Question

On the last day of school, Mrs. Wonderful gave jelly beans to her class. She gave each boy as many jelly beans as there were boys in the class. She gave each girl as many jelly beans as there were girls in the class. She brought $400$ jelly beans, and when she finished, she had six jelly beans left. There were two more boys than girls in her class. How many students were in her class?

$26$

$28$

$30$

$32$

$34$

Solution:

Let $b$ be the number of boys in the class and $g$ be the number of girls. From the problem, we get that $b = g + 2.$

If each boy gets $b$ jelly beans, then Mrs. Wonderful will give out a total of $b^2$ jelly beans to all the boys. Similarly, she will give out $g^2$ jelly beans to all the girls.

Therefore, $\begin{gather*} b^2 + g^2 = 400 - 6 \\ (g + 2)^2 + g^2 = 394 \\ 2g^2 + 4g + 4 = 394 \\ g^2 + 2g - 195 = 0 \\ (g + 15)(g - 13) = 0. \end{gather*}$ Since $g$ cannot be negative, we get that $g = 13.$ This means that $b = 15,$ so $b + g = 28.$

Thus, B is the correct answer.

2009 AMC 8 Problem 23

Solution:

Problem 23 in Other Years