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

Answer: 252

Concepts:percentage linear equation

Difficulty rating: 1750

Solution:

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

2008 AIME I Problem 1

Answer: 252

Difficulty rating: 1750

Solution:

Problem 1 in Other Years