2015 AIME II Problem 2

Question

In a new school

40

percent of the students are freshmen,

30

percent are sophomores,

20

percent are juniors, and

10

percent are seniors. All freshmen are required to take Latin, and

80

percent of the sophomores,

50

percent of the juniors, and

20

percent of the seniors elect to take Latin. The probability that a randomly chosen Latin student is a sophomore is

\frac{m}{n},

where

m

and

n

are relatively prime positive integers. Find

m + n.

Answer

Suppose the school has

100

students and count how many students in each class take Latin

The Latin students number

40 + 24 + 10 + 2,

and the answer comes from the sophomores’ share of that total

Assume the school has

100

students. The Latin students are then

40

freshmen,

30(0.8) = 24

sophomores,

20(0.5) = 10

juniors, and

10(0.2) = 2

seniors, for a total of

76.

The probability that a random Latin student is a sophomore is

\frac{24}{76} = \frac{6}{19},

so

m + n = 6 + 19 = 25.

Problem 2 in other years