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.
Show 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.
Step 1: Set up the fixed sum.
Four numbers have an average of 25 under statement (S1), so their total is fixed at 100. Statements (S2) and (S3) put an upper and a lower bound of 40 and 20 respectively on each individual number. Test each option with a counter-example approach.
Step 2: Try to break "(S1) and (S2) imply (S3)".
Take the four numbers \(0, 40, 40, 20\). They sum to 100, matching S1, and none exceeds 40, matching S2. But 0 is less than 20, so S3 fails here. Since S1 and S2 can hold while S3 fails, this option is not necessarily true.
Step 3: Try to break "(S2) and (S3) imply (S1)".
Take \(20, 20, 20, 20\). Every number is between 20 and 40, so S2 and S3 both hold, but the average is 20, not 25, so S1 fails. This option is not necessarily true either.
Step 4: Try to break "(S1) and (S3) imply (S2)".
With the sum fixed at 100 and every number required to be at least 20, the largest any single number can become is found by pushing the other three down to their floor of 20 each: \(20+20+20=60\), leaving \(100-60=40\) for the fourth number. So no number can ever exceed 40 while S1 and S3 both hold, which is exactly what S2 requires. No counter-example is possible, so this option is necessarily true.
Step 5: Try to break "(S1) implies (S3)".
Take \(0, 30, 30, 40\). This sums to 100, so the average is 25 and S1 holds, but the first number is 0, which is below 20, so S3 fails. This option is not necessarily true.
Final Answer:
Only the pair "(S1) and (S3) together imply (S2)" is necessarily correct.
Was this answer helpful?
0
Top Questions on Analytical Aptitude - Seating Arrangement
