Question

The Amaco Middle School bookstore sells pencils costing a whole number of cents. Some seventh graders each bought a pencil, paying a total of $1.43$ dollars. Some of the $30$ sixth graders each bought a pencil, and they paid a total of $1.95$ dollars. How many more sixth graders than seventh graders bought a pencil?

$1$

$2$

$3$

$4$

$5$

Solution:

The number of seventh graders that bought a pencil is $143$ divided by the price of a pencil. Similarly, the number of sixth graders that bought a pencil is $195$ divided by the price of a pencil.

This means that the price of a pencil divides both $143$ and $195.$ Prime factorizing, we get $143 = 11 \cdot 13$ and $195 = 3 \cdot 5 \cdot 13.$ The only numbers that divide both $143$ and $195$ are $1$ and $13.$

If $1$ cent was the price of the pencil, that means $195$ sixth graders bought pencils, which is not possible. Therefore, the price of a pencil is $13$ cents.

This means that $143 \div 13 = 11$ seventh graders bought a pencil, and $195 \div 13 = 15$ sixth graders bought a pencil. Therefore, $4$ more sixth graders than seventh graders bought pencils.

Thus, D is the correct answer.

2009 AMC 8 Problem 11

Solution:

Problem 11 in Other Years