37½% of the candidates in an examination were girls. 75% of the boys and 62½%…

Concept

When a subgroup forms a known fraction of the total, its complement gives the fraction represented by the other subgroup.

A failure count equals subgroup size multiplied by the failure rate; therefore, subgroup size equals failure count divided by the failure rate.

Application

1. Girls are 37½% = 3/8 of all candidates. Since 62½% = 5/8 of the girls passed, the fraction of girls who failed is 1 − 5/8 = 3/8.

2. Thus, (3/8) × number of girls = 342, so number of girls = 342 × (8/3) = 912.

3. Because 912 is 3/8 of all candidates, total candidates = 912 × (8/3) = 2,432.

4. Boys form the remaining 5/8 of the total, so number of boys = (5/8) × 2,432 = 1,520.

5. Since 75% of the boys passed, 25% = 1/4 failed. Hence boys who failed = (1/4) × 1,520 = 380.

Cross-check

• For the girls, (3/8) × 912 = 342, which reproduces the given failure count.

• For the boys, 1,140 passed and 380 failed, and 1,140 + 380 = 1,520.

Therefore, the number of boys who failed was 380.