In a class of 60 students, girls are twice as many as boys. A, who is a boy,…

CONCEPT

In rank problems, a person ranked rth from the top has r − 1 people ahead and N − r people behind. When the group is split into categories, first find each category total and then partition it into people ahead, the ranked person when applicable, and people behind.

A ratio such as girls : boys = 2 : 1 divides the class into three equal ratio-parts. Category counts and rank-side counts must be kept separate.

APPLICATION

1. Let the number of boys be b. Then the number of girls is 2b, so b + 2b = 60 and 3b = 60.

2. Therefore b = 20 boys and there are 40 girls.

3. A is seventeenth from the top, so 17 − 1 = 16 students are ahead of him.

4. Of those 16 students, 9 are girls; hence 16 − 9 = 7 boys are ahead of A.

5. Because A is also a boy, the boys are partitioned as 7 ahead + 1 for A + boys after A. Thus the boys after A are 20 − 7 − 1 = 12.

CROSS-CHECK

1. A has 60 − 17 = 43 students after him.

2. Because A is a boy, all 40 girls consist of 9 ahead and 40 − 9 = 31 after.

3. Therefore the students after A contain 43 − 31 = 12 boys.

Therefore, 12 boys ranked after A.