$37 \frac{1}{2} \%$ of the candidates in an examination were girls, $75 \%$ of the boys and $62 \frac{1}{2} \%$ of the girls passed and 342 girls failed. The number of boys failed was:
(a)350
(b)360
(c)370
(d)380
Answer
Answer (as printed): D
Explanation
Let the total number of candidates be $x$. Then, $\left(100-62 \frac{1}{2}\right) \%$ of $37 \frac{1}{2} \%$ of $x=342$ $$\begin{aligned} & \Leftrightarrow \frac{75}{2} \times \frac{1}{100} \times \frac{75}{2} \times \frac{1}{100} \times x=342 \\ & \Leftrightarrow \frac{9 x}{64}=342 \Leftrightarrow x=\left(\frac{342 \times 64}{9}\right)=2432 . \end{aligned}$$ Number of boys failed $=(100-75) \%$ of $\left(100-37 \frac{1}{2}\right) \%$ of $2432=\left(\frac{25}{100} \times \frac{125}{2} \times \frac{1}{100} \times 2432\right)=380$.
Explanation as extracted from the printed page; notation may be imperfect.