1986 AJHSME Problems/Problem 25

Problem

Which of the following sets of whole numbers has the largest average?

Solution

Solution 1

From to there are (see floor function) multiples of , and their average is

$\begin{align*} \frac{2\cdot 1+2\cdot 2+2\cdot 3+\cdots + 2\cdot 50}{50} &= \frac{2(1+2+3+\cdots +50)}{50} \\ &= \frac{2\cdot \frac{50\cdot 51}{2}}{50} \\ &= \frac{2\cdot 51}{2} \\ &= 51 \\ \end{align*}$ (Error compiling LaTeX. Unknown error_msg)

Similarly, we can find that the average of the multiples of between and is , the average of the multiples of is , the average of the multiples of is , and the average of the multiples of is , so the one with the largest average is

Solution 2

You can just add the first term and the last term, and see which one is the biggest. 2+100=102, 3+99=102, 4+100=104, 5+100=105, 6+96=102. Therefore, the answer is multiples of five because it has the highest number.