2008 AMC 8 Problem 20

Question

The students in Mr. Neatkin’s class took a penmanship test. Two-thirds of the boys and

\frac{3}{4}

of the girls passed the test, and an equal number of boys and girls passed the test. What is the minimum possible number of students in the class?

\ 12

\ 17

\ 24

\ 27

\ 36

Let the common number of passing boys and passing girls be

p

.

Then total boys and girls are

\frac{3p}{2}

and

\frac{4p}{3}

, so choose the smallest

p

that makes both whole.

Let

p

be the common number of boys and girls who passed.

Since

\dfrac{2}{3}

of the boys passed, the number of boys is

\dfrac{3p}{2}

. Since

\dfrac{3}{4}

of the girls passed, the number of girls is

\dfrac{4p}{3}

.

The smallest positive

p

that makes both counts whole is

6

. Then there are

9

boys and

8

girls, for a minimum total of

17

students.

Thus, B is the correct answer.

Problem 20 in other years