Question

Consider the following statements about four numbers:
(S1) The average of the four numbers is 25
(S2) Each number is at most 40
(S3) Each number is at least 20
Choose the option that is necessarily correct.

• A. (S1) and (S2) together imply (S3)
• B. (S2) and (S3) together imply (S1)
• C. (S1) and (S3) together imply (S2)
• D. (S1) implies (S3)

Hint:

The sum of the four numbers is fixed at 100 by S1; work out the largest a single number can be while the other three stay at their minimum allowed by S3.

Answer 1

The Correct Option is C

Step 1: Understanding the Question.
We have four numbers whose average is 25, so their sum is 100. We are given three statements about bounds on the individual numbers and need to find which implication among the options must always hold.

Step 2: Key Formula or Approach.
To check whether "X and Y imply Z" is necessarily true, try to construct a counter-example that satisfies X and Y but violates Z. If no such counter-example can exist, the implication is necessarily true. Here, S1 fixes the sum of the four numbers at 100 (since \(4 \times 25 = 100\)).

Step 3: Detailed Explanation.
Check option (C): does S1 (sum = 100) and S3 (each number \(\geq 20\)) together force S2 (each number \(\leq 40\))?
Suppose one of the numbers, say \(a\), is made as large as possible while the other three, \(b, c, d\), are each kept at their minimum allowed value of 20 (since S3 requires each number to be at least 20). Then
\[ a + b + c + d = 100 \] \[ a + 20 + 20 + 20 = 100 \] \[ a = 40 \] So the largest that any single number can be, while still keeping the other three numbers at least 20 and the total sum at 100, is exactly 40. It is not possible to push any number above 40 without pushing at least one of the other numbers below 20 (which S3 forbids). So S1 and S3 together force every number to be at most 40, which is exactly S2. Option (C) is necessarily correct.
Now check why the other options fail:
Option (A): S1 (sum = 100) and S2 (each \(\leq 40\)) do not force S3 (each \(\geq 20\)). For example, the numbers \(0, 40, 40, 20\) sum to 100 and are all at most 40, so S1 and S2 hold, but the first number is 0, which is less than 20, so S3 fails. This is a valid counter-example, so option (A) is false.
Option (B): S2 (each \(\leq 40\)) and S3 (each \(\geq 20\)) do not force S1 (average = 25). For example, \(20, 20, 20, 20\) satisfies S2 and S3, but its average is 20, not 25, so S1 fails. Option (B) is false.
Option (D): S1 alone (average = 25) does not force S3 (each \(\geq 20\)). For example, \(0, 30, 30, 40\) sums to 100 and gives an average of 25, but the first number is 0, which is below 20, so S3 fails. Option (D) is false.

Step 4: Final Answer.
S1 and S3 together necessarily imply S2.
\[ \boxed{\text{Option (C)}} \]