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

Step 1: Set up the numbers.
Let the four numbers be $a, b, c, d$ with sum $a+b+c+d = 100$ from (S1), since the average is 25.
(S2) says each number is at most 40, and (S3) says each number is at least 20.

Step 2: Test (S1) and (S3) together.
If $a, b, c, d$ are all at least 20, then any three of them sum to at least 60.
Since all four sum to exactly 100, the fourth number is at most $100 - 60 = 40$, so every number is automatically at most 40, which is (S2).

Step 3: Rule out the other combinations.
(S1) with (S2) fails, since 40, 40, 40, and minus 20 gives sum 100 with every number at most 40, yet one number is below 20.
(S2) with (S3) only bounds each number between 20 and 40, so the sum could range from 80 to 160, and the average need not be 25.
(S1) alone does not force (S3), since 10, 10, 10, 70 averages 25 but has a number below 20.

Final Answer:
Only (S1) and (S3) together always force (S2).\[ \boxed{\text{(S1) and (S3) together imply (S2)}} \]