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:

Check whether the other two numbers force a bound on the fourth number.

Answer 1

The Correct Option is C

The average condition fixes the total sum of the four numbers at 100, and the question asks which pair of the remaining statements always forces the third one.

1. (S1) and (S2) imply (S3): false, because 40, 40, 40, and minus 20 add up to 100 and stay at or below 40, yet the last value breaks the at least 20 rule.
2. (S2) and (S3) imply (S1): false, since any four values between 20 and 40 give a sum anywhere from 80 to 160, so the average is not pinned to 25.
3. (S1) and (S3) imply (S2): true, since if the other three numbers are each at least 20, they already total at least 60, leaving at most 40 for the remaining one out of a fixed sum of 100.
4. (S1) implies (S3) alone: false, since 10, 10, 10, and 70 average to 25 without every value reaching 20.

Only the third combination holds in every case, so "(S1) and (S3) together imply (S2)" is the correct choice.