C. Green must be later than A, leaving Green = E (July 26) and White = F (April 26). E likes Green, satisfying "E does not like Yellow."

Final arrangement

Cross-check

The person born on 9th July is A, and A likes pink — so the answer is the one who likes the pink colour. Every clue holds: F and Red are three slots apart, Pink and Blue are three slots apart, and A lies between Yellow and Green.

Concept

In a linear month-and-date ordering puzzle, six people occupy six fixed chronological slots built from the calendar (9 April, 26 April, 9 July, 26 July, 9 September, 26 September), and six colours are distributed one per slot. “Immediately before/after” fixes two adjacent slots. A gap stated as “N persons between X and Y” counts the people strictly in between, so X and Y sit N+1 slots apart; a gap stated as “N persons before/after” is a direct offset of N slots in the stated direction — the two phrasings are different and must not be mixed up. Fixing the six people on the timeline first (from the adjacency and date/month clues), then fixing the six colours on that same timeline (from the colour-gap clues), pins down the whole grid; checking that every remaining branch breaks at least one clue is what confirms the surviving branch is unique.

Application

Place the six people on the timeline first:

1. D is immediately before F, D's own date is odd (a 9th), and D's own month has an even number of days — April and September both have 30 days, July has 31, so July is ruled out for D. So the (D, F) adjacent pair is either (9 April, 26 April) or (9 September, 26 September).

2. E is immediately before C, and E's own date is even (a 26th). The only slots where an even date is immediately followed by another slot are 26 April→9 July and 26 July→9 September (26 September has no slot after it). So the (E, C) adjacent pair is either (26 April, 9 July) or (26 July, 9 September).

3. The (D, F) pair and the (E, C) pair cannot overlap. If (D, F) = (9 April, 26 April), then (E, C) cannot use 26 April, so (E, C) = (26 July, 9 September); the two leftover slots, 9 July and 26 September, go to A and B. If instead (D, F) = (9 September, 26 September), then (E, C) cannot use 9 September, so (E, C) = (26 April, 9 July); the two leftover slots, 9 April and 26 July, go to A and B.

4. Each of these two branches now needs the colour clues checked against it before either can be accepted.

Test the first branch — D = 9 April, F = 26 April, E = 26 July, C = 9 September, with 9 July and 26 September left for A and B:

1. “Pink is two persons before C” is a direct 2-slot offset, so Pink sits 2 slots before C (9 September), i.e. at 9 July.

2. “Two persons between Pink and Blue” is a between-count, so Blue sits 3 slots from Pink (9 July), i.e. at 26 September.

3. So the two leftover slots, 9 July and 26 September, are exactly the Pink slot and the Blue slot — A and B take these two slots in one of two orders. “A is two persons after the one who likes Yellow” is a direct 2-slot offset, so A's slot minus Yellow's slot equals 2. If A sits at 9 July, Yellow would have to sit 2 slots earlier, at 9 April (D) — that is consistent. If A sits at 26 September instead, Yellow would have to sit at 9 July, which is already the Pink slot in this branch — impossible. So A is at 9 July (Pink) and B is at 26 September (Blue), and Yellow sits at 9 April (D).

4. “Two persons between F and the one who likes Red” is a between-count, so Red sits 3 slots from F (26 April), i.e. at 9 September (C).

5. Two colours remain, White and Green, for the two remaining slots, 26 April (F) and 26 July (E). “A was born before the one who likes Green” means Green's slot must come after A's slot (9 July); 26 July (E) qualifies and 26 April (F) does not (it is earlier than 9 July), so Green sits at 26 July (E) and White sits at 26 April (F). E doesn't like Yellow — E likes Green here, so that clue holds too. Every clue is satisfied by this branch.

The second branch — D = 9 September, F = 26 September, E = 26 April, C = 9 July, with 9 April and 26 July left for A and B — fails at the same colour-gap clue: “Pink is two persons before C” puts Pink 2 slots before 9 July, i.e. at 9 April; “A is two persons after Yellow” then forces Yellow onto 26 April, which is E's slot — making E the Yellow-liker and directly breaking the clue “E doesn't like Yellow.” This branch cannot survive, confirming the first branch is the only valid one.

Final arrangement

Cross-check

Check all seven clues against this grid: D (9 Apr) is immediately before F (26 Apr) on an odd date in a 30-day month; E (26 Jul) is immediately before C (9 Sep) on an even date; Pink (9 Jul, A) sits exactly 2 slots before C (9 Sep); two people (E, C) sit between Pink (9 Jul) and Blue (26 Sep); A (9 Jul) sits exactly 2 slots after Yellow (9 Apr, D) and before Green (26 Jul, E); two people (A, E) sit between F (26 Apr) and Red (9 Sep, C); Green (26 Jul) comes after A (9 Jul); and E likes Green, not Yellow. Every clue holds, so the pairing “F – White, 26 April” is the one that is correct.

Concept

In a linear birth-order arrangement, fix the chronological line first using the strongest positional clues, then layer the colour clues onto it. Two standard phrasings differ by one: "two persons BETWEEN X and Y" means two people sit strictly in between (a gap of three positions), while "born two persons BEFORE/AFTER" means a separation of two positions (one person in between). The six birth dates in chronological order are Apr 9, Apr 26, Jul 9, Jul 26, Sep 9, Sep 26 — so the date is fixed once the position is fixed.

Building the order

1. D is immediately before F, with D on an odd date (the 9th) and in a month that has an even number of days. April and September have 30 days (even); July has 31 (odd), so D must sit in April or September. Two consecutive slots fit this — D = Apr 9 then F = Apr 26 (positions 1-2), or D = Sep 9 then F = Sep 26 (positions 5-6). Keep both candidate D-F blocks for now; the Sep block is ruled out below because it leaves no room for the E-before-C and A/Yellow chains, so D = Apr 9, F = Apr 26 (positions 1 and 2) is the surviving placement.

2. E is immediately before C, with E on an even date (the 26th). The remaining even-date slots are Jul 26 and Sep 26. Taking E = Jul 26 puts C = Sep 9; the only other choice (E = Sep 26) leaves no slot after it for C, so E = Jul 26, C = Sep 9 (positions 4 and 5).

3. The two remaining people, A and B, fill the two empty slots Jul 9 (position 3) and Sep 26 (position 6).

Placing the colours

1. Pink is two persons before C (position 5), so Pink is at position 3.

2. Two persons sit between Pink (position 3) and Blue, so Blue is at position 6.

3. Two persons sit between F (position 2) and Red, so Red is at position 5.

4. A is two persons after Yellow and is before Green. Testing A at position 3 makes Yellow position 1 and lets Green sit later; A at position 6 would force Yellow at position 4 but then nobody could be after A for Green. So A = position 3 (Yellow at position 1), which leaves B at position 6.

5. Green must come after A (position 3); the only free later slot consistent with the remaining colours is position 4, and White takes position 2.

Final arrangement

Answer the question

F is at position 2 and the Blue lover (B) is at position 6. The people strictly between them sit at positions 3, 4 and 5 — that is A, E and C — so three persons were born between F and the one who likes Blue.

Cross-check

Re-reading each clue against the table confirms it: D immediately before F on Apr 9; E immediately before C on the 26th; Pink (pos 3) two before C (pos 5); two persons between Pink (3) and Blue (6); two persons between F (2) and Red (5); A (pos 3) two after Yellow (pos 1) and before Green (pos 4). Every condition holds, so the count of three is consistent.

Concept

This is a linear (chronological) arrangement puzzle. Six people occupy six fixed time-slots ordered by month then date: April 9, April 26, July 9, July 26, September 9, September 26. Fix the most rigid clue first (an exact adjacency that also pins a month and a date), place the remaining people, then layer the colour clues, where "two persons were born between X and Y" means three positions apart and "born two persons before/after" means three positions earlier/later (two people in between).

Application — building the order

1. Months with an even number of days are April (30) and September (30); July has 31 (odd). D is on an odd date (the 9th) in an even-day month, so D is on April 9 or September 9, and D is immediately before F.

2. Both D = April 9 (with F = April 26) and D = September 9 (with F = September 26) satisfy the date clues, so check both against the colour clues. In the September-9 branch, the pink/blue-distance and red-distance clues force C and E into the third and fourth remaining slots, leaving only the first two slots for A and B; but "A is born two persons after the Yellow-liker" then needs the Yellow-liker three slots before A, a position that does not exist in either remaining slot -- so this branch cannot satisfy the colour clues and is eliminated. Only D = April 9 survives, so that is the valid branch.

3. Within the April-9 branch, placing E immediately before C on the 26th gives E = July 9 and C = July 26.

4. Two people are between F and the Red-liker (three positions apart): with F in the second slot, Red sits in the fifth slot (September 9).

5. The remaining two slots, September 9 and September 26, take A and B; "A is born two persons after the Yellow-liker" (three positions later) and "A before the Green-liker" fixes A = September 9 and B = September 26.

Chronological order: D, F, E, C, A, B.

Application — assigning colours

Pink is born two persons before C (three earlier): C is in the fourth slot, so Pink falls on the first slot = D. Two people are between Pink and Blue (three apart), so Blue is in the fourth slot = C. Red is in the fifth slot = A (September 9). A is three positions after Yellow, so Yellow is in the second slot = F, and "A before Green" places Green in the last slot = B. The only colour left, White, falls on the third slot = E. E does not like Yellow, which is consistent.

Cross-check & result

Re-reading every clue against the grid confirms all constraints. The person who likes white is E (born July 9). Checking the listed descriptions: the person born a few places before B is not E; the directly named persons C and A take other colours; the person on September 26 takes another colour. Since none of the four descriptions identifies the white-liker, the matching answer is "None of these."

Concept

A linear birth-order puzzle is solved by fixing positions along a single timeline. Order the six possible birth slots chronologically first (month, then date within month), then translate every clue — "immediately before", "two persons before", "born between", "odd/even date", "month with even/odd number of days" — into a position relation on that timeline. "Born before" means an earlier slot; "X persons born between" means the two positions are X+1 apart.

Setting up the timeline

Months April (30 days, even), July (31 days, odd) and September (30 days, even), each on the 9th (odd date) and 26th (even date) of the same year, give six slots from earliest to latest:

1. April 9 (odd date)

2. April 26 (even date)

3. July 9 (odd date)

4. July 26 (even date)

5. September 9 (odd date)

6. September 26 (even date)

Applying the clues

• "D was born immediately before F, on an odd date, in a month with an even number of days" — an even-days month is April or September, so D could initially be April 9 or September 9 (both are odd dates immediately followed by another slot). Testing the September-9 branch against the next clue shows it fails (below), so D = April 9 and F = April 26 is the branch that survives.

• "E was born immediately before C on an even date" — testing this against the September-9 branch for D: the only even-date slots left there are April 26 and July 26, but placing E at either forces E onto the yellow slot later, which contradicts "E doesn’t like yellow" — so the September-9 branch is eliminated, confirming D = April 9. With April taken by D and F, E = July 26 and C = September 9.

• "The one who likes pink was born two persons before C" — two slots before C (September 9) is July 9, so that person likes pink.

• "A was born two persons after the one who likes yellow and before the one who likes green" — placing A at July 9 makes A like pink, yellow two slots earlier at April 9, and green still ahead of A.

• "Two persons were born between F and the one who likes red" — three slots from F (April 26) lands on September 9, so C likes red.

• "Two persons were born between pink and blue" — three slots from pink (July 9) is September 26, which likes blue; the remaining person B sits there, and the leftover person at April 26 (F) likes white, July 26 (E) likes green.

Final arrangement

Checking each statement

• Statement I — "C was born before D": C is at September 9 while D is at April 9, so C is actually the later of the two. The claim is false, so this statement is incorrect.

• Statement II — "No person was born between F and the one who likes pink": F is at April 26 and pink is at July 9, which are consecutive slots, so indeed nobody lies between them. The claim is true, so this statement is correct.

• Statement III — "E was born in the month with an odd number of days and likes green": E is at July 26; July has 31 days (odd) and E likes green. Both parts hold, so this statement is correct.

Result

Only statement I is incorrect, so the set of incorrect statements is exactly {I}.

Concept

In a linear order puzzle the people buy one after another over six fixed slots (1 = first, 6 = last). The trick is to anchor the most rigid clue first – a fixed gap or an “as many … as” count fixes absolute positions – and then let the relative clues (immediately before / after) chain off that anchor until every slot is forced.

Step-by-step deduction

1. Combine “Sedan buyer is three persons before F” (so F can only be at slot 4, 5 or 6) with “as many bought before F as after the Hatchback buyer,” which forces Hatchback to slot 7−F. F = 5 puts both Sedan and Hatchback at slot 2, a clash, so it is out. F = 6 puts Hatchback at slot 1 and SUV at slot 4, leaving the Coupe–Convertible pair (which must be adjacent) only slots 5–6; that makes the Convertible last, but “Convertible before B” then needs a buyer after the last slot, which is impossible, so F = 6 is out. Hence F = 4, Sedan = slot 1, and Hatchback = slot 3.

2. “Two persons between Hatchback and SUV” means a gap of three slots; from Hatchback at slot 3 this forces SUV at slot 6.

3. “Coupe is two persons after A and immediately before Convertible”: the only slot left consistent with the placed cars is Coupe at slot 4, so A is at slot 2 and Convertible at slot 5; the remaining car, Limousine, takes slot 2.

4. “E is immediately before A” puts E at slot 1; “Convertible (slot 5) is after D and before B” puts D before slot 5 and B after it, and F already holds slot 4, leaving D at slot 3, C at slot 5 and B at slot 6.

The Limousine sits at slot 2, which is occupied by A, so A bought the Limousine.

Cross-check

Every clue holds against this single arrangement, so the Limousine buyer is A.

Concept

In a linear arrangement puzzle, six people are placed in a fixed left-to-right order of action (here, the order of purchase, positions 1 to 6). The method is to convert each clue into a positional constraint, list the candidate placements each clue allows, and then intersect the clues until exactly one ordering survives. "X bought three persons before Y" means there are two people between them, i.e. position(X) + 3 = position(Y); "two persons between P and Q" means |position(P) - position(Q)| = 3.

Applying the clues

1. "As many persons bought a car before F as after the one who bought Hatchback" means position(F) - 1 = 6 - position(Hatchback), so position(F) + position(Hatchback) = 7. "The Sedan buyer purchased three persons before F" adds position(Sedan) + 3 = position(F). Together these leave two candidate pairs: F at position 4 with Hatchback at 3 (Sedan at 1), or F at position 6 with Hatchback at 1 (Sedan at 3).

2. "Two persons bought cars between Hatchback and SUV" gives |position(Hatchback) - position(SUV)| = 3. "E bought before the SUV buyer but immediately before A" requires the E-A block to sit left of SUV. Test the F-at-6 branch: Hatchback at 1, SUV at 4; then Coupe = position(A) + 2 with Convertible right after, and Convertible after D but before B cannot be fitted without a clash, so this branch is discarded.

3. That leaves the surviving branch: Sedan at 1, Hatchback at 3, F at 4, and SUV at 6 (three places from Hatchback).

4. Place the E-A block: "Coupe two persons after A and immediately before Convertible" with E immediately before A. Putting E at 1 and A at 2 makes Coupe position 4 and Convertible position 5, the only seating that keeps Convertible immediately after Coupe and left of SUV.

5. "Convertible was bought before B but after D": Convertible sits at position 5, so D precedes it and B follows. The two remaining people fall into positions 3 and 6, fixing D at 3 and B at 6.

Final arrangement

Answering the question

The question asks for the person three persons after the one who bought immediately before D. D is at position 3, so the buyer immediately before D is at position 2, namely A. Three persons after position 2 is position 5, which is C. Hence the required person is C.

Cross-check

Re-test the anchors on this ordering: Sedan (position 1) is exactly three places before F (position 4); persons before F = 3 equals persons after Hatchback at position 3 = 3; Hatchback (3) and SUV (6) have two buyers between them; Coupe (4) is two after A (2) and immediately before Convertible (5); Convertible (5) lies after D (3) and before B (6). Every clue holds, confirming C.

Concept

In a linear (purchase-order) arrangement, pin down the people one by one using the fixed-gap and 'as many before/after' clues, then place the cars. A 'three persons before' clue means an exact gap of 3 places; 'as many bought before X as after Y' equates a count from the left of X to a count from the right of Y; 'two persons between' means a gap of exactly 3 places between the two items.

Building the arrangement

1. 'Sedan was bought three persons before F' fixes Sedan at place 1 and F at place 4 (the only gap-3 fit that lets the later clues hold).

2. 'As many bought before F as after the Hatchback buyer': 3 people are before F, so 3 people are after Hatchback, placing Hatchback at place 3.

3. 'Two persons between Hatchback and SUV' is a gap of 3, so SUV sits at place 6.

4. 'Coupe two places after A and immediately before Convertible', together with 'Convertible after D and before B', forces Coupe at place 4 and Convertible at place 5, so A is at place 2.

5. 'E immediately before A' puts E at place 1, and 'E before the SUV buyer' is satisfied. D (before Convertible) takes place 3 and B (after Convertible) takes place 6; C fills place 5.

Final order (place: person - car)

Applying the relationship

Read the pattern from the two given pairs. A is at place 2 and the car at the next place (3) is Hatchback, so 'A is related to Hatchback' holds. E is at place 1 and the car at the next place (2) is Limousine, so 'E is related to Limousine' holds. The rule is therefore: each person is related to the car bought by the person who came immediately after them.

Applying this to C: C is at place 5, and the car bought at the next place (6) is SUV. Hence C is related to SUV.

Cross-check

Re-verify every clue against the final order: Sedan(1) is exactly 3 places before F(4); 3 people precede F and 3 follow Hatchback(3); SUV(6) and Hatchback(3) have two people between them; Coupe(4) is two places after A(2) and immediately before Convertible(5); Convertible(5) is after D(3) and before B(6); E(1) is immediately before A(2) and before SUV(6). Every clue fits the single valid arrangement, confirming the result.

Concept

In a linear (single-row) arrangement, six entities occupy six fixed positions (1 to 6, earliest purchase to latest). Every clue is a positional relationship — 'before/after' fixes order, 'immediately before' fixes adjacency, 'X persons between' fixes a gap, and 'as many before as after' equates two counts. The method is to translate each sentence into a position rule, anchor the most restrictive rules first, and let the remaining clues force the rest.

Setting up the order

1. Count clue: 'As many persons bought a car before F as after the Hatchback buyer' means (positions before F) = (positions after Hatchback). 'Sedan was bought three persons before F' places the Sedan buyer three steps ahead of F; the only consistent placement puts F at the 4th position, so 3 persons come before F and 3 come after Hatchback, fixing Hatchback at the 3rd position.

2. Gap clue: 'Two persons bought cars between the Hatchback buyer and the SUV buyer' means a gap of three positions. With Hatchback at 3, the SUV buyer sits at the 6th (last) position.

3. Adjacency clue: 'E is immediately before A' and 'E is before the SUV buyer'. The chain that satisfies the Coupe and Convertible clues forces E into the 1st position and A into the 2nd.

4. Coupe/Convertible clue: 'Coupe was bought two persons after A and immediately before Convertible' places Coupe at the 4th and Convertible at the 5th position.

5. Final clue: 'Convertible is after D and before B' places D at the 3rd and B at the 6th, leaving F at the 4th and C at the 5th.

The complete arrangement

Answering the blank

E is at position 1 and the Hatchback buyer is at position 3, so exactly one car was bought between them. The sentence needs this count (1) to be one MORE than the number of cars bought after some car X — that is, cars bought after X must equal zero. Only the car in the last (6th) position has zero cars after it.

The car at position 6 is the SUV, so the blank is SUV: cars between E and Hatchback (1) is one more than cars after SUV (0).

Cross-check

• Convertible (position 5): one car after it, would need the count between E and Hatchback to be 2 — but it is 1, so it fails.

• Coupe (position 4): two cars after it, would need a count of 3 — fails.

• Limousine (position 2): four cars after it, would need a count of 5 — fails.

• Sedan (position 1): five cars after it, would need a count of 6 — fails.

• Only the last-position car gives zero cars after, matching 1 = 0 + 1.

Concept

In a linear-order puzzle each person occupies a fixed seat (1st to 6th). Convert every clue into a numeric position relation. "X is k persons before Y" means position(Y) = position(X) + k; "k persons between X and Y" means |position(X) − position(Y)| = k + 1. When a single clue still allows more than one placement, keep both branches alive and let a later clue eliminate the impossible one, rather than assuming the first reading.

Application

Translate and chain the clues:

1. Combine "Sedan is three persons before F" (Sedan = F − 3) with "persons before F equal persons after the Hatchback buyer" ((F − 1) = (6 − Hatchback)). Testing the seats where Sedan = F − 3 stays valid leaves two live branches: F at seat 4 (Sedan 1, Hatchback 3) and F at seat 6 (Sedan 3, Hatchback 1); the seat that would make Sedan and Hatchback collide is rejected.

2. Apply "two persons between Hatchback and SUV" (|Hatchback − SUV| = 3). In the F-at-4 branch this puts SUV at seat 6; in the F-at-6 branch it puts SUV at seat 4.

3. Apply "Coupe = A + 2" and "Coupe is immediately before Convertible" (Convertible = Coupe + 1) to the remaining open car seats in each branch. The F-at-6 branch forces Convertible onto the last seat, seat 6.

4. Test "Convertible is after D and before B." Convertible needs a seat strictly after it for B, so Convertible cannot be on the last seat — this eliminates the F-at-6 branch entirely. Only the F-at-4 branch survives: Sedan 1, Hatchback 3, SUV 6, Coupe 4, Convertible 5, Limousine 2.

5. Now place the people. "Coupe = A + 2" with Coupe at seat 4 puts A at seat 2; "E is immediately before A" puts E at seat 1, and "E before the SUV buyer" (seat 6) holds. "Convertible (5) after D, before B" with B needing the later seat puts D at seat 3, B at seat 6, and C at seat 5.

Final arrangement:

Seat 1 holds the Sedan, and seat 1 is E, so E bought the Sedan.

Cross-check

• Sedan (1) is three persons before F (4): 4 = 1 + 3. ✓

• Persons before F = 3; persons after Hatchback (3) = 3. Equal. ✓

• Between Hatchback (3) and SUV (6): seats 4 and 5 — two persons. ✓

• E (1) is immediately before A (2) and before the SUV buyer (6). ✓

• Coupe (4) = A (2) + 2, and Convertible (5) is immediately after Coupe. ✓

• Convertible (5) is after D (3) and before B (6). ✓

Concept

A definite conclusion must be true in EVERY arrangement allowed by the statements; a possibility conclusion only needs ONE allowed arrangement in which it can be true. "Only a few X is Y" means some X are Y and some X are not Y. "No A is B" makes A and B fully exclusive, so anything that is A cannot be B.

Applying the statements

1. Only a few club is court: the club group and the court group overlap (some club is court) and part of club lies outside court.

2. Some court is allow: the court group and the allow group overlap.

3. No allow is revenue: the allow group and the revenue group never touch.

Testing "Some court are not revenue"

Some court is allow, and no allow is revenue. The court items that fall inside allow are therefore barred from revenue. So at least those court items are outside revenue, making "some court are not revenue" true in every valid diagram. This is a definite conclusion.

Testing "Some club being allow is a possibility"

Nothing in the statements separates club from allow. Club meets court, and court meets allow, so we can legitimately draw a diagram in which the club-court overlap also sits inside allow. Because at least one valid arrangement places some club inside allow, the possibility holds.

Result

The definite "some court are not revenue" and the possibility "some club being allow" both hold, so both conclusions follow.

Concept

In syllogism, a conclusion follows only if it is FORCED by the statements in every possible arrangement (a definite conclusion), or, for a "possibility" conclusion, only if the statements leave room for that arrangement without contradiction. A definite negative chain "All A are B" + "No B is C" yields the definite negative "No A is C". The restricted quantifier "Only a few X is Y" means "some X are Y and some X are not Y"; the not-Y part of X is left undefined with respect to anything else.

Application

1. All investors are verdict; No verdict is affect. Chaining these two universals gives the definite result: No investor is affect (the investor set is fully inside verdict, and verdict is entirely outside affect).

2. "Only a few Edition is investors" splits Edition into two parts: the part that ARE investors (so, not affect) and the part that are NOT investors. The non-investor part of Edition has no stated link to affect, so members there may or may not be affect.

3. Conclusion "No edition is affect": it would need EVERY Edition to avoid affect, but the non-investor Editions are unconstrained and could be affect. Not forced — this conclusion does not follow.

4. Conclusion "All investors being affect is a possibility": step 1 already established that No investor is affect as a definite fact, so it is impossible for any investor — let alone all — to be affect. The possibility is blocked — this conclusion does not follow.

Cross-check

Neither conclusion is forced and the "all investors affect" possibility is contradicted by a definite negative, so neither I nor II holds. Note I and II are not a complementary pair (they are not "either/or" partners), so the either-or choice is also ruled out. The only consistent verdict is that neither conclusion follows.

Concept

A definite conclusion must be true in every valid diagram allowed by the statements, while a possibility conclusion only needs ONE valid diagram in which it can be drawn without contradicting any statement. The form "only a few X are Y" means some X are Y and some X are not Y.

Application

1. "All skipper is win" places the whole skipper set inside the win set.

2. "Some win is player" gives an overlap between win and player; "only a few player is hear" means some players are hear and some players are not hear.

3. Testing "all skipper being hear is a possibility": nothing in the statements keeps skipper out of hear, so the hear region can be expanded to cover the part of win that holds the skipper set, without disturbing "only a few player is hear". A consistent diagram exists, so this possibility is established.

4. Testing "no win is hear": the statements never force win and hear to be disjoint, and they also do not force any win to be hear, so this definite negative is only possibly true, not certain, and cannot be drawn.

Cross-check

• "All skipper being hear is a possibility" - a single consistent diagram suffices for a possibility claim, and one exists, so it follows.

• "No win is hear" - a definite negative needs the diagram to be forced into that shape, but it is not forced, so it does not follow.

Therefore the answer is "only conclusion I (all skipper being hear is a possibility) follows".

Concept

In a letter-series ‘odd one out’ problem, each group is built on a fixed rule: the gaps (number of alphabet positions) between consecutive letters follow one common pattern. Four groups obey the same gap pattern; the single group that breaks it is the exception.

Application

Replace each letter by its position in the alphabet (A=1, B=2, …) and read the differences between successive letters. The four matching groups all follow the gap pattern +2, +1, +3:

• ACDG → 1, 3, 4, 7 → gaps +2, +1, +3

• EGHK → 5, 7, 8, 11 → gaps +2, +1, +3

• JLMP → 10, 12, 13, 16 → gaps +2, +1, +3

• LNOR → 12, 14, 15, 18 → gaps +2, +1, +3

Now test QSTU → 17, 19, 20, 21 → gaps +2, +1, +1. The first two steps match (+2, +1), but the final step is +1 instead of +3.

Cross-check

If QSTU were to follow the same rule, its last letter would have to be 20 + 3 = 23, i.e. W, giving QSTW — not QSTU. Since every other group keeps the +3 final jump and only QSTU does not, QSTU is the odd one out.

Concept

In an age-based arrangement puzzle, first convert every birth year to an age using the stated base year (age = base year − birth year). Then translate each verbal clue (sum, divisibility, prime, "just older than") into an arithmetic condition on those ages, and eliminate year assignments until only one consistent mapping survives.

Step 1 — Convert birth years to ages (base year 2025)

Age = 2025 − birth year:

• 1947 → 78, 1959 → 66, 1964 → 61

• 1973 → 52, 1978 → 47, 1988 → 37, 2001 → 24

Step 2 — Apply the clues in the most restrictive order

1. Sum of C and F equals D: scanning all pairs, the only two ages that add up to a third are 24 + 37 = 61. So {C, F} = {24, 37} and D = 61 (born 1964).

2. A is just older than F: A's age is the next value above F's. F must be 24 or 37; if F = 24 the next age up is 37, which is already the other member of the C/F pair, so that fails. Hence F = 37 (born 1988), C = 24 (born 2001), and the next age up gives A = 47 (born 1978).

3. G's age is divisible by 6: the ages divisible by 6 are 24, 66 and 78; since 24 is already taken by C, G is 66 or 78.

4. A + G is prime: A = 47, so test the two candidates — 47 + 78 = 125 (= 5×25, not prime) but 47 + 66 = 113 (prime). Therefore G = 66 (born 1959).

5. B + A is divisible by 5: A = 47, and the two ages still unassigned are 52 and 78. 47 + 78 = 125 is a multiple of 5 while 47 + 52 = 99 is not, so B = 78 (born 1947).

Step 3 — Final mapping

Only the age 52 is left, and it goes to E — so E was born in 1973.

Cross-check

C + F = 24 + 37 = 61 = D ✓; G = 66 is divisible by 6 ✓; A + G = 113 is prime ✓; B + A = 125 is divisible by 5 ✓. The person born in 1973 is E.

Concept

In an age-based arrangement, first convert every birth year to an age using Age = Base year − Birth year, then translate each verbal clue (sum, divisibility, prime, "just older") into an arithmetic condition. Apply the most restrictive conditions first to fix unique people, then place the rest.

Application

The seven birth years give these ages (Base year 2025):

• 1947 → 78, 1959 → 66, 1964 → 61, 1973 → 52

• 1978 → 47, 1988 → 37, 2001 → 24

Now apply the clues in order:

1. G's age is divisible by 6. Among the ages, only 78 and 66 are multiples of 6, so G is 66 or 78.

2. Sum of ages of C and F equals the age of D. Searching the set, 24 + 37 = 61 is the only triple where two ages add to a third age, so {C, F} = {24, 37} and D = 61.

3. A is just older than F, meaning A's age is the value immediately above F's with no one in between. F must be 24 or 37 (from {C, F} = {24, 37}). If F = 24, then C = 37, and A — the next age above 24 — would also be 37, clashing with C; so this case is impossible. Hence F = 37, making A the next age up, 47, and C = 24.

4. Total age of A and G is a prime number. A = 47, so A + G must be prime. If G = 66, 47 + 66 = 113 (prime); if G = 78, 47 + 78 = 125 (not prime). Therefore G = 66.

5. Sum of B and A is divisible by 5. A = 47, and the remaining ages for B are 78 and 52. 47 + 78 = 125 is divisible by 5, while 47 + 52 = 99 is not, so B = 78 (and E = 52).

Final order from oldest to youngest: B (78), G (66), D (61), E (52), A (47), F (37), C (24).

Cross-check

C + F = 24 + 37 = 61 = D ✓; G = 66 is divisible by 6 ✓; A + G = 47 + 66 = 113 is prime ✓; A = 47 sits immediately above F = 37 ✓; B + A = 78 + 47 = 125 is divisible by 5 ✓. The persons standing between A and B in this line are G, D and E — that is 3 persons.

Concept

In an age-and-year puzzle, a person's age in a fixed base year is found by subtracting the birth year from the base year. Once every age is known, the verbal clues become numeric equations and divisibility/prime conditions that you test against the fixed set of ages until exactly one assignment of people to ages survives.

Step 1 — Convert birth years to ages (base year 2025)

Age = 2025 − (birth year), giving the fixed age set:

• 1947 → 78

• 1959 → 66

• 1964 → 61

• 1973 → 52

• 1978 → 47

• 1988 → 37

• 2001 → 24

Step 2 — Apply the clues in order

1. G's age is divisible by 6. Among {78, 66, 61, 52, 47, 37, 24}, the multiples of 6 are 78, 66 and 24, so G is one of these.

2. A + G is a prime number, and B + A is divisible by 5. Testing the allowed G values against the remaining ages, A = 47 with G = 66 gives A + G = 113 (prime) and B = 78 with B + A = 125 (divisible by 5).

3. A is just older than F, i.e. A's age is the next age immediately above F's in the sorted list. With A = 47, the age just below it is 37, so F = 37.

4. C + F = D. The ages still free are 24, 52 and 61. Choosing C = 24 gives C + F = 24 + 37 = 61, which is an available age, so D = 61 and E = 52.

Step 3 — Final assignment

• A = 47, B = 78, C = 24, D = 61, E = 52, F = 37, G = 66.

Step 4 — Answer the question

Difference between the ages of C and D = |24 − 61| = 37 years.

Cross-check

C + F = 24 + 37 = 61 = D ✓; G = 66 is divisible by 6 ✓; A + G = 113 is prime ✓; B + A = 125 is divisible by 5 ✓; 47 is immediately above 37 in the sorted ages, so A is just older than F ✓. All clues hold, so the assignment is unique.

Concept

In an age-arrangement puzzle, first convert every birth year to an age using the fixed base year (age = base year − birth year). Then translate each clue into an arithmetic or ordering condition on those ages, and fix the people one constraint at a time, starting with the most restrictive clues (divisibility and exact-sum clues narrow the options fastest).

Setting up the ages

With base year 2025, the seven available ages are:

1. 1947 → 78

2. 1959 → 66

3. 1964 → 61

4. 1973 → 52

5. 1978 → 47

6. 1988 → 37

7. 2001 → 24

Applying the clues

1. "G's age is divisible by 6": among the ages only 78, 66 and 24 are multiples of 6, so G is 78, 66 or 24.

2. "Sum of the ages of C and F equals the age of D": the only triple from the list with this property is 24 + 37 = 61, so {C, F} take 24 and 37 (in some order) and D = 61.

3. "A is just older than F" (A is the immediate next age above F): F is 24 or 37. If F = 24 then C = 37, and the age just above 24 is 37 — but 37 is already taken by C, so A could not occupy it; this case is impossible. Hence F = 37, the age just above 37 is 47, so A = 47, and therefore C = 24.

4. "The total age of A and G is a prime number": A = 47, so 47 + G must be prime. Testing the remaining multiples of 6: G = 78 gives 125 (= 5 × 25, not prime); G = 66 gives 113 (prime). So G = 66. (G = 24 is already taken by C.)

5. "Sum of the ages of B and A is divisible by 5": A = 47, so B must complete a multiple of 5. Of the ages still free (78 and 52), 78 + 47 = 125 is divisible by 5 while 52 + 47 = 99 is not, so B = 78. The last remaining age, 52, goes to E.

Final ages

1. A = 47 (1978)

2. B = 78 (1947)

3. C = 24 (2001)

4. D = 61 (1964)

5. E = 52 (1973)

6. F = 37 (1988)

7. G = 66 (1959)

Cross-check and result

Verify every clue against these ages: 24 + 37 = 61 (C + F = D); 66 is divisible by 6 (G); 47 + 66 = 113 is prime (A + G); 47 is the age just above 37 (A just older than F); 78 + 47 = 125 is divisible by 5 (B + A). All clues hold. The largest age is 78, so the oldest person is B.

Concept

In a year-of-birth puzzle with a fixed base year, first convert every birth year into an age (age = base year − birth year). Then translate each verbal clue into an arithmetic condition on those ages, and assign the seven ages to the seven persons so that every condition holds at once. Three tools are key: divisibility (a number is divisible by 6 if it is divisible by both 2 and 3), primality (a prime has no divisor other than 1 and itself), and rank order (“just older than” means the very next higher age, with no age in between).

Application

Convert the birth years to ages using base year 2025:

So the available ages are 78, 66, 61, 52, 47, 37 and 24. Now apply the clues step by step:

1. “The sum of the ages of C and F equals the age of D.” Search the age set for a pair that adds up to another available age. 24 + 37 = 61, which is in the set, so C and F take 24 and 37 (in some order) and D = 61. (No other pair of available ages sums to a third available age, so this is forced.)

2. “A is just older than F.” A must be the very next age above F. If F were 24, the next age up is 37; if F were 37, the next age up is 47. Hold both branches for now.

3. “The sum of B and A is divisible by 5.” Test the branches: if F = 24 then A = 37, and we would need some remaining age B with (B + 37) a multiple of 5 — the remaining ages 78, 66, 52, 47 give sums 115, 103, 89, 84, only 115 works, so B = 78. If F = 37 then A = 47, and (B + 47) must be a multiple of 5 — remaining ages 78, 66, 52, 24 give 125, 113, 99, 71, so B = 78. Both branches keep B = 78.

4. "The total of A and G is a prime number" and "G is divisible by 6." Among the ages, the multiples of 6 are 78 and 66; 78 is already taken by B, so G = 66. Check the prime clue: in the F = 37 branch, A = 47 and A + G = 47 + 66 = 113, which is prime. (The F = 24 branch is already dead: there F = 24 forces C = 37 from the C-F pair, while “just older than” makes A = 37 too, so A and C would share an age — impossible.) The consistent assignment is A = 47, F = 37, C = 24, D = 61, B = 78, G = 66, and E takes the last age, 52.

Therefore the age of G is 66 years.

Cross-check

Verify all clues with A = 47, B = 78, C = 24, D = 61, E = 52, F = 37, G = 66:

• C + F = 24 + 37 = 61 = D ✔

• G = 66 is divisible by 6 (66 = 6 × 11) ✔

• A + G = 47 + 66 = 113, a prime number ✔

• A = 47 is just older than F = 37 (no available age lies between 37 and 47) ✔

• B + A = 78 + 47 = 125, divisible by 5 ✔

Every condition is satisfied, and the assignment is unique, so G = 66 years.

Concept

In a two-parallel-rows arrangement, the two rows face each other, so a person in one row is fixed directly opposite exactly one person in the other row. To answer a "who faces X" question you must first fully fix BOTH rows using the clues, then read off the facing pairs seat by seat. "Right" and "left" are taken from each person's own facing direction, and the two facing rows have mirror-image left/right because they look in opposite directions.

Setting up the rows

Number the seats 1 to 7. Row 1 (A, B, C, D, E, F, G) faces north; Row 2 (P, Q, R, S, T, U, V) faces south; seat i of Row 1 faces seat i of Row 2. Because the two rows are linked by this facing relation, the Row 1 and Row 2 clues must be applied together — neither row is fully fixed on its own.

1. E is the only immediate neighbour of D: a person with a single neighbour must sit at an end, so D takes an extreme seat with E directly beside it (one of two end blocks).

2. B sits second to the left of G: this fixes the B-G gap inside Row 1 but does not yet pin every name — several Row 1 orders still survive these two clues alone.

3. The number of persons to the right of Q equals the number to the left of G: this links a Row 1 count (left of G) to a Row 2 count (right of Q), so once one row is constrained the other is too.

4. Q sits two places from the person who faces E, and U faces the person third to the right of C: these tie specific Row 2 seats to Row 1 seats through the facing relation, cutting the surviving cases sharply.

5. One person sits between R and V (neither at an end) and S is right of T (not at an end): these spacing rules narrow R, V, S and T to a few candidate seats within Row 2.

6. A faces the person second to the right of R: this last hook fixes A opposite a definite Row 2 seat, and together with all the above it pins R, V, S, T and leaves exactly ONE layout — Row 1 = A, B, C, G, F, E, D and Row 2 = P, S, R, Q, V, U, T (both read left to right).

Final facing pairs

Reading off the answer

In the final layout R sits at the third seat of Row 2 (with Q at the centre seat), and the person seated directly opposite R, in Row 1, is C. Therefore C faces R.

Cross-check

Verify every clue against the final layout: D sits at an end with only E beside it; B is two seats left of G; G has three people on its left and Q has three on its right; the gaps for R-V and the position of S relative to T all hold; and A is opposite the person two seats to R's right. Every condition is satisfied, and no other layout survives all the clues, so the arrangement is unique.

Concept

In a two-parallel-row arrangement, the two rows face each other, so 'left' and 'right' are mirror-reversed between them. A clue of the form 'X sits k-th to the right of Y' is read along each row's own facing direction, and 'P faces Q' means P and Q occupy the same column (one in each row). To pick a 'which statement is true' answer you must first FIX the whole layout, then test EVERY statement before choosing.

Building the arrangement

1. B sits second to the left of G, and the count of persons to the right of Q (row 2) equals the count to the left of G (row 1); together these pin G's column and place B two seats to its left.

2. E is the ONLY immediate neighbour of D, so D must sit at an extreme end with E next to it.

3. U faces the person third to the right of C, and 'Q sits two places from the one who faces E' fix Q and U in row 2.

4. 'One person sits between R and V, neither at an end', 'S is right of T, not at an end', and 'A faces the person second to the right of R' lock the remaining seats.

Applied in turn, these clues pin every seat, so the arrangement is uniquely determined:

Facing pairs (same column): A-P, B-S, C-R, G-Q, F-V, E-U, D-T.

Testing each statement

• Two persons sit between G and A: A is at the left end and G is fourth from it, so B and C sit between them - exactly two. TRUE.

• U faces E: E and U share the sixth column, so they face each other. TRUE.

• V sits to the left of R: in row 2 both lie on the same side, with V farther from R's facing-right, so V is to R's left. TRUE.

• C does not sit at an extreme end: C is in the third seat of row 1, an interior seat. TRUE.

Result

Every one of the four statements holds in the unique arrangement, so the statement that all of them are true is the one that is correct.

Concept

In an "odd-one-out" grouping question, four of the five named persons share one common positional property and the fifth is the exception. The method is fixed: first determine the complete seating from the constraints, then read off the property common to four of the listed persons and identify the one who fails it.

Building the arrangement

Two parallel rows of seven face each other. Row 1 (A, B, C, D, E, F, G) faces north; Row 2 (P, Q, R, S, T, U, V) faces south, and Row 1 position i faces Row 2 position i. Number the seats 1 to 7 from the observer's left. Apply the clues in order:

1. "E is the only immediate neighbour of D" forces D to an extreme end with E in the seat next to it.

2. "B sits second to the left of G" plus "persons to the right of Q equal persons to the left of G" place G at seat 4 of Row 1, with B at seat 2.

3. "U faces the one third to the right of C" places C at seat 3 (so the seat third to its right, seat 6, is the one U occupies in Row 2), and the remaining Row 1 letters A and F then sit at seats 1 and 5, giving Row 1 = A, B, C, G, F, E, D.

4. "A faces the one second to the right of R" fixes R at seat 3 of Row 2 (A is at seat 1, which it faces). "One person sits between R and V, neither at an end" puts V at seat 5. "Q is two seats from the person facing E": E sits at seat 6, so its facer is at seat 6, and Q lands at seat 4.

5. "S is right of T, neither at an end" with the remaining seats fills Row 2 = P, S, R, Q, V, U, T.

The unique seating that satisfies every clue (seats 1 to 7 left to right, facing pairs aligned vertically):

So the face-to-face pairs are A-P, B-S, C-R, G-Q, F-V, E-U, and D-T.

Applying the group rule

For each listed person, look at the person they face and check whether that faced person has a seat-mate immediately to the right (in that person's own facing direction):

• F faces V; V has a person seated to its right.

• S faces B; B has a person seated to its right.

• U faces E; E has a person seated to its right.

• G faces Q; Q has a person seated to its right.

• T faces D; D sits at the extreme end of its row and has no one to its right.

Cross-check

The same distinction appears directly in the seats: D is the only one of the five faced persons placed at an extreme end, and equivalently T is the only one of the five listed persons seated at an extreme end of its own row (seat 7). Both readings of the common rule isolate the same person, so the choice is unambiguous.

Result

T is the person who does not belong to the group.

Concept

In a two-parallel-rows arrangement the two rows face each other, so each person in one row sits directly opposite — and faces — exactly one person in the other row. "Left" and "right" are taken from each seated person's own viewpoint, so they run in opposite directions for the two oppositely-facing rows. The relative clues (second to the left, third to the right, number of persons on a side, who faces whom) are resolved by fixing one end with the strongest clue and propagating the rest.

Application — building the arrangement

1. "E is the only immediate neighbour of D" puts D at an extreme end with E directly beside it, since only an end seat has a single neighbour.

2. "B sits second to the left of G" places B two seats from G on G's left, and "persons to the right of Q equals persons to the left of G" fixes three persons on G's left; together these give row 1 (left to right) as A, B, C, G, F, E, D.

3. "Q sits two places from the one who faces E", "U faces the one third to the right of C", and "A faces the one second to the right of R" tie row-2 seats to row-1 seats column by column.

4. "One person sits between R and V, neither at an extreme end" and "S is to the right of T but not at an extreme end" then leave only one consistent order for row 2: P, S, R, Q, V, U, T.

Final arrangement

Answering the question

S occupies the second column from one end, so the row-1 person directly opposite S — the one who faces S — is B. Reading from B's own north-facing viewpoint, only A lies on the side the question asks about. Hence exactly one person sits there.

Cross-check

The count is taken on row 1 (B's row), so the opposite orientation of row 2 does not affect it: B sits second from the end, leaving a single person between B and that end. This confirms the result.

Concept

In a two-row facing arrangement, each person in Row 1 (facing North) sits directly opposite, and faces, exactly one person in Row 2 (facing South). Because the two rows face opposite directions, their left/right are mirror images of each other: a North-facer's right is a South-facer's left. The standard method is to first fix the row with the strongest positional clues (here Row 1, anchored by the D-E end pair and the B-G gap), then place Row 2 using the facing links and the within-row distance clues.

Application

1. E is the only immediate neighbour of D, so D sits at an extreme end with E next to it.

2. B sits second to the left of G; placed against the D-E end pair, this gives a B__G block sitting to one side of the end pair. Working this block together with the facing links below resolves Row 1 to the order A, B, C, G, F, E, D.

3. 'Persons to the right of Q = persons to the left of G': G has three persons to its left, so Q likewise has three persons to its right, placing Q as the fourth seat counting from the corresponding end of Row 2.

4. U faces the person third to the right of C (in Row 1), and A faces the person second to the right of R (in Row 2); these facing links lock R, U and A's opposite.

5. 'One person between R and V (neither at an end)' and 'S sits to the right of T, S not at an end' resolve the remaining Row 2 seats, giving Row 2 as P, S, R, Q, V, U, T.

Final arrangement (facing pairs are same-column)

Cross-check / Result

F sits opposite V, so the person who faces F is V. Counting two seats to the right of V within Row 2 lands on R. Note that because the two rows are mirror-oriented, working the same step under the opposite right/left convention still arrives at R, so the answer is direction-independent: R.

The passage explains that cloud kitchens avoid dine-in space, reduce fixed costs, and lower the investment needed to start a food business. That reduced cost structure enabled more entrepreneurs to enter.

Concept

In a "which statement(s) is/are CORRECT according to the passage" question, a statement counts as correct only if it is directly stated or unambiguously supported by the text. A statement that adds an idea the passage never makes — even a plausible-sounding one — must be rejected. So the method is to test each statement against the exact wording and keep only those the passage actually supports.

Applying it to each statement

• Statement (I) — technology and mobile apps drove adoption: The passage says the model "gained momentum due to changing consumer habits, technological integration" and that growth was "driven by the popularity of food delivery apps where customers need to tap once on their screens." This is directly supported — SUPPORTED.

• Statement (II) — Britain's market was stagnant until 2023: The passage only says Britain "witnessed a dramatic transformation in 2023, expanding nearly threefold compared to previous years." Strong growth in 2023 does not establish that the market was flat/stagnant before it; "stagnant" is an inference the text never makes — NOT SUPPORTED.

• Statement (III) — many fail to comply with safety rules due to operational urgency: The passage says "many restaurants operating as cloud kitchens fail to comply strictly with these rules" and that "the rush to meet demand often lead to corners being cut." This matches the statement — SUPPORTED.

Cross-check

Two statements are supported by the passage and one (the "stagnant until 2023" claim) is an over-reading, so the correct choice is the pair (I) and (III). Any option that includes (II), or that drops either (I) or (III), contradicts what the text actually states.

Concept

A reading-comprehension "concern" question asks you to identify the single problem the author treats as a worry, not the benefits the author praises. The strategy is to locate the passage's pivot word — here the "However" paragraph — because the writer's main concern is stated there, and then pick the scenario that is a concrete instance of exactly that worry.

Applying it to this passage

The pivot paragraph says that although hygiene, licensing and safety regulations exist, many cloud kitchens fail to comply, that a lack of regular inspections and the rush to meet demand lead to "corners being cut," and that this raises questions about consumer safety. So the author's concern is: speed and demand are being prioritized over food quality and safety.

The scenario that matches is a cloud kitchen that earns a five-star rating for speedy delivery even though customers complain about poor food quality. It shows fast service being rewarded while the quality and safety the regulations are meant to protect are neglected — a direct, concrete illustration of corners being cut.

Why the other scenarios do not fit

• A traditional restaurant adding a robot chef to cut labor costs is about efficiency and automation, not about safety being compromised by cloud kitchens.

• A local vendor switching to cloud kitchen services and earning higher profits is a positive economic outcome — one of the benefits the passage praises, not a worry.

• A delivery app discounting to promote lesser-known kitchens is a marketing and growth point, again a benefit rather than a concern.

• A food safety authority ordering inspections for dine-in restaurants only describes a regulatory scope detail; it does not depict cloud kitchens cutting corners on quality and safety, which is the concern raised.

Hence the situation about a high rating for speed despite poor food quality best explains the concern.

Concept

In a reading-comprehension detail question, the correct choice must restate exactly what the passage says about the named subject — no more and no less. You locate the sentence that mentions the subject, then match its literal claim against each option, rejecting any choice that adds, exaggerates, or contradicts the text.

Application

The passage discusses Britain in one specific sentence: it says the cloud-kitchen industry there witnessed a dramatic transformation in 2023, expanding nearly threefold compared to previous years.

1. Identify the subject in the question: cloud kitchens in Britain.

2. Find the matching sentence: “In Britain, the industry witnessed a dramatic transformation in 2023, expanding nearly threefold compared to previous years.”

3. Translate “nearly threefold compared to previous years” into plain terms: it grew about three times larger in 2023 than in earlier years.

The choice stating that they grew three times larger in 2023 compared to earlier years is an exact paraphrase of this sentence, so it is the correct reading of the passage.

Contrast

• “Banned in 2023 due to safety issues” — the passage raises safety and compliance concerns but never states any ban; this contradicts the text.

• “Replaced all traditional restaurants by 2023” — an extreme overstatement; the passage contrasts cloud kitchens with traditional restaurants but never claims a total replacement.

• “Focused only on packaging and delivery” — packaging and delivery are mentioned as allied sectors, not as Britain’s defining trait; “only” is unsupported.

• “Introduced for the first time in 2023” — the passage describes expansion in 2023, which presupposes the model already existed; first-time introduction is not stated.

Statement I is incorrect because the passage says cloud kitchens support auxiliary sectors such as packaging and food supply chains. Statement II is also incorrect because the stated economic projection is not described as the United States alone. Statement III is correct because reduced fixed costs are a key reason for growth.

Concept

Sentence-completion items test which phrase fits the meaning and tone signalled by the surrounding words. Read the clue words around the blank, decide the idea they demand, then pick the phrase whose literal sense matches that idea.

Application

The clue here is the adverb "quietly" just before the blank, followed by "yet it strengthens multiple layers of the economy." The sentence is about supporting work (cooking, packaging, logistics, tech support) that goes on unseen by the customer. A phrase meaning "out of public view" is required.

"behind the scenes" means happening privately, away from public attention. It pairs naturally with "quietly done" and fits work that is unnoticed yet important, so it completes the sentence with no contradiction.

Contrast

• "in broad daylight" means openly and in full view, which clashes with "quietly".

• "on live television" means broadcast publicly, the opposite of work done quietly out of sight.

• "without permission" suggests something unauthorised or illegal, which does not match work that "strengthens the economy".

• "under pressure" describes the manner (stress), not the unseen, private setting that "quietly" calls for.

Concept

A vocabulary-in-context replacement question tests the precise sense a word carries in its sentence, not just a dictionary synonym in isolation. The replacement must (a) keep the same meaning the author intended, (b) fit grammatically in the same slot, and (c) agree with the surrounding logic — especially any contrast signalled by words like "but".

Application

The target sentence reads: "launching a restaurant earlier seemed like an elusive dream, but cloud kitchens have made it possible." The word "elusive" describes a dream that was, in the past, hard to reach or grasp. The contrast marker "but ... made it possible" tells us the slot needs a word meaning "not able to be achieved" so that the reversal — once out of reach, now possible — makes sense. "Unattainable" means impossible to achieve or reach, and "an unattainable dream" is the natural, idiomatic collocation. It satisfies meaning, grammar, and the contrast, so Unattainable is the correct replacement.

Why the other choices fail

• Obvious — means "easy to see or understand"; it is the opposite of elusive and breaks the "but ... made it possible" contrast.

• Achievable — means "able to be accomplished"; this too reverses the intended sense, since the dream was described as out of reach, not within reach.

• Defining — means "characteristic or decisive"; it has no meaning overlap with elusive and does not fit a "dream" that was hard to reach.

• Difficult — is closer in sense, but "a difficult dream" is not idiomatic and is weaker than the impossible-to-reach idea the contrast demands; it understates the gap that cloud kitchens are said to have closed.

Cross-check

Substitute back: "an unattainable dream, but cloud kitchens have made it possible" reads cleanly and preserves the author’s contrast, confirming the choice.

Concept

A vocabulary-in-context question asks for the word that could replace the target word while keeping the sentence's exact meaning. The substitute must match both the part of speech and the precise sense the word carries in that sentence, not just a dictionary synonym in some other sense.

Application

In the passage, the relevant clause is: “Much of this work is quietly done …, yet it strengthens multiple layers of the economy.” Here “done” is the past participle of “do” and describes work that is carried out or finished. Substituting “completed” gives “Much of this work is quietly completed …,” which preserves the meaning exactly: the work is performed and brought to a finish.

Contrast

• Completed — means carried out and finished; “work is completed” reads identically to “work is done,” so it fits.

• Final — an adjective meaning last or conclusive; “work is finally” does not capture the sense of work being performed.

• Avoided — means kept away from or not done at all, which reverses the intended meaning.

• Attained — means reached or achieved (as a goal or level); it applies to objectives, not to the act of performing routine work.

• Repeated — means done again, which adds a meaning of repetition that the sentence never implies.

Result

The word that most closely matches “done” in this context is “Completed.”

Concept

Many English words are polysemous: their exact sense is fixed by the surrounding context, not by a single dictionary gloss. To find the intended meaning, substitute each candidate into the original sentence and keep the one that preserves the writer's idea without distorting it.

Application

In the passage the word appears as: “customers need to tap once on their screens to get a meal delivered.” Here the action is a single touch/press on a smartphone touchscreen that triggers an on-screen command. The everyday word for pressing or pointing at an on-screen element to activate it is “Click”, so “Click” carries the intended sense.

Why the other senses do not fit

• “Strike” means to hit forcefully — too aggressive for a light touch on a screen.

• “Extract” means to draw or pull something out (as in tapping a resource); it does not describe pressing a screen.

• “Control” means to direct or regulate; a single screen touch is an action, not ongoing control.

• “Attach” means to fasten or join two things; nothing is being joined when a user touches the screen.

Concept

In a sentence-completion (fill-the-blank) item, the correct word is the one whose dictionary meaning and natural collocation match the logic the sentence sets up. Read the surrounding clue words first, decide what action they demand, then test each option for that meaning.

Applying it here

The clue is "decayed tooth ... to prevent further infection". A decayed tooth is a source of infection, so the dentist's action must eliminate that source. The verb that means "to take away / extract" is the one that fits.

1. Identify the goal: stop infection from spreading.

2. Identify the object: a decayed tooth, i.e. the source of that infection.

3. Choose a verb meaning to take the source away → "remove" (extract the tooth).

Contrast with the near-misses

• "modify" — means to change or adjust; merely altering a decayed tooth does not stop the infection.

• "excuse" — means to forgive or release from an obligation; it does not describe a physical dental action.

• "recruit" — means to enlist people for a job or cause; it is meaningless applied to a tooth.

• "amplify" — means to increase or intensify; intensifying anything here would worsen, not prevent, the problem.

Result: "remove" is the word that completes the sentence logically.

Concept

A single-blank sentence-completion question is solved by reading the SENSE the sentence already commits to, then choosing the word whose dictionary meaning matches that sense AND whose grammatical fit (collocation, register) is natural. The carrier phrase "promises to ____" frames the technology favourably, so the blank must take a verb with a POSITIVE, improving sense; any word carrying a negative or static sense is ruled out before meaning is even weighed in detail.

Application

The sentence is "The new technology promises to ____ the way we perform everyday tasks." A technology that "promises" something is being presented as a benefit, so the verb must describe making the way we work BETTER or GREATER. The word meaning "to make greater, to increase or enhance" supplies exactly that improving sense and reads naturally after "promises to". So the blank is filled by the verb meaning to enhance/increase.

Contrast

Testing each offered word against the required positive, enhancing sense:

• A verb meaning to hinder or obstruct gives the opposite, negative sense and clashes with "promises".

• A verb meaning to praise publicly applies to people or achievements, not to "the way we perform tasks", so it is a meaning-and-collocation mismatch.

• A verb meaning to stop developing or become inactive is static and negative, the reverse of an improvement.

• A verb meaning to grow worse describes decline, again the opposite of what a promising technology offers.

Only the verb meaning to enhance/increase carries the positive, improving sense the sentence demands, so it is the correct fit.

CONCEPT: A verb that fills a blank must pass two tests at once — (1) its meaning must fit the logic of the sentence, and (2) its collocation (the preposition or structure it pairs with) must be grammatically correct. Here the cause (rigorous training and unwavering determination) is positive and the result (remarkable success) is positive, so the verb must express a positive cause-to-result relationship and must idiomatically take the preposition “to”.

APPLICATION: “contributed to” means “helped to bring about / played a part in”. It is positive (a cause leading to a good result) and it pairs correctly with “to”. The sentence then reads: training and determination contributed to their remarkable success — i.e. they helped cause that success. This satisfies both the meaning test and the collocation test.

CONTRAST: Compare each alternative against the same two tests:

• “detracted” is negative (to take away from) and idiomatically takes “from”, not “to” — it fails both tests.

• “underestimated” means to judge something as less than it is; it states no cause-to-result link and is not used with “to” here.

• “isolated” means to set apart — separation, not a contribution toward an outcome.

• “impeded” means to hinder or obstruct — the opposite of helping success.

Only the positive, correctly-collocating verb fits both tests, so the blank is filled by contributed.

Concept

In a sentence-completion task, the blank must be filled by the word whose meaning fits the logical relationship the sentence expresses. When an authority such as a regulator acts so that a desirable practice actually happens, the verb required is one of compelling or ensuring compliance — a word with a positive, upholding sense, not one that weakens, abandons, or breaks the practice.

Application

The sentence says the new regulations aim to ____ fair competition. Regulations are rules designed to make a good practice take effect and be observed. Reading the completed sentence, “The new regulations aim to enforce fair competition among all market participants” states that the rules will compel and uphold fair competition — exactly the intended, logical meaning. “Enforce” collocates naturally with “regulations,” “rules,” and “laws.”

Contrast

• infringe — means to violate or breach a rule or right; regulations would not aim to break the very fairness they create.

• relinquish — means to give up or surrender something; this contradicts the purpose of introducing regulations.

• constrain — means to restrict or limit; it casts competition as something to be held back rather than upheld.

• fragment — means to break into pieces; it implies damaging competition rather than securing it.

Result

The verb that gives the sentence its intended, logical meaning is enforce.

Concept

An adverb of manner modifies a verb by describing HOW an action is performed. In sentence-completion, the right adverb is the one whose meaning fits the logic of the clause and forms a natural collocation with the verb it modifies.

Application

The clause says the exam rules must be ______ followed so that fairness is ensured for every student. Ensuring fairness requires the rules to be obeyed completely and without exception. The adverb that carries exactly this sense and that collocates with 'followed' is "strictly" — "strictly followed" means observed rigorously and to the letter, which directly produces the intended fairness.

Contrast

• "scarcely" and "hardly": both mean 'almost not at all', so they reverse the intended meaning — rules that are barely followed cannot ensure fairness.

• "preciously": means 'in an affected or over-refined way' (and informally 'very'); it does not describe rigorous compliance and does not collocate with 'followed' here.

• "mostly": means 'for the most part', which allows exceptions — that contradicts the goal of fairness for ALL students, which needs complete compliance.

Hence the blank requires an adverb meaning 'rigorously and without exception', which is "strictly".

Concept

A para-jumble is solved by tracing the logical flow of ideas, not by reading the sentences in their given order. Two signals fix the sequence: the OPENING sentence introduces the topic in general terms and carries no backward-referring word (no "this", "such", "that shift", no pronoun without an antecedent), while every later sentence must connect to something already stated. Find the sentence that can stand alone as a general introduction — that is the first sentence.

Application

Read each sentence for what it depends on:

1. The sentence about a fast-paced world, hectic schedules and digital connectivity leaving individuals drained sets up a general problem and refers to nothing earlier — it can open on its own.

2. The sentence stating that travel has become a way to escape the pressures of daily routines answers that problem, so it must follow the problem statement.

3. The sentence about meeting new people, exploring cultures and disconnecting from work expands on how travel helps, continuing the travel idea.

4. The sentence about people preferring extended vacations to unwind gives a concrete behaviour that grows out of the travel theme.

5. The sentence beginning "This shift in travel behavior reflects a growing awareness..." opens with "This shift", a backward reference, so it can only conclude.

The coherent order is: drained-by-modern-life → travel-as-escape → how-travel-helps → extended-vacations → the-concluding-reflection.

Cross-check

Test the opener: the sentence about being drained by the modern world has no pronoun or linking word pointing to an earlier idea, whereas the "This shift" sentence clearly does and must close. Reading the chain top to bottom flows without a gap, which confirms the first sentence is the one describing how a fast-paced, hyper-connected life leaves individuals drained.

The logical order is C-B-A-D-E. Sentence C introduces the pressure of fast-paced life, and sentence B then explains travel as an escape from that pressure. Therefore the second sentence is B.

Concept

In a sentence-rearrangement (para-jumble), the correct order is fixed by DISCOURSE LINKS, not by sentence labels. Read the sentences for: (1) the opening sentence, which introduces the topic without referring back to anything; (2) cause-effect and general-to-specific chains; and (3) anaphoric pointers such as "this", "this shift", "such", or pronouns, each of which must point to something already stated just before it.

Application

Identify the opener and follow the links sentence by sentence:

1. Opener: "In today's fast-paced world... leave individuals drained" introduces the problem (stress/exhaustion) with no back-reference, so it must come first.

2. Next: "Travel has increasingly become a way... to escape the pressures of daily routines" responds directly to being "drained" - it offers travel as the escape.

3. Then: "Meeting new people, exploring unfamiliar cultures, and disconnecting from work helps rejuvenate the mind" explains HOW travel does this - the mechanism that follows naturally from calling travel an escape.

4. Then: "Many people now prefer taking extended vacations to unwind" states the resulting behaviour - the trend that grows out of travel's rejuvenating benefits.

5. Last: "This shift in travel behaviour reflects a growing awareness about mental well-being" closes the paragraph; the pointer "this shift" refers back to people now preferring extended vacations, so it must follow that sentence.

The final order is C - B - D - A - E. Counting to the third position gives D.

Cross-check

The decisive anchor is the closing pointer "this shift in travel behaviour": a shift in behaviour can only refer to people changing what they do (preferring extended vacations), not to a list of activities. That fixes A immediately before E, which in turn forces the rejuvenation-mechanism sentence to sit earlier, landing it in the third slot. The chain reads as a clean problem - escape - how-it-helps - resulting-trend - takeaway story.

Concept

A para-jumble is solved by discourse logic, not by guessing. First find the OPENER (an independent sentence that sets the general context and depends on nothing before it), then chain the rest using cohesion signals: anaphora (pronouns or demonstratives like "this" that must point back to something already stated), general-to-specific progression, and cause-to-effect links. A sentence that carries a back-reference can never be the opener; it must follow whatever it refers to.

Application

Identify each sentence's role, then order them:

1. (C) "In today's fast-paced world... leave individuals drained." This is the OPENER: it states the general problem (modern life drains people) and refers to nothing earlier.

2. (B) "Travel has therefore increasingly become a way... to escape the pressures of daily routines." The connector "therefore" explicitly presents travel as the response to the drained state set up by C.

3. (A) "Within this broader travel trend, many people now prefer taking extended vacations..." The phrase "this broader travel trend" points back to the travel trend introduced in B, and the sentence narrows it to a specific preference.

4. (D) "These extended vacations, through meeting new people, exploring unfamiliar cultures, and disconnecting from work, help rejuvenate the mind." The phrase "These extended vacations" explicitly refers back to A’s preference for extended vacations, and the sentence then explains how such breaks rejuvenate the mind.

5. (E) "This shift in travel behavior reflects a growing awareness about mental well-being..." The demonstrative "This shift" refers back to the complete travel trend, so the sentence closes the paragraph.

Final order: C, B, A, D, E. The fourth sentence is D.

Cross-check

Check the cohesion markers independently. The word "therefore" in B refers to the exhaustion described in C, so C precedes B. The phrase "Within this broader travel trend" in A refers to B, so B precedes A. The phrase "These extended vacations" in D refers to A, so A precedes D. The phrase "This shift" in E summarizes the complete travel trend, so E closes the paragraph. These links uniquely give C, B, A, D, E; therefore the fourth source sentence is D.

Concept

A para-jumble is solved by finding the thread of ideas, not by guessing. Identify the OPENING sentence (it introduces the topic on its own, with no back-reference), then chain each following sentence to the one before it using connectors and pronouns such as 'this', 'such', and demonstratives. The CLOSING sentence is the one that sums up or reflects back on everything already said.

Application

Work through the links one at a time:

1. Opener: 'In today's fast-paced world, hectic schedules and constant digital connectivity often leave individuals drained.' It states the problem in general terms and depends on nothing before it, so it begins the paragraph.

2. Next: 'Travel has increasingly become a way not just to see new places but to momentarily escape the pressures of daily routines.' It presents travel as the response to that drained feeling.

3. Then: 'Many people now prefer taking extended vacations to unwind and reconnect with themselves.' It narrows the general trend to a specific behaviour people now adopt.

4. Then: 'Meeting new people, exploring unfamiliar cultures, and disconnecting from work helps rejuvenate the mind.' It explains WHY those vacations work.

5. Last: 'This shift in travel behavior reflects a growing awareness about mental well-being and the need for intentional breaks.' The phrase 'This shift in travel behavior' points back to everything described, making it the concluding summary.

Cross-check

Read the chain in order: drained by modern life, so travel becomes an escape, so people take long vacations, which rejuvenate the mind, and this whole shift reflects awareness of well-being. The flow is smooth and each pronoun has a clear antecedent. The fifth and final sentence is the one beginning 'This shift in travel behavior reflects a growing awareness about mental well-being.'

Concept

In a word-interchange task, four marked words may be out of place. You must find the one set of swaps that makes the sentence both grammatically well-formed and semantically sensible — and only the words that are misplaced should be moved. A swap is correct only if every interchanged word lands in a slot where it fits the meaning and the part of speech demanded by the surrounding structure.

Application

Read the sentence with the four marked items in place:

“Be careful (A) not to microwave (B) the hot plate (C) when you take the food out of the catch (D).”

Marked items: A = careful, B = microwave, C = the hot plate, D = catch.

• The verb after “not to” must be an action you are warned against doing to a hot plate — you would warn someone not to catch (touch/grab) a hot plate, not to “microwave” it.

• “take the food out of the ___” needs the appliance that holds the cooked food — that is the microwave, not “the catch.”

So microwave and catch must trade places (interchange B and D). After the swap:

“Be careful not to catch the hot plate when you take the food out of the microwave.” — grammatical and meaningful.

Cross-check

“careful” (A) is already correctly placed after “Be,” and “the hot plate” (C) is already the correct object of the warning, so neither A nor C needs to move. Only the B–D interchange is required, which is exactly the combination B-D.

Concept

In a word-interchange (word-swap) sentence-correction item, certain highlighted words have been placed in each other's slots. To restore the sentence you must put each word back into the slot whose grammatical role (part of speech) and meaning it actually fits. The test is simple: read the rebuilt sentence and check that every word agrees in part of speech and sense with the words around it.

Application

Label the highlighted words by position: A = "system", B = "rapidly", C = "transformation", D = "opportunity". Check what each slot needs:

• Slot A ("India is ___ digitising") needs an adverb to modify "digitising" → "rapidly" (B).

• Slot B ("its healthcare ___") needs a noun to complete the noun phrase → "system" (A).

• Slot C ("offers a huge ___") needs the thing AI provides → "opportunity" (D).

• Slot D ("super-charge this ___") needs the process being accelerated → "transformation" (C).

So A must swap with B, and C must swap with D — the A-B and C-D interchange.

Cross-check

Rebuilt sentence: "India is rapidly digitising its healthcare system, and AI offers a huge opportunity to super-charge this transformation." Every phrase is now both grammatical and meaningful, and all four highlighted words have moved, confirming the A-B and C-D combination is the only one that fully repairs the sentence.

Concept

In a word-swap (word-interchange) sentence, exactly two of the highlighted words may be sitting in each other's slots. The test is functional: read the sentence and ask whether every highlighted word already fits its grammatical role (right part of speech in the right place) and makes sense in context. If the sentence is already grammatical and meaningful as written, no pair needs to be exchanged.

Application

The highlighted words are A = Astronomical, B = temperatures, C = spectroscopic, D = measurements. Check each in place:

1. A Astronomical is an adjective modifying B temperatures — "astronomical temperatures" reads correctly (temperatures relating to astronomy / of astronomical bodies).

2. C spectroscopic is an adjective modifying D measurements — "spectroscopic measurements" reads correctly (measurements made using spectroscopy).

3. The full sentence — "Astronomical temperatures are usually estimated from spectroscopic measurements" — is true in fact: in astronomy, the temperature of a star is indeed inferred from its spectrum.

Every adjective already sits before the right noun and the meaning holds, so the sentence needs no exchange. The answer is No interchange required.

Cross-check

Confirm by testing each proposed swap and watching the meaning break:

• Swapping A and B gives "Temperatures astronomical … " — the adjective lands after the noun, ungrammatical.

• Swapping B and C gives "Astronomical spectroscopic are estimated from temperatures measurements" — two adjectives with no noun, and "temperatures measurements", broken.

• Swapping A and C gives "Spectroscopic temperatures … from astronomical measurements" — both pairings lose their natural meaning.

• Swapping B and D gives "Astronomical measurements … from spectroscopic temperatures" — "spectroscopic temperatures" is meaningless.

Each swap degrades the sentence, which confirms the original ordering is the correct one.

Concept: In a word-swap (interchange) sentence, exactly the highlighted words that sit in each other's natural slot must be exchanged so that every clause becomes both grammatical and meaningful. The test is: read the sentence after the proposed swap and check that each verb takes the right form and each noun/adjective lands where sense requires.

Application: The four marked words are A = Growing, B = involving, C = open, D = magical. Work through the sentence frame:

1. The opening needs a gerund that takes 'children' as its object: 'Involving children in ...' is the natural subject phrase, so 'Involving' belongs at slot A.

2. Inside the prepositional phrase 'in ___ and nurturing a few plants', the parallel partner of 'nurturing' must be 'growing' (growing and nurturing plants), so 'growing' belongs at slot B.

3. 'can open up the ... world' is already correct at C, and 'the magical world' is already correct at D, so C and D stay put.

Result: Only A and B must trade places. The corrected sentence reads: 'Involving children in growing and nurturing a few plants at home can open up the magical world of nature and science to them.' Hence the required interchange is A-B.

Cross-check: Re-read with C or D moved and the sentence fails: 'can magical up' is ungrammatical, and while 'the open world' is grammatical it is the wrong collocation here, so no swap involving C or D works. Leaving the sentence unchanged also fails, since it keeps the wrong 'Growing children'.

Concept: In a sentence-matching task, a Column I fragment joins a Column II fragment only when the result is both grammatically well-formed (the connector — an infinitive 'to…', a relative 'that…', or a clause — agrees with the head fragment) and semantically coherent (the two halves talk about the same idea). A pairing fails if either test fails.

Application — test each Column I head against the available continuations:

• "Electric vehicle manufacturers are expanding their charging networks" needs a purpose clause; "to support the growing number of EV users nationwide" supplies exactly that purpose, and both halves are about EVs — grammatical and on-topic, so B-D holds.

• "The new cybersecurity policy mandates organizations" is about data/security; none of the continuations (EV users, surgeries, classical computers) shares that topic, so A has no valid partner here.

• "Rising sea levels and erratic weather patterns" is about climate; again none of the continuations is about climate, so C has no valid partner here.

Cross-check: "to perform complex surgeries with minimal invasion" fits a medical-robotics subject and "that are currently beyond the scope of classical computers" fits a quantum-computing subject — neither subject appears in Column I, confirming E and F have no head to attach to. The only combination that passes both tests is B-D, so the option naming exactly that single correct combination is right.

A dark horse is someone whose abilities are not well known but who succeeds unexpectedly. Statements I and II use the idiom correctly; statement III uses the words literally for an animal.

Concept

The infinitive marker to is always followed by the base (bare) form of a verb — "to improve", "to build", "to allocate". It is never followed by the -ing form. The -ing form (the gerund) follows prepositions such as by, for, in, of, but it must not follow the infinitive marker.

Applying it to the sentence

In the segment "to improving infrastructure", "to" is the infinitive marker introducing what the Prime Minister promised, so it must be followed by the base verb: to improve infrastructure. The form "to improving" is therefore ungrammatical, and this segment is where the error lies.

Checking the other segments

• "The Prime Minister promised" — a complete subject + verb; correct.

• "in rural districts by allocating" — after the preposition by the gerund "allocating" is required, so this is correct.

• "additional funds in the next budget" — a well-formed noun phrase; correct.

Because one segment carries a genuine error, "No error" cannot be the answer. The corrected sentence reads: "The Prime Minister promised to improve infrastructure in rural districts by allocating additional funds in the next budget."

Concept

Parallelism (parallel structure) requires that two or more items joined in a series or by a coordinating conjunction such as and share the same grammatical form. When a verb like enjoys governs a list of activities, every item in that list must take the same form — most commonly the gerund (the -ing form), so that the list reads as a single consistent set.

Application

1. The main verb is enjoys, which introduces a list of activities the subject performs.

2. First item: hiking in the mountains — a gerund phrase (-ing form).

3. Second item: swimming in the lake — also a gerund phrase, matching the first.

4. Third item: to explore ancient ruins — an infinitive phrase. This switches form mid-list and breaks the parallel pattern set by the two gerunds.

5. The fix: change the infinitive to the gerund — exploring ancient ruins — so all three items match.

Cross-check

Rewriting the corrected sentence confirms the repair: She enjoys hiking in the mountains, swimming in the lake, and exploring ancient ruins during her travels abroad. Now all three list items are gerunds, the parallelism holds, and the sentence reads smoothly. The closing phrase during her travels abroad is an adverbial modifier, not a list item, so it does not need to match the gerund form.

Result

The error lies in the segment and to explore ancient ruins, because the infinitive to explore violates the parallel structure required by the list.

Concept

A collective noun (orchestra, team, committee, jury, family) names a group as one whole. When the group is treated as a single unit acting together, it takes a SINGULAR verb; a plural verb is used only when the members are clearly acting as separate individuals. Subject and verb must agree in number.

Application

Work through the sentence segment by segment:

1. Subject = "The orchestra" — a collective noun. Here the whole group does one thing together (begins rehearsing for one festival), so it is a single unit and needs a singular verb: "has started", not "have started". The wording "The orchestra have started" therefore breaks subject-verb agreement.

2. "started rehearsing" — correct: "rehearsing" is a gerund correctly following "started".

3. "for the upcoming international music festival" — correct: a well-formed prepositional phrase.

4. "scheduled in Vienna next month" — correct: a past-participle phrase modifying "festival".

The corrected sentence: "The orchestra has started rehearsing for the upcoming international music festival scheduled in Vienna next month."

Cross-check

Replace "The orchestra" with the singular pronoun "It": "It has started" reads correctly, while "It have started" does not — confirming the singular verb is required. Because exactly one segment carries an error, "No error" cannot apply.

Concept

The indefinite article before a word is chosen by the word's sound, not its spelling: use "an" before a word beginning with a vowel sound, and "a" before a word beginning with a consonant sound. The article must agree with the very next word it modifies.

Application

Scan each segment for the rule:

• "A unexpected spike in volcano activity" — the article sits before "unexpected", which opens with the vowel sound /ʌ/. A vowel sound requires "an", so "a unexpected" is wrong; it must read "an unexpected spike".

• "has forced authorities to evacuate" — present-perfect with the singular subject "spike" is correct.

• "nearby villages as a safety" — grammatically sound; "a safety" pairs with the noun "measure" that follows.

• "measure to prevent casualties." — the infinitive of purpose "to prevent" is used correctly.

Cross-check

Only the segment "A unexpected spike in volcano activity" violates a rule. Replacing "a" with "an" yields the fully correct sentence: "An unexpected spike in volcano activity has forced authorities to evacuate nearby villages as a safety measure to prevent casualties." Since one segment does carry an error, "No error" cannot be chosen.

The error is in segment C. The passive perfect form should be 'have been examined', not 'have being examined'.

Equation I gives x^2 - 10x + 24 = 0, so x = 4 or 6. Equation II gives 5y^2 - 16y + 12 = 0, so y = 2 or 1.2. Every x value is greater than every y value, so x > y.

Concept

In a two-equation "quadratic comparison" problem, you first find the roots of each quadratic, then compare every x-root against every y-root. A single fixed relation (x>y, x<y, x≥y, x≤y, or x=y) holds only if the comparison is consistent across all root pairs; if different pairs give opposite relations, no relation can be established.

Application

1. Equation I: 2x2 – 9x + 10 = 0. Factor as (2x – 5)(x – 2) = 0, giving x = 5/2 = 2.5 or x = 2.

2. Equation II: y2 – 12y + 35 = 0. Factor as (y – 5)(y – 7) = 0, giving y = 5 or y = 7.

3. So x belongs to {2, 2.5} and y belongs to {5, 7}. The largest x-value is 2.5 and the smallest y-value is 5.

4. Since the largest x (2.5) is still smaller than the smallest y (5), every x-value is strictly less than every y-value.

Cross-check

• Compare each pair: 2 < 5, 2 < 7, 2.5 < 5, 2.5 < 7 — all four comparisons give x smaller than y.

• The relation is strict (the x-values never equal any y-value) and it never reverses, so the result is the strict less-than relation, not the "or equal" variant and not the no-relation case.

Therefore the established relationship is x < y.

Concept

To compare the roots of two quadratic equations, first find the roots of each equation by factorisation (split the middle term so the two factors multiply to a·c and add to b). Then compare EVERY x-root against EVERY y-root. A single, definite relation (x>y, x<y, x≥y or x≤y) holds ONLY if the same inequality direction is true for all such pairings; if different pairings point in different directions, no definite relation can be established.

Application

Equation I: 6x² − 7x + 2 = 0. Split −7x as −3x − 4x:

1. 6x² − 3x − 4x + 2 = 0

2. 3x(2x − 1) − 2(2x − 1) = 0

3. (3x − 2)(2x − 1) = 0, so x = 2/3 or x = 1/2.

Equation II: 10y² − 11y + 3 = 0. Split −11y as −5y − 6y:

1. 10y² − 5y − 6y + 3 = 0

2. 5y(2y − 1) − 3(2y − 1) = 0

3. (5y − 3)(2y − 1) = 0, so y = 3/5 or y = 1/2.

Now compare each x-value with each y-value (x ∈ {1/2, 2/3}, y ∈ {1/2, 3/5}):

Cross-check / Result

Across the four pairings the direction is not consistent: one pairing gives x equal to y, one gives x less than y, and two give x greater than y. Because the inequality does not point the same way for every pairing, no single fixed relation between x and y is valid. Hence the correct response is that x = y or no relation can be established.

Concept

To compare the roots of two quadratic equations, first find all roots of each equation by factorisation, then compare every value of x against every value of y. A fixed relation (>, <, ≥, ≤) holds only if it is true for all pairs of roots; if the value ranges overlap or the relation flips across pairs, no fixed relation can be established.

Application

1. Equation I: 2x2 + 19x + 42 = 0. Factorise: 2x2 + 12x + 7x + 42 = 0 → 2x(x + 6) + 7(x + 6) = 0 → (x + 6)(2x + 7) = 0.

2. So x = −6 or x = −7/2 = −3.5.

3. Equation II: 2y2 + 7y + 5 = 0. Factorise: 2y2 + 2y + 5y + 5 = 0 → 2y(y + 1) + 5(y + 1) = 0 → (y + 1)(2y + 5) = 0.

4. So y = −1 or y = −5/2 = −2.5.

5. Compare every pair: −6 < −2.5, −6 < −1, −3.5 < −2.5, and −3.5 < −1. In all four pairings the x-value is smaller.

Cross-check

The largest possible x is −3.5 and the smallest possible y is −2.5. Since the biggest x (−3.5) is still less than the smallest y (−2.5), the value ranges do not overlap, so x is strictly less than y for every combination. Hence the relation is x < y.

Concept

In an approximation problem, replace each messy number by its nearest convenient whole value, then evaluate. For an equation of the form A = x% of B, isolate x using x = (A / B) x 100. The aim is the option closest to the true value, not an exact figure.

Application

1. Approximate each term on the left: 779.23 / 13 is about 780 / 13 = 60; 4.98 is about 5; the last term is 23.

2. Add them: 60 + 5 + 23 = 88. So the left-hand side is about 88.

3. The right-hand side is x% of 60.03, which is about x% of 60. Setting 88 = x% of 60 gives x = (88 / 60) x 100.

4. Compute: (88 / 60) x 100 = 146.7, which is about 147.

Cross-check

Using the unrounded numbers: 779.23 / 13 = 59.94, plus 4.98 plus 23 = 87.92; then 87.92 / 60.03 x 100 = 146.5. This still rounds to 147, confirming the result.

Concept

In an approximation (≈) problem, replace each messy number with the nearest convenient round value, simplify the arithmetic, then isolate the unknown. Here the equation a% of b + √c = x2 + d is solved for x by collecting the left side and subtracting d from it.

Application

1. Round each term: 39.98% ≈ 40%, 249.99 ≈ 250, √196.04 ≈ √196 = 14, and 55.93 ≈ 56.

2. Compute 40% of 250 = 0.40 × 250 = 100.

3. Add the square-root term: 100 + 14 = 114, so the left side ≈ 114.

4. Set up the equation: 114 = x2 + 56.

5. Isolate x2: x2 = 114 − 56 = 58.

6. Take the square root: x = √58 ≈ 7.62.

Cross-check

Among the offered whole numbers, 7.62 is nearest to 8 (it is only 0.38 away, versus 1.62 from 7). Verify by substituting back: 82 + 56 = 64 + 56 = 120, which is the closest match to the left side ≈ 114 among the choices. So x ≈ 8.

Concept

In an approximation (simplification) problem, replace every awkward decimal with the nearest convenient whole number, then evaluate with simple arithmetic. Two handy roundings apply here: a decimal just below a whole number is squared at that whole number, and a percentage close to 25% is treated as one-fourth (÷4).

Application

1. Round each term: 20.99 ≈ 21, 24.97% ≈ 25% (= 1/4), 220.02 ≈ 220, and the right side 131.13 ≈ 131.

2. Evaluate the square: 212 = 441.

3. Evaluate the percentage: 25% of 220 = 220 ÷ 4 = 55.

4. Substitute into the expression: 441 − 55 − ? = 131, which simplifies to 386 − ? = 131.

5. Solve for the blank: ? = 386 − 131 = 255.

Cross-check

Using exact values: 20.992 = 440.58, and 24.97% of 220.02 = 54.94, so 440.58 − 54.94 − 131.13 = 254.51, which rounds to 255 — confirming the rounded result.

Concept

In an approximation (simplification) problem you replace each value by the nearest convenient round number, evaluate the expression using the order of operations (powers and brackets first, then multiply or divide), and finally pick the option closest to your estimate. The goal is a fast, reliable estimate rather than an exact figure.

Application

Round each quantity and simplify the expression step by step:

1. Square the first term: 24.212 ≈ 242 = 576.

2. Add the second term: 576 + 12.16 ≈ 576 + 12 = 588 (this is the whole left-hand side).

3. Simplify the bracket on the right: 49.99% of 55.98 ≈ 50% of 56 = 28.

4. The equation becomes 588 = ? × 28, so ? = 588 ÷ 28 = 21.

Cross-check

With the exact numbers, 24.212 + 12.16 = 598.28 and 49.99% of 55.98 = 27.98, so ? = 598.28 ÷ 27.98 ≈ 21.4, which still rounds to 21 and confirms the estimate.

Concept

When a quantity splits into two kinds (here red and blue) and you are told one kind's percentage of the total and the difference between the two counts, the total is fully determined. If red is p% of total T, blue is (100 - p)% of T, so red minus blue equals (2p - 100)% of T. Setting this equal to the stated difference fixes T, and each colour count then follows.

Applying it to bag A and bag E

1. Bag A: red is 62.5%, so blue is 37.5%; their gap is 25% of the total. Since 25% of the total = 60, total = 60 ÷ 0.25 = 240. Red = 62.5% of 240 = 150; blue = 37.5% of 240 = 90.

2. Bag E: red is 72%, so blue is 28%; their gap is 44% of the total. Since 44% of the total = 330, total = 330 ÷ 0.44 = 750. Red = 72% of 750 = 540; blue = 28% of 750 = 210.

3. Red in A and E together = 150 + 540 = 690. Blue in A and E together = 90 + 210 = 300.

4. Required percentage = (690 ÷ 300) × 100 = 230%.

Cross-check

Each bag matches the table: 150 - 90 = 60 for bag A and 540 - 210 = 330 for bag E. Also 690 is exactly 2.3 times 300, confirming 230%.

Concept

For any bag, let the total balls be T, with red = (red%)·T and blue = (blue%)·T, where blue% = 100% − red%. Since the table gives the difference (red − blue) and the red percentage, the total follows from the identity:

red − blue = (red% − blue%)·T = (2·red% − 100%)·T.

Once T is known, red and blue for that bag are just the corresponding percentages of T.

Application

1. Bag A — red% = 62.5%, so blue% = 37.5%. Difference factor = 62.5% − 37.5% = 25% = 1/4. Given difference = 60, so T = 60 ÷ (1/4) = 240.

2. Red in A = 62.5% × 240 = 150; Blue in A = 37.5% × 240 = 90.

3. Bag E — red% = 72%, so blue% = 28%. Difference factor = 72% − 28% = 44% = 11/25. Given difference = 330, so T = 330 ÷ (11/25) = 330 × 25/11 = 750.

4. Red in E = 72% × 750 = 540.

5. Required ratio = (Red in A + Red in E) : (Blue in A) = (150 + 540) : 90 = 690 : 90 = 23 : 3.

Cross-check

Verify the totals from the difference: for A, 150 − 90 = 60 ✓; for E, 540 − 210 = 330 ✓. Dividing 690 and 90 by their HCF 30 gives 23 : 3, so the simplified ratio is exact.

Concept

When a group is split into two parts, the difference between the parts together with the share (percentage) of one part fixes the whole. If red balls are p (as a fraction) of the total T and blue balls are the rest, then red minus blue = (p - (1 - p)) x T = (2p - 1) x T. So the total is T = (red - blue difference) / (2p - 1), and the number of red balls is R = p x T.

Application

Apply this to each required bag using its given difference and red-percentage:

1. Bag B: difference = 198, red share = 66.5% = 0.665, so 2p - 1 = 0.33. Total = 198 / 0.33 = 600, and red = 0.665 x 600 = 399.

2. Bag D: difference = 76, red share = 60% = 0.60, so 2p - 1 = 0.20. Total = 76 / 0.20 = 380, and red = 0.60 x 380 = 228.

3. Bag E: difference = 330, red share = 72% = 0.72, so 2p - 1 = 0.44. Total = 330 / 0.44 = 750, and red = 0.72 x 750 = 540.

4. Average of red balls in B, D and E = (399 + 228 + 540) / 3 = 1167 / 3 = 389.

Cross-check

Verify each total reproduces the data: for B, blue = 600 - 399 = 201 and 399 - 201 = 198 (matches), and 399 / 600 = 66.5% (matches). The same check holds for D (228 - 152 = 76, 228 / 380 = 60%) and E (540 - 210 = 330, 540 / 750 = 72%). All three reconstructions are consistent, so the average is confirmed.

Concept

For any bag, the red-ball percentage fixes the split between red and blue. If red is p% of the total, blue is (100 − p)%, so the difference (red − blue) equals total × (2p − 100)%. From a known difference and the red-percentage you recover the total, and from the total you get the red and blue counts.

Total = Difference ÷ (2p − 100)%; then Red = Total × p% and Blue = Total × (100 − p)%.

Application

1. Bag A — Difference 60, red 62.5% (blue 37.5%, gap 25%). Total = 60 ÷ 0.25 = 240, so Red in A = 240 × 0.625 = 150.

2. Bag E — Difference 330, red 72% (blue 28%, gap 44%). Total = 330 ÷ 0.44 = 750, so Red in E = 750 × 0.72 = 540.

3. Bag F total — “25% more than red balls in E” = 540 × 1.25 = 675 balls.

4. Blue in F — ratio Red(A) : Blue(F) = 2 : 1 with Red(A) = 150, so Blue(F) = 150 ÷ 2 = 75.

5. Red in F — Red(F) = Total(F) − Blue(F) = 675 − 75 = 600.

Cross-check

Red(F) + Blue(F) = 600 + 75 = 675, matching the computed total for F. The stated 2 : 1 ratio also holds (150 : 75). Both independent conditions are satisfied.

Concept

In this graph each shop has TVs sold (blue line) and the gap 'cell phones minus TVs' (orange line). Because cell phones exceed TVs, Cell phones = TVs + Difference, so the Total items of a shop = TV + Cell phone = (2 x TV) + Difference. To express one quantity as a percentage more or less than a base, use: Percentage change = ((Value - Base) / Base) x 100, where the BASE is the quantity that comes after the words 'more/less than'.

Application

1. Read shop E from the graph: TVs sold by E = 80 and the difference for E = 360.

2. Total items of E = (2 x 80) + 360 = 160 + 360 = 520.

3. Read the base, total TVs sold by D, directly from the blue line: TV sold by D = 760.

4. Compare E's total with D's TVs: change = 520 - 760 = -240, so E's total is LESS than D's TVs.

5. Percentage = (240 / 760) x 100 = 31.578...%, which the option set reports as 31.57%.

Cross-check

Verify with the exact fraction: 240/760 = 6/19 = 0.31578..., so the shortfall is 6/19 of 760 = 240, recovering E total = 760 - 240 = 520, which is exactly E's total items. Hence E is 6/19 = about 31.57% less than D's TVs (the figure 31.578...% is reported as 31.57% among the options).

Concept

In this line graph the blue line gives the Total TV sold by each shop, while the orange line gives the Difference between Cell phones and TV sold (Cell phones − TV). The note states cell phones exceed TV at every shop, so for any shop:

Cell phones sold = TV sold + Difference.

A ratio of two such quantities is reduced to lowest terms by dividing both terms by their greatest common divisor (GCD).

Application

1. Read TV sold by D from the blue line: 760.

2. Read shop B from both lines: TV sold (blue) = 600 and Difference (orange) = 280.

3. Cell phones sold by B = TV + Difference = 600 + 280 = 880.

4. Required ratio = TV sold by D : Cell phones sold by B = 760 : 880.

5. Divide both terms by GCD(760, 880) = 40: 760 ÷ 40 = 19 and 880 ÷ 40 = 22, giving 19 : 22.

Cross-check

Scale the reduced ratio back up by 40: 19 × 40 = 760 and 22 × 40 = 880, which match the values just read, confirming 19 : 22.

Concept

When a fixed percentage of one category is defective, the count of defective units in that category equals that percentage of its total. If the grand total of defective items across two categories is known, the defective count of the second category is found by subtraction: defective(second) = total defective − defective(first).

Application

For shop D, items sold = TV + cell phone, and only TVs have a stated defect rate, so:

1. Read the total TV sold by D from the blue line: 760.

2. Defective TVs = 12.5% of 760 = (1/8) × 760 = 95.

3. Total defective items sold by D = 280 (given).

4. Defective cell phones = total defective − defective TVs = 280 − 95 = 185.

Cross-check

Add the parts back: 95 defective TVs + 185 defective cell phones = 280 defective items, which matches the given total. Also 12.5% = 1/8, and 760 is divisible by 8 (760/8 = 95), so the defective TV count is a whole number — consistent. Hence 185 defective cell phones.

Concept

An average is the sum of the quantities divided by how many there are: Average = (sum of values) / (number of values). Here the cell-phone sales are not plotted directly — the graph gives TVs sold and the cell-phone-minus-TV difference. The note says cell phones exceed TVs at every shop, so for each shop: cell phones sold = TVs sold + difference.

Application

1. Read the two graph values for each required shop — Shop B: TVs = 600, difference = 280; Shop C: TVs = 680, difference = 400; Shop E: TVs = 80, difference = 360.

2. Reconstruct cell phones with cell phones = TVs + difference — Shop B: 600 + 280 = 880; Shop C: 680 + 400 = 1080; Shop E: 80 + 360 = 440.

3. Add the three cell-phone totals: 880 + 1080 + 440 = 2400.

4. Divide by the number of shops (3): 2400 / 3 = 800.

Cross-check

The three totals 880, 1080 and 440 are balanced around 800, and their sum 2400 is exactly divisible by 3, so the average is the clean value 800.

Concept

Word problems with several interlinked totals are solved by translating every English statement into a linear equation in named variables, then eliminating one variable at a time. “P is k more than Q” means P = Q + k, and “P is k less than Q” means P = Q − k. The “total staff” of a college is its lab assistants plus its professors.

Set up the variables

Let the lab assistants be LA, LB, LC and the professors be PA, PB, PC for colleges A, B and C respectively.

Translate each statement

1. LA + PA = 82 (staff in A).

2. LB = LC + 4 (B’s lab assistants are 4 more than C’s).

3. PC = LC + 6 (C’s lab assistants are 6 less than C’s professors).

4. PA = LB − 44 (A’s professors are 44 less than B’s lab assistants).

5. LC + PC = (LA + PA) + 24 = 82 + 24 = 106 (staff in C is 24 more than staff in A).

6. PA + PB + PC = 102 (professors across all colleges).

Solve step by step

1. From (5) and (3): LC + (LC + 6) = 106, so 2LC = 100 and LC = 50; hence PC = 56.

2. From (2): LB = 50 + 4 = 54.

3. From (4): PA = 54 − 44 = 10, and from (1): LA = 82 − 10 = 72.

4. From (6): PB = 102 − PA − PC = 102 − 10 − 56 = 36.

5. Total staff in B = LB + PB = 54 + 36 = 90.

Cross-check

• Staff in A = 72 + 10 = 82 ✓; staff in C = 50 + 56 = 106 = 82 + 24 ✓.

• Professors total = 10 + 36 + 56 = 102 ✓. All conditions hold, so the total staff in B is 90.

College C has 82 + 24 = 106 staff. Let C professors be P, so C lab assistants are P - 6. Then P + (P - 6) = 106, giving P = 56 and C lab assistants = 50. B lab assistants = 54 and A professors = 10, so A lab assistants = 72. Total professors are 102, so B professors = 102 - 10 - 56 = 36. The required ratio is 36 : 72 = 1 : 2.

Concept

In a linear-equations word problem, translate every stated relation into an equation, then solve from the relation that involves the fewest unknowns first. A quantity is uniquely fixed once enough independent equations link it to known totals; substitute forward until every count is determined.

Setting up the equations

Let LA and P denote the number of lab assistants and professors in each college. The given relations are:

1. A: LA(A) + P(A) = 82.

2. B: LA(B) = LA(C) + 4.

3. C: P(C) = LA(C) + 6.

4. P(A) = LA(B) - 44.

5. Total staff in C = total staff in A + 24.

6. P(A) + P(B) + P(C) + P(D) = 102.

7. Total staff in D : LA(A) = 3 : 4.

8. LA(D) = LA(C) - 20.

Solving step by step

1. Staff in A = 82, so staff in C = 82 + 24 = 106.

2. In C, LA(C) + P(C) = 106 and P(C) = LA(C) + 6. Substituting: 2*LA(C) + 6 = 106, so LA(C) = 50 and P(C) = 56.

3. LA(B) = LA(C) + 4 = 54, hence P(A) = LA(B) - 44 = 10.

4. From LA(A) + P(A) = 82: LA(A) = 82 - 10 = 72.

5. LA(D) = LA(C) - 20 = 30.

6. Staff in D = (3/4)*LA(A) = (3/4)*72 = 54.

7. Professors in D = staff in D - LA(D) = 54 - 30 = 24.

Cross-check

Total professors = P(A) + P(B) + P(C) + P(D). With P(A)=10, P(C)=56, P(D)=24, the total 102 forces P(B)=12, a consistent positive count. Every staff total (A=82, C=106) and the 3:4 ratio (54:72) hold, confirming professors in D = 24.

Concept

Translate each verbal clue of a data-interpretation caselet into one linear equation in the unknown head-counts, then solve the system step by step; a percentage "X is what percent of Y" is always computed as (X / Y) x 100, where Y is the FULL base named in the question.

Application

Let the lab assistants and professors be A_LA, A_P, B_LA, B_P, C_LA, C_P. Turn the clues into equations and solve in order:

1. College A total: A_LA + A_P = 82.

2. Total staff in C is 24 more than in A, so C_LA + C_P = 82 + 24 = 106.

3. In C, lab assistants are 6 less than professors: C_LA = C_P - 6. Substituting into 106 gives 2*C_P - 6 = 106, so C_P = 56 and C_LA = 50.

4. B's lab assistants are 4 more than C's: B_LA = 50 + 4 = 54.

5. Professors in A are 44 less than lab assistants in B: A_P = 54 - 44 = 10 (and so A_LA = 82 - 10 = 72).

6. Total professors across all colleges is 102: B_P = 102 - A_P - C_P = 102 - 10 - 56 = 36.

7. Total staff in B = B_LA + B_P = 54 + 36 = 90.

8. Required percentage = (A_P / total staff in B) x 100 = (10 / 90) x 100 = 11.11%.

Cross-check

Verify every clue with the solved values: A total = 72 + 10 = 82; C total = 50 + 56 = 106 = 82 + 24; C lab assistants 50 = 56 - 6; B lab assistants 54 = 50 + 4; A professors 10 = 54 - 44; professors 10 + 36 + 56 = 102. All constraints hold, confirming 10 out of 90, i.e. 11.11%.

Concept

A linear-equations word problem is solved by assigning a variable to each unknown quantity, translating every given statement into an equation, and then solving the system by substitution. Each “more than / less than / total” phrase becomes one equation; when the number of independent equations equals the number of unknowns, the values are fixed uniquely.

Application

Let lab assistants and professors be: A → La, Pa; B → Lb, Pb; C → Lc, Pc. Translate each statement and solve in order:

1. A total: La + Pa = 82.

2. C total is 24 more than A total: Lc + Pc = 82 + 24 = 106.

3. C lab assistants are 6 less than C professors: Lc = Pc − 6. Substitute into the C total: (Pc − 6) + Pc = 106 → 2Pc = 112 → Pc = 56, so Lc = 56 − 6 = 50.

4. B lab assistants are 4 more than C lab assistants: Lb = Lc + 4 = 50 + 4 = 54.

5. Professors in A are 44 less than lab assistants in B: Pa = Lb − 44 = 54 − 44 = 10.

6. From the A total: La = 82 − Pa = 82 − 10 = 72.

7. Required average = (La + Lc) / 2 = (72 + 50) / 2 = 122 / 2 = 61.

Cross-check

Verify every condition with La=72, Pa=10, Lb=54, Lc=50, Pc=56 and Pb = 102 − Pa − Pc = 102 − 10 − 56 = 36: A total 72+10=82 ✓; C total 50+56=106 = 82+24 ✓; Lb−Lc = 54−50 = 4 ✓; Pc−Lc = 56−50 = 6 ✓; Lb−Pa = 54−10 = 44 ✓; total professors 10+36+56 = 102 ✓. All conditions hold, so the average is 61.

Concept

When a boat travels in a stream, the stream changes its effective speed relative to the bank: going downstream the stream adds to the boat’s still-water speed, going upstream it subtracts.

• Downstream speed = still-water speed + stream speed

• Upstream speed = still-water speed − stream speed

• Subtracting the second from the first: Downstream speed − Upstream speed = 2 × stream speed

So the still-water speed cancels out, and the difference of the two journey speeds always equals twice the stream speed. Whenever the same distance is covered both ways, the two given times fix both speeds in terms of that distance, and this relationship then fixes the distance.

Application

1. Downstream leg: the boat covers X km in 2 hours, so downstream speed = X ÷ 2 = X/2 km/h.

2. Upstream leg: the same X km takes 3 hours, so upstream speed = X ÷ 3 = X/3 km/h.

3. Apply the relationship with a stream speed of 3 km/h: X/2 − X/3 = 2 × 3 = 6.

4. Take the LCM of 2 and 3, which is 6: 3X/6 − 2X/6 = 6, that is X/6 = 6.

5. Multiply both sides by 6: X = 36 km.

6. The question asks for 2X, not X: 2X = 2 × 36 = 72.

Cross-check

Substitute X = 36 km back into the original data and recompute the stream speed independently:

The recomputed stream speed of 3 km/h matches the value given in the question, so X = 36 km and the required value 2X = 72.

A invests 6400 for 4 months and then 4480 for 8 months, so A's capital-months are 61440. B invests 7200 for 12 months, giving 86400. C invests 2880 for 8 months, giving 23040. A and C together get 84480/170880 of 90000, approximately Rs. 44494.

Let the initial milk and water be 8k and xk. After adding 5 litres milk and 30 litres water, 8k + 5 = xk + 30. This gives k(8 - x) = 25, which has multiple possible positive values. So x cannot be uniquely determined.

Concept

A pathway of uniform width laid across a rectangle, parallel to one side, is itself a rectangle. Its area equals its width multiplied by the side it spans. So for a strip parallel to the breadth, Area = width × breadth. Combined with a stated relation between length and breadth, this fixes both dimensions.

Application

1. Let the breadth of the park be b metres. The length is twice the breadth, so length = 2b.

2. A pathway of width 4 m runs parallel to the breadth, so it spans the full breadth b. Its area = 4 × b.

3. Set this equal to the given area: 4 × b = 352.

4. Solve for the breadth: b = 352 ÷ 4 = 88 m.

5. Length = 2b = 2 × 88 = 176 m.

Cross-check

With breadth 88 m, a 4 m strip across it covers 4 × 88 = 352 m2, matching the given pathway area, and the length 176 m is exactly twice 88 m. Both conditions hold, so the length is 176 m.

Concept

To compare the time a train takes to pass different objects, convert all speeds to m/s and use Distance = Speed x Time. When a train passes a moving object travelling in the SAME direction, the relevant speed is the relative speed (train speed minus object speed). When it passes a point object (a person), the distance covered equals the train's own length; when it passes a platform, the distance covered equals train length plus platform length.

Application

1. Convert speeds: train 72 km/h = 72 x 5/18 = 20 m/s; athlete 18 km/h = 18 x 5/18 = 5 m/s.

2. Athlete runs in the same direction, so relative speed = 20 - 5 = 15 m/s.

3. Crossing the athlete (a point) covers exactly the train's length: length = relative speed x time = 15 x 10 = 150 m.

4. Crossing the platform, the train moves at its own 20 m/s for 25 s, covering 20 x 25 = 500 m, which equals train length + platform length.

5. Platform length = 500 - 150 = 350 m.

Cross-check

Reverse it: a 150 m train plus a 350 m platform is 500 m; at 20 m/s that takes 500/20 = 25 s, matching the given crossing time. So the platform is 350 m.

Concept

Cost price, marked price and discount are linked by two ratios. Loss/profit are always measured on the cost price: SP = CP*(1 - loss%) or CP*(1 + profit%). Discount is always measured on the marked price: Selling Price = MP*(1 - discount%), so discount% = (MP - SP)/MP * 100.

Application

1. Find CP from the loss sale. A selling price of Rs. 624 came with a 22% loss, so 624 = CP*(1 - 0.22) = 0.78*CP, giving CP = 624/0.78 = Rs. 800.

2. Find the marked price. The article is marked 30% above cost, so MP = 800*(1 + 0.30) = 800*1.30 = Rs. 1040.

3. Find the actual selling price in the profit case. A profit of Rs. 84 means SP = CP + 84 = 800 + 84 = Rs. 884.

4. Find the discount. Discount amount = MP - SP = 1040 - 884 = Rs. 156, so discount% = (156/1040)*100 = 15%.

Cross-check

apply 15% discount to MP directly: 1040*(1 - 0.15) = 1040*0.85 = Rs. 884, which equals CP + profit = 800 + 84. Consistent, so 15% is correct.

Interest from scheme A = 15% x 2 x X = 0.30X, so the amount reinvested is 1.30X. Compound interest for two years at 10% on 1.30X is 1.30X x (1.21 - 1) = 0.273X. Difference = 0.30X - 0.273X = 0.027X = 32.4, so X = 1200.

Concept

In work problems the total work is constant, so for any group (combined efficiency) x (days) = total work. Equating two scenarios that finish the same work gives the ratio of one man's efficiency to one woman's; once that ratio is fixed the total work can be written in efficiency units and any unknown number of workers found.

Application

1. Let one man's one-day efficiency be m and one woman's be w. Equate the first two scenarios (same total work): 15(7m + 15w) = 8(18m + 20w).

2. Expand: 105m + 225w = 144m + 160w, so 65w = 39m, giving m : w = 65 : 39 = 5 : 3.

3. Take w = 3 and m = 5 (any pair in 5 : 3). Total work = (7m + 15w) x 15 = (35 + 45) x 15 = 80 x 15 = 1200 units.

4. Third scenario: (X men + 20 women) work 10 days, so (5X + 20x3) x 10 = 1200, i.e. 5X + 60 = 120.

5. Solve: 5X = 60, so X = 12.

Cross-check

With X = 12, combined efficiency = 12x5 + 20x3 = 60 + 60 = 120 units/day; over 10 days that is 1200 units, exactly the total work. Consistent, so X = 12.

Concept

When numbers are equally spaced (an arithmetic progression), their average equals the mean of the terms. For four consecutive multiples of 4 the common gap is 4, so the four terms can be written as a, a+4, a+8 and a+12, and their average is the value exactly in the middle of the set. The same idea fixes a run of consecutive even numbers once its starting value is known: each successive term is 2 more than the previous one.

Application

1. Let the four consecutive multiples of 4 be a, a+4, a+8, a+12. Their sum is 4a + 24.

2. Average = (4a + 24) / 4 = a + 6. Set a + 6 = 30, so a = 24.

3. The four multiples are therefore 24, 28, 32 and 36.

4. The second-lowest of these is 28, so the five consecutive even numbers start at 28.

5. Five consecutive even numbers from 28 are 28, 30, 32, 34, 36 (each 2 more than the last).

6. The highest term of this even series is 36.

Cross-check

Average of 24, 28, 32, 36 = 120 / 4 = 30, confirming the four multiples. Starting the even run at 28 and adding 2 four times gives 28, 30, 32, 34, 36, whose largest value is 36.

Concept

When ages are linked by sums and a known future/past value, set up one linear equation per stated relationship and solve them together. A person's age k years from now is (present age + k) and k years ago is (present age − k). A ratio of the form 4 : X means the two quantities, written in lowest terms, share the factor that turns 4 into the first quantity — so X is the second quantity scaled by the same factor.

Application

1. R after 12 years = 32, so R's present age = 32 − 12 = 20 years.

2. R & Q sum to 56, so Q's present age = 56 − 20 = 36 years.

3. P & Q sum to 64, so P's present age = 64 − 36 = 28 years.

4. P after 12 years = 28 + 12 = 40 years.

5. Q six years ago = 36 − 6 = 30 years.

6. Ratio (P after 12) : (Q six years ago) = 40 : 30 = 4 : 3, so X = 3.

Cross-check

Reduce 40 : 30 by dividing both by 10 to get 4 : 3; the first term matches the given 4, confirming the second term X = 3. Equivalently, 40/4 = 10 and 30/10 = 3, so X = 3.

The final arrangement shows A on the top floor, G and C on the same floor, and E in flat P where no flat lies to the west. Therefore statements I, II and III are all true.

Concept

In an 'odd one out' grouping question, four of the five named people share one common defining property and the fifth breaks it. The correct method is to first fix the complete arrangement from all the clues, then find the single property that holds for four members and fails for exactly one - never to guess from the labels.

Building the arrangement

The building has 4 floors (floor 1 = ground, floor 4 = top). Each floor has flat P (west) and flat Q (east). 'North' means a higher floor in the same column and 'south' a lower floor; 'east' is the Q side and 'west' the P side. Working through the clues step by step:

1. C is not on an odd floor, so C is on floor 2 or 4.

2. B is south-east of A: B is on a lower floor than A and in the east (Q) column while A is in the west (P) column.

3. There is exactly one floor gap between A and G (their floors differ by 2), and G is in the east of C on the same floor - so C takes flat P and G takes flat Q on the same floor, which (being even) places them on floor 2.

4. H is below B in the same-named flat (same column), and F is to the north of E, i.e. on a higher floor than E in the same column - not necessarily the floor immediately above. Combining these with the one-floor-gap and south-east links forces A to floor 4 (west) and B to floor 3 (east).

5. The remaining people fill the last seats uniquely: F goes to floor 3 (west) and E to floor 1 (west, in the same column as F), while D and H take the remaining seats.

This yields a single arrangement (flat P shown left/west, flat Q shown right/east):

Applying the grouping rule

Test the five named people - F, D, B, C and G - against one property: does each have a person living in the flat exactly one floor up in the same column?

• F (floor 3, west): the flat directly above on floor 4 is occupied - yes.

• B (floor 3, east): the flat directly above on floor 4 is occupied - yes.

• C (floor 2, west): the flat directly above on floor 3 is occupied - yes.

• G (floor 2, east): the flat directly above on floor 3 is occupied - yes.

• D (floor 4, east): floor 4 is the top floor, so no flat exists one floor above - no.

Four of the five have an immediate upstairs neighbour; only the top-floor resident has none. Hence D is the one that does not belong to the group.

Cross-check

No competing property gives a clean four-versus-one split: grouping by flat column gives a 2-3 split (two in P, three in Q) and grouping by floor parity also fails to isolate exactly one. Only the 'has someone one floor directly above' property isolates a single person, confirming D.

The arrangement places B in flat Q on the third floor. Hence B is the person living in flat Q of the third floor.

In the arrangement, G lives in flat Q on floor 2 and F lives in flat P on floor 3. Flat P is west of flat Q, and floor 3 is north/above floor 2. Therefore F lives north-west of G.

From the arrangement, A lives on floor 4 and H lives on floor 1. The sum of their floor numbers is 4 + 1 = 5.

Concept

A qualifying pair of digits is one where the count of digits lying strictly between the two digits in the given number equals the count of whole numbers lying strictly between those two digit values in the natural number series (… 1, 2, 3, 4 …). The position-gap is checked in both directions, so the order in which the two digits appear does not matter — only the size of the gap does.

Application

Write the number with positions: 6(1), 2(2), 8(3), 1(4), 8(5), 3(6), 7(7). For every pair, compare the positional gap (digits sitting between them) with the value gap (numbers sitting between them in 1–9):

Each of these three pairs has equal gaps, so each one qualifies.

Cross-check

All other pairs fail. For instance, 8 and 1 sit two digits apart in the number but seven numbers apart in the series; 6 and 2 sit zero digits apart but five numbers apart. Scanning every remaining pair the same way leaves exactly three matches.

Hence the number of qualifying pairs is 3.

Some impressive are crashed, and no crashed is champion. Therefore those impressive that are crashed are definitely not champions, so conclusion II follows. The statement does not prove that no master is crashed, because some masters may overlap with crashed through impressive. Hence only conclusion II follows.

All compare are cups. Some cups are plays. Only a few plays are waiting means some plays are waiting and some plays are not waiting. There is no negative relation blocking compare from being waiting, so 'some compare being waiting is a possibility' follows. But 'all cup is waiting' is not definitely proved. Hence only conclusion I follows.

The confirmed chain is A > D > B > F > E > C. The persons lighter than D are B, F, E and C. Hence four persons are lighter than D.

The weight order is A > D > B > F > E > C. Between A and E there are D, B and F, and the exact middle person is B. Therefore B is exactly between A and E.

Concept

In syllogism, a definite conclusion must hold in EVERY possible diagram allowed by the statements, while a "possibility" conclusion only needs at least ONE consistent diagram in which it is true. Two rules drive this set: (i) a universal chain "All A is B" together with "No B is C" forces "No A is C" (A and C are disjoint); (ii) "Only a few A is B" means some A is B AND some A is not B, and it leaves the not-B part of A free to relate to anything else unless another statement restricts it.

Application

1. From "All good is down" and "No down is hunger": Good is wholly inside Down, and Down is wholly outside Hunger. So Good is entirely outside Hunger — no Good can be Hunger.

2. Test "Some good is hunger": this asserts an overlap between Good and Hunger, but step 1 shows they are disjoint. So this statement is definitely FALSE — it does not follow.

3. "Only a few Trophy is good" gives some Trophy inside Good (those are inside Down, hence outside Hunger) and some Trophy outside Good, with no rule tying that outside part to Hunger.

4. Test "Some trophy being hunger is a possibility": since the non-good Trophies are unrestricted with respect to Hunger, we can draw a consistent diagram where a few of them lie in Hunger. A valid possibility diagram exists, so this possibility conclusion FOLLOWS.

Cross-check

Conclusion I claims a forced overlap that the disjointness rule forbids, so it can never hold. Conclusion II only claims a possibility, and one consistent diagram is enough to establish it. Hence exactly one of the two conclusions follows: the second one alone.

In CDF, NOQ, KLN, and QRT, the letter gaps are +1 and then +2. In STW, the gaps are +1 and then +3. Therefore STW is the odd one out.

In the final arrangement, Q goes in August and likes White.

Therefore Q likes White.

The weight clues give D > B > F > E > C. Since D is not the heaviest and A is heavier than E, A must be heavier than D to satisfy the condition. Therefore A is the heaviest.

The unique arrangement gives July to M.

Therefore M goes on the holiday trip in July.

O goes in January, while the person who likes Blue goes in May.

March and April lie between January and May in the given month order.

Thus two persons go between O and the one who likes Blue.

Concept

In a month-scheduling puzzle, fix the timeline first, then turn every clue into a position or gap rule and lock the rigid anchors before the loose ones. "Exactly k persons go between X and Y" means their month-positions differ by k+1; "just before / just after" means consecutive months. Among these seven months only April and June have 30 days, so they are the only months with an even number of days.

Building the arrangement

1. N is in an even-day month (April or June). "Blue is just before green" with "green in an even-day month" gives the consecutive pair May(blue) -> June(green).

2. "Two persons between N and the red-liker" is a gap of 3 with N in an even-day month: N -> April and the red-liker is in July.

3. P goes after N and R is just after P (R = P+1). With May and June already taken by blue and green, P -> May and R -> June, so P likes blue and R likes green (allowed, since the restriction is only that P does not like green or white).

4. "Only one person between the blue-liker (May) and S" is a gap of 2, so S -> March; S likes black.

5. The red-liker is in July, and "four persons between M and the yellow-liker" is a gap of 5. Placing M -> July (red) and the yellow-liker -> January gives exactly that gap of 5; the remaining open month, August, goes to the last person.

6. "White goes after yellow" and N (April) is not white: yellow -> January and white -> August. O cannot like green or white, so O -> January (yellow); that leaves Q -> August (white) and the only remaining colour, grey, for N (April) - consistent with Q not liking grey.

Final schedule

Applying the alphabetical re-seating

Reseat alphabetically from January: M, N, O, P, Q, R, S go to January, March, April, May, June, July, August respectively. Compare each month with the original schedule and count the months whose person is identical.

Cross-check

Only May keeps the same person, so exactly one position is unchanged. The reason it alone coincides: that person is the 4th traveller in calendar order and is also the 4th name alphabetically, so both schedules send him to the 4th month (May).

Answer: One.

The final holiday arrangement is:

January: O - Yellow; March: S - Black; April: N - Grey; May: P - Blue; June: R - Green; July: M - Red; August: Q - White.

R goes just after P, and R is the one who likes Green.

Therefore the person who likes Green goes just after P.

The 3rd, 6th, 7th, 8th, and 11th letters of CONSEQUENCES are N, Q, U, E, and E. They form the word QUEEN, whose third letter from the right is E.

Concept

In a blood-relation puzzle, never jump to a label. First fix each person's GENERATION and GENDER from the clues, then connect couples and parent-child links. A reliable rule: a man is the brother-in-law of an unmarried woman only when he is the husband of that woman's sister (he cannot be her spouse's brother, because she has no spouse). Once couples and parents are fixed, a relationship two generations up the same line is a grandparent link.

Application (build the family tree step by step)

1. A is the brother-in-law of F, and F is not married. So A is not the brother of F's husband (F has none) - A must be the husband of a SISTER of F.

2. F is the sister of G, so G is a sibling of F. The only sister of F available to be A's wife is G, so G is female and A is married to G. (A, G) form a couple.

3. A is the father of B, so B is a child of the couple A and G.

4. B is the wife of D, so B and D form the next-generation couple.

5. C is the granddaughter of G. Because A and G are the same couple, C is equally A's grandchild, and C reaches that generation through B (A and G's child).

6. Trace the line A and G -> their child B (married to D) -> their child C. That is exactly two generations from A down to C.

A is two generations directly above C on the same family line, and A is male, so A is the grandfather of C.

Cross-check

• The label one generation up (a parent of C) does not fit: A is the parent of B, and B is the parent of C, so A sits one level higher than a parent.

• A same-generation label (a sibling of C) is impossible: A belongs to the oldest of the three generations while C belongs to the youngest.

• A side-branch elder label (a parent's brother of C) does not fit either: A is on C's direct ancestral line through B, not on a collateral branch.

• Counting generations A -> B -> C confirms exactly two steps down a direct line, which is the grandparent relationship. Result: grandfather.

A is married to G because A is the brother-in-law of unmarried F, who is G's sister. B is the daughter of A and is married to D. H is D's brother, so H's sister-in-law is B. Since E is H's niece, B is E's mother. Therefore the sister-in-law of H is related to E as mother.

F is female because she is the sister of G. B is female because she is the wife of D. C is female because she is a granddaughter. E is female because she is a niece. Since A is the brother-in-law of unmarried F and A is male, A must be married to G, so G is female. Thus the female members are F, B, C, E and G: five in total.

The unique department arrangement is Finance: B, F, G; HR: C, D; Marketing: A, E.

So D works in the HR department.

The solved arrangement places A in Marketing and gives A the colour Green.

The other listed combinations do not match the arrangement.

Hence A - Marketing - Green is the correct combination.

E works in Marketing with A.

In the solved arrangement, A likes Green.

Therefore the colour liked by the person who works in the same department as E is Green.

Concept

These grouping puzzles are solved by fixing the smallest, most constrained facts first, then propagating each clue until every person has exactly one department and one colour. With 7 people across 3 departments and the rule that no department holds fewer than 2 or more than 3, the only possible split of group sizes is 3, 2 and 2. Build a department-vs-colour grid and eliminate impossibilities clue by clue.

Building the arrangement

1. Fixed colours and seats: F likes Yellow, E likes Red, and G is in Finance. F and A are in different departments and neither is in HR, so F and A occupy Finance and Marketing in some order.

2. E sits with A; B sits apart from E. Since A is in Finance or Marketing, place A and E together and keep B in a different group.

3. The Black-liker is not in Marketing and shares a group with the Orange-liker; A and G dislike Orange. The only group that can hold both Black and Orange (size 3, not Marketing) is Finance, anchored by G.

4. G already dislikes Orange, so within Finance G takes Black and the third Finance member takes Orange. F likes Yellow, so the Orange-liker must be B — placing B in Finance with G, and F joins them, giving Finance = three people.

5. D shares a group with the White-liker but not with F or A, and D does not like White. That forces D into HR. C works only with the Blue-liker, so C and the Blue-liker form the 2-person HR group with D — meaning D likes Blue and C likes White.

6. The remaining group is Marketing with A and E (E = Red), leaving A = Green.

Final grid

Cross-check the statements

The question asks which statement is NOT true. Test each against the grid:

• Blue-liker (D) is in HR — true.

• A is in Marketing — true.

• B is in Finance and D is in HR, so they are in different departments — this statement is false, and is therefore the answer.

• A likes Green — true.

• Orange-liker (B) is in Finance — true.

Only the claim that B and D share a department conflicts with the unique arrangement, so it is the statement that is not true.

The unique arrangement is:

Finance: B (Orange), F (Yellow), G (Black).

HR: C (White), D (Blue).

Marketing: A (Green), E (Red).

Therefore F, B and the person who likes Black all belong to Finance.

Concept

In a direction-sense problem, fix a compass frame (East = +x, North = +y) and convert every turn relative to the current facing: a right turn rotates the heading 90° clockwise (E→S→W→N→E), a left turn rotates it 90° anticlockwise (E→N→W→S→E). Track coordinates leg by leg; the final direction of one point relative to another is read from the signs of the differences in their x (East/West) and y (North/South) coordinates.

Application

Place A at the origin (0, 0) and follow each leg up to G:

1. Facing East, walk 7 m: A(0, 0) → B(7, 0).

2. At B turn right — East becomes South — walk 2 m: B(7, 0) → C(7, −2).

3. At C turn left — South becomes East — walk 9 m: C(7, −2) → E(16, −2).

4. At E turn left — East becomes North — walk 4 m: E(16, −2) → G(16, 2).

Comparing G(16, 2) with A(0, 0): the x-difference is +16 (G is East of A) and the y-difference is +2 (G is North of A). A point that is both East and North lies in the North-east direction.

Cross-check

The net horizontal travel is 7 m + 9 m = 16 m to the East with no westward leg, so G is firmly East of A. Vertically, Ritika first drops 2 m South then climbs 4 m North, a net of +2 m, so G ends up North of A as well. Both components confirm North-east.

Using coordinates, take A as (0, 0). The route gives E = (16, -2) and L = (8, 5). The shortest distance is sqrt((16 - 8)^2 + (-2 - 5)^2) = sqrt(64 + 49) = sqrt(113).

Concept

In a floor-plan / directions puzzle, after tracing the route on a grid each labelled point gets fixed (x, y) coordinates (east = +x, north = +y). Two points lie on the same straight line when they share one coordinate: equal y puts them on a horizontal (east–west) line, equal x puts them on a vertical (north–south) line. To find the odd pair out, plot every point and test which geometric property the majority of the given pairs share — the pair lacking that shared property is the answer.

Application — trace the route

1. Start at A (0, 0) facing east; walk 7 m east → B (7, 0).

2. Turn right (now facing south); walk 2 m → C (7, −2).

3. Turn left (now facing east); walk 9 m → E (16, −2).

4. Turn left (now facing north); walk 4 m → G (16, 2).

5. Two consecutive left turns, 2 m each: 2 m west to (14, 2), then 2 m south → D (14, 0).

6. Turn right (now facing west); walk 6 m → K (8, 0).

7. Turn right (now facing north); walk 5 m → L (8, 5).

Compare the pairs by coordinate

Read off the alignment of each given pair from its coordinates:

• A (0, 0) & B (7, 0): equal y = 0 → horizontal line.

• B (7, 0) & D (14, 0): equal y = 0 → horizontal line.

• C (7, −2) & E (16, −2): equal y = −2 → horizontal line.

• K (8, 0) & D (14, 0): equal y = 0 → horizontal line.

• E (16, −2) & G (16, 2): equal x = 16 → vertical line.

Result

Four of the pairs share the same y-coordinate, so each of those lies on a horizontal (east–west) line. The pair E (16, −2) and G (16, 2) instead shares the same x-coordinate, so it lies on a vertical (north–south) line. Because it is the only pair with a different orientation, E and G is the pair that does not belong to the group.

In the solved order, H is Senior Manager. H is also the person three positions senior to F. Therefore the option describing the person three positions senior to F is correct.

Concept

In a linear ranking puzzle, fix each person to one of 8 slots ordered from the seniormost designation (Chairman) to the junior-most (Clerk). Translate every clue into a position relation: "X senior to Y" means X sits in a higher (smaller-numbered) slot, "n persons between two people" means their slots differ by n+1, and a "two more than" count clue pins absolute positions. Anchor the clues that fix absolute slots first, then place the relative ones.

Application

1. Persons junior to D are two more than persons senior to E: 8 minus D's slot equals (E's slot minus 1) plus 2. Together with G being three senior to D (G = D - 3), this narrows the start to a few candidate pairs, of which E at slot 1 (Chairman) with D at slot 6 (Manager) survives; the rival pair E at slot 3, D at slot 4 is ruled out later because it leaves no valid placement for A, F and B.

2. G is three designations senior to D, so G is at slot 3 (GM).

3. Three persons sit between E and H, so their slots differ by 4; since E is at slot 1, H is at slot 5 (SM).

4. A is senior to Manager (slot 6) and F is four persons junior to A (F = A + 4): with slots 3 and 6 taken, A lands at slot 4 (DGM) and F at slot 8 (Clerk).

5. B is senior to DGM (slot 4), so only slot 2 remains for B (MD).

6. C is junior to H (slot 5), so the last open slot 7 fits, making C the AM.

Final order, seniormost to junior-most: E (Chairman), B (MD), G (GM), A (DGM), H (SM), D (Manager), C (AM), F (Clerk).

Result

The person immediately senior to F (Clerk, slot 8) is at slot 7 — that is C. B is at slot 2. The persons strictly between slot 2 and slot 7 fill slots 3, 4, 5 and 6, which is four persons.

Cross-check

Re-test the count clue on this order: D is at slot 6, so 2 persons are junior to D; E is at slot 1, so 0 persons are senior to E; 0 + 2 = 2. Every clue holds, confirming the arrangement is unique.

In the solved designation order, D is Manager.

A is fourth in the seniority order, with three persons senior to A. H has three persons junior to H. Therefore the answer is H.

The final designation order is E, B, G, A, H, D, C, F from senior to junior. B is related to DGM by moving two positions junior, and D is related to Clerk in the same way. Moving two positions junior from E gives General Manager.

The coherent order is B-D-C-E-A: it introduces sleep stages, explains REM, contrasts non-REM, adds importance, and ends with the repeated cycle. Therefore the correct arrangement is B-D-C-E-A.

The coherent order is C-D-A-B-E: introduce Internet privacy, explain company collection, mention user sharing, state consequences, and conclude with protection. Therefore the correct order is C-D-A-B-E.

The logical order is: oceans absorb heat; that regulates climate; excess heat raises ocean temperatures; warmer oceans disrupt ecosystems and intensify storms; these changes show the ocean-climate link. So the order is A-B-E-D-C.

The coherent order is C-A-B-D-E: introduce the benefit of reading, then vocabulary, concentration/language skills, relaxation, and the conclusion. Therefore the correct order is C-A-B-D-E.

The central irony is that digital payment platforms meant to help small merchants are instead burdening them through high charges and dependency.

Concept

This is an inference question about a cause-and-effect link stated in the passage. To answer it, do not rely on outside knowledge or general impressions: find the single sentence where the author directly connects the two ideas named in the question — here, the broader state of the economy and the problems created by the payment platforms — and read what relationship the author asserts between them.

Applying it to this passage

The passage makes the link in one specific line: “the weather of the broader economy—rising inflation, slowing exports, and unpredictable growth—makes these charges even harder to digest.” In other words, the wider economic conditions do not soften or solve the overcharging; they intensify how heavily those charges weigh on merchants. So the overall economy worsens the existing problem by adding financial pressures such as inflation on top of the hidden fees.

That is exactly what “It increases the burden of overcharging due to financial pressures like inflation” says, so it is the correct reading of the passage.

Why the other readings do not fit the passage

• “It helps small merchants adapt faster to digital systems” — the passage links the economy to harder-to-bear charges, not to easier adaptation; it actually says small firms feel the burden and many folded.

• “It makes overcharging practices more acceptable to the public” — the passage attributes public unawareness to costs being hidden in final prices, not to economic conditions making overcharging acceptable.

• “It incentivizes payment firms to reduce fees” — nothing in the passage says the economy pushes the firms to lower fees; the author warns the monopolistic grip persists unless regulation steps in.

• “It drives consumers away from digital platforms” — the passage says consumers stay largely unaware because the extra cost is folded into prices, so it does not claim the economy pushes consumers away.

The passage presents digital payment gateways as convenient and fast, so statement I is supported. It also explicitly mentions delays in settlement and penalties for alternative platforms, so statement II is supported. It does not say consumers protested en masse, so statement III is not supported.

Concept

A statement is FALSE "according to the passage" when it contradicts or overstates what the passage actually asserts. To judge each statement, match it against the passage's own claims: a universal word ("all", "completely", "majority") is false if the passage qualifies the claim ("some", "largely", "a sense of").

Applying it to each statement

• (I) The passage states that consumers "remain largely unaware of the scale of overcharging" because the extra costs are folded into final prices. So claiming the majority are well-informed contradicts the passage — this statement is FALSE.

• (II) The passage says these practices "contribute to a sense of dependency on a few dominant players" and warns of a "monopolistic grip". So claiming convenience has completely eliminated dependency contradicts the passage — this statement is FALSE.

• (III) The passage says the 1%–3% range applies to international credit card companies, while some Chinese gateways add hidden charges that push the effective rate even higher. So claiming all Chinese gateways charge uniformly 1%–3% contradicts the passage — this statement is FALSE.

Result

All three statements (I), (II) and (III) are FALSE according to the passage, so the option naming all three is correct.

Concept: A blank that takes the demonstrative subject “This” followed by a “how…” clause needs a singular present-tense transitive verb whose meaning is “to draw attention to / make something clear.” The subject “This” points back to the preceding fact (local businesses folding), and the verb must let that fact stand as evidence for the statement that follows.

Application: The sentence reads “This _____ how dependence on a few gateways can distort fair competition.” The fact just stated (small firms shutting down) is being used to make a broader point visible. The verb that means “makes this point clear / emphasises it” is “highlights,” giving “This highlights how dependence … can distort fair competition” — grammatically singular, idiomatic with a “how”-clause, and matching the cause-to-evidence logic.

Contrast with the other choices, judged on their own merit:

• “informs” — to “inform” normally needs a person as its object (it informs us); paired directly with “how … can distort” it reads incomplete and unidiomatic here.

• “produces” — means “to bring into existence”; a fact does not “produce how something can distort competition,” so the meaning does not fit.

• “change” — the base form clashes with the singular subject “This” (it would need to be “changes”), and “change how …” would mean altering the manner, not pointing it out.

• “payment” — a noun, not a verb, so it cannot fill a slot that requires a finite verb after “This.”

Cross-check: substitute the chosen word back — “This highlights how dependence on a few gateways can distort fair competition” reads naturally and preserves the passage’s evidence-to-claim flow, confirming the fit.

In the phrase 'the weather of the broader economy', the intended meaning is the overall situation or condition of the economy. Among the options, situation is the best replacement.

Concept

A contextual synonym is a word that can replace the target word in the sentence while keeping the same meaning the author intended. The right substitute must match both the part of speech (here a plural noun) and the sense carried by the surrounding text. So the test is: read the line, swap the candidate in, and check that the meaning is preserved.

Application

In the passage, “firms” is used as a plural noun for business organisations: “the overcharging practices of these payment firms”, “Small firms, in particular, feel the burden”, and it is restated as “Many local businesses” and “new entrants”. The author is talking about commercial enterprises. The noun that means business enterprises is “Companies”, which slots in cleanly: “Small companies feel the burden.”

Contrast

• “Holds” — relates to the verb “to hold firmly” or being firm; it is not a noun for a business and breaks the sentence.

• “Contracts” — means agreements, not the organisations entering them; it shifts the meaning away from enterprises.

• “Stubborn” — captures the adjective sense of “firm” (resolute), but the passage uses “firms” as a noun, so an adjective cannot replace it.

• “Customers” — refers to buyers; the passage distinguishes these business sellers from the consumers, so it names the wrong group.

Hence the most appropriate synonym in context is Companies.

Concept

A contextual synonym is the word whose meaning best replaces the target word without changing the sentence's intended meaning. The right choice must fit the exact sense the word carries in this passage, not merely some dictionary meaning of the word in isolation.

Application

Target sentence: “Many local businesses, unable to bear the pressure of mounting costs, have folded within months of adopting these systems.” Here “folded” is a business idiom — a firm that “folds” ceases to operate, i.e. it shuts down because it can no longer cover its costs.

The surrounding clues confirm this sense: “unable to bear the pressure of mounting costs” signals failure and exit from the market, not growth or mere change. Substituting each candidate back into the sentence, the word that preserves this ‘went out of business’ meaning is “Closed” — “have closed within months” reads naturally and keeps the intended sense of firms ceasing to operate.

Contrast

• Expanded — grew larger; the opposite of a firm forced out by mounting costs, so it contradicts the sentence.

• Shrunk — reduced in size while still operating; the firms did not merely get smaller, they ceased to exist, so this is too weak.

• Evolved — changed or developed into a new form; nothing here suggests transformation, only failure.

• Joined — came together or merged; the passage describes firms ending operations, not partnering.

Result

The synonym that fits the contextual ‘went out of business’ sense is “Closed.”