Question

One day the Beverage Barn sold $252$ cans of soda to $100$ customers, and every customer bought at least one can of soda. What is the maximum possible median number of cans of soda bought per customer on that day?

$2.5$

$3.0$

$3.5$

$4.0$

$4.5$

Solution:

Order the $100$ purchases from least to greatest. The median is the average of the $50$ th and $51$ st entries. To maximize it, make the first $49$ entries as small as possible, so set them all equal to $1$ .

If the $50$ th entry were at least $4$ , then the total would be at least $49+51\cdot4=253$ , too many cans. So the $50$ th entry is at most $3$ .

This maximum is attainable: use $49$ entries equal to $1$ , the $50$ th entry equal to $3$ , and the final $50$ entries equal to $4$ . The total is $49+3+50\cdot4=252$ , and the median is $(3+4)/2=3.5$ .

Thus, C is the correct answer.

2014 AMC 8 Problem 24

Solution:

Problem 24 in Other Years