06/05/2020 (2011: National Sprint Round, Problem 15)

Q: Seven numbers form an arithmetic sequence. The median of the seven numbers is five. The mean of the four largest numbers is seven more than the mean of the three smallest numbers. What is the ratio of the smallest number to the largest number? Express your answer as a common fraction.

A: - 1/11

So, let's say that the middle number is a, and the common difference is b. That would mean the sequence is: a - 3b, a - 2b, a - b, a, a + b, a + 2b, a + 3b.

We know that the median of the numbers is 5, so that means that a = 5, so now the sequence is:

5 - 3b, 5 - 2b, 5 - b, 5, 5 + b, 5 + 2b, 5 + 3b

We also know that the mean of the four largest numbers is seven more than the mean of the three smallest numbers. The sum of the four largest numbers is 20 + 6b and the sum of the three smallest numbers is 15 - 6b. That means the average of the four largest numbers is (20 + 6b) ÷ 4 = 5 + 1.5b, and that the average of the three smallest numbers is (15 - 6b) ÷ 3 = 5 - 2b.

We subtract the averages we got above from each other and get the difference between them to be 3.5b.

We know that the difference is also equal to 7, so we get: 7 = 3.5b, which simplifies to: 2 = b.

Because a = 5 and b = 2, and the smallest and largest numbers are a - 3b and a + 3b, respectively, that means the smallest number is 5 - 3(2) and the largest number is 5 + 3(2). Those simplify to -1 and 11 respectively. We find -1 ÷ 11 = - 1/11, so that's our answer.