One number, n, is selected at random from a set of 10 integers. What is the…
2023
One number, n, is selected at random from a set of 10 integers.
What is the probability that ¼ n + 13 = 0 ?
Statement 1:The largest integer in the set is 13.
Statement 2:The arithmetic mean of the set is zero.
- A.
statement 2 alone is sufficient, but statement 1 alone is not sufficient to answer the question
- B.
statement 1 alone is sufficient, but statement 2 alone is not sufficient to answer the question
- C.
both statements taken together are sufficient to answer the question, but neither statement alone is sufficient
- D.
statements 1 and 2 together are not sufficient, and additional data is needed to answer the question
Attempted by 2 students.
Show answer & explanation
Correct answer: D
Concept: In a Data Sufficiency question, a statement (or combination of statements) is sufficient only if it lets you pin down ONE definite value for what is asked. Since the 10 numbers form a set, all 10 integers are distinct — so the probability that the selected integer equals a target value is 1/10 if that value is a member of the set, and 0 if it is not. So a statement is sufficient here only if it lets you decide, with certainty, whether the target value belongs to the set.
Application: First solve the equation in the question: ¼n + 13 = 0 → n = -52. So the real question is: “is -52 one of the 10 integers?” Now test each statement against that question.
Statement 1 alone (largest integer in the set is 13): this only bounds the set from above — every element is at most 13 — it says nothing about which values below 13 are actually chosen. -52 could be included or excluded while still keeping 13 as the largest value, so statement 1 alone cannot settle the question.
Statement 2 alone (arithmetic mean of the set is zero): this only fixes the sum of the 10 integers at zero. Countless different collections of 10 distinct integers sum to zero, some containing -52 and some not, so statement 2 alone cannot settle the question either.
Statements 1 and 2 together (largest = 13 and sum = 0): combining the two constraints still leaves many different valid sets of 10 distinct integers — the two conditions restrict the largest value and the total, but not which specific numbers below 13 make up the rest, so whether -52 is present remains undecided even with both facts.
Cross-check: build one valid set under both statements that includes -52 and one that excludes it, to confirm the ambiguity is real, not just a suspicion.
Includes -52: {13, 12, 11, 10, 9, 8, 7, -3, -15, -52} — ten distinct integers, largest is 13, sum = (13+12+11+10+9+8+7) + (-3-15-52) = 70 − 70 = 0. Both statements hold, and -52 IS a member.
Excludes -52: {13, 12, 11, 10, 9, 8, 7, 6, 5, -81} — ten distinct integers, largest is 13, sum = 81 − 81 = 0. Both statements hold, and -52 is NOT a member.
Since two sets satisfying both statements give two different answers to “is -52 in the set?”, the probability cannot be pinned down from the two statements even taken together — more data beyond what is given is required to answer the question.