Directions: Read the information and answer the following questions. Four…

2023

Directions: Read the information and answer the following questions.

Four stations are in a rectangular form ABCDA. The distance between A and B is (P2 + 100) km and that between B and C is (20P + 100) km. Train Y starts from station A and travels to D via stations B and C. The total distance travelled by the train is 2700 km in 25 hours. The table below shows the time to cross a pole and the time to cross a platform.

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Note: The average time taken by all the trains to cross the platform is (Q + 6) seconds and the length of the platform is 400 metres.

Find the ratio P : Q.

  1. A.

    5:4

  2. B.

    2:1

  3. C.

    1:8

  4. D.

    2:3

  5. E.

    5:3

Attempted by 1 students.

Show answer & explanation

Correct answer: A

Concept

Two independent relations fix the two unknowns. Distance: for a journey along the sides of a rectangle, the total length is the sum of the side-lengths covered, with opposite sides equal (CD = AB). Average: the mean of a set of values equals their sum divided by the count, avg = (sum of values) / (number of values); setting this mean equal to the given expression yields a linear equation in the second unknown.

Application — finding P (distance)

Train Y runs A → B → C → D, i.e. three sides of rectangle ABCD. Since CD = AB, the path length is 2·AB + BC.

  1. AB = P2 + 100 and BC = 20P + 100 (given).

  2. 2·AB + BC = 2700 → 2(P2 + 100) + (20P + 100) = 2700.

  3. Expand: 2P2 + 200 + 20P + 100 = 2700 → 2P2 + 20P − 2400 = 0.

  4. Divide by 2: P2 + 10P − 1200 = 0 → (P + 40)(P − 30) = 0.

  5. A length must be positive, so P = 30 (reject P = −40).

Application — finding Q (average)

From the table the times to cross the platform are P (for one train), Q, and P + 6. Their average is given as Q + 6:

  1. (P + Q + (P + 6)) / 3 = Q + 6.

  2. Multiply by 3: 2P + Q + 6 = 3Q + 18.

  3. Rearrange: 2P − 12 = 2Q → Q = P − 6.

  4. With P = 30: Q = 30 − 6 = 24.

Result

P : Q = 30 : 24 = 5 : 4.

Cross-check

  • Distance: AB = 302 + 100 = 1000, BC = 20·30 + 100 = 700, so 2·1000 + 700 = 2700 km. ✓

  • Average: platform times 30, 24, 36 → mean = 90/3 = 30 = Q + 6 = 24 + 6. ✓

  • The platform length (400 m) and the pole-crossing entries are extra data for this set and are not needed for the P : Q ratio.

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