2008 AIME I Problem 1

Question

Of the students attending a school party,

60\%

of the students are girls, and

40\%

of the students like to dance. After these students are joined by

20

more boy students, all of whom like to dance, the party is now

58\%

girls. How many students now at the party like to dance?

Answer

Let

x

be the original number of students; the number of girls does not change when the

20

boys arrive

Solve

0.6x = 0.58(x + 20),

then count

40\%

of the original students plus all

20

new boys

Let

x

be the number of students originally at the party, so

0.6x

are girls and

0.4x

like to dance. When the

20

boys arrive, the number of girls is unchanged but the total becomes

x + 20,

so

0.6x = 0.58(x + 20).

Then

0.02x = 11.6,

giving

x = 580.

The number of students who now like to dance is

0.4 \cdot 580 + 20 = 232 + 20 = 252.

Problem 1 in other years