The larger of two real numbers is 7 times the sum of 5 and the smaller number. The sum of the real numbers is at most 60. What are the largest two real numbers that can make these statements true?

Question

The larger of two real numbers is 7 times the sum of 5 and the smaller number. The sum of the

real numbers is at most 60. What are the largest two real numbers that can make these statements

true?

Answer 1

Let a,b be two real numbers. Without loss of generality, we can assume a ≥ b. Hence, a = 7(b+5) = 7b+35 and a+b ≤ 60. Now

60 ≥ a+b = 7b+35+b = 8b+35 ⇒ 25≥8b ⇒ 25/8 ≥ b

So a = 7b+35 ≤ 7(25/8)+35 = 455/8. Thus. we have b ≤ 25/8 and a ≤ 455/8. Since we want the largest possible numbers that make the statements true, take a = 455/8 and b = 25/8.