Problem Solving

A foreign language club at Washington Middle School consists of \(n\) students, \(\dfrac25\) of whom are boys. All of the students in the club study exactly one foreign language. \(\dfrac13\) of the girls in the club study Spanish and \(\dfrac56\) of the remaining girls study French. If the rest of the girls in the club study German, how many girls in the club, in terms of \(n,\) study German?

A. \(\dfrac{2n}5\)

B. \(\dfrac{n}3\)

C. \(\dfrac{n}5\)

D. \(\dfrac{2n}{15}\)

E. \(\dfrac{n}{15}\)

Answer: E

Source: Manhattan GMAT

$$Total\ number\ of\ \ students=n$$ $
$$Total\ number\ of\ boys\ =\ \frac{2}{5}\ n$$
$$Number\ of\ girls=total\ students-boys$$
$$=\frac{1}{1}n-\frac{2}{5}n=\frac{5n-2n}{5}=\frac{3n}{5}$$
$$Number\ of\ girls\ studying\ \operatorname{span}ish\ =\ \frac{1}{3}\cdot\frac{3n}{5}=\frac{1n}{5}$$
$$Number\ of\ girls\ studying\ french\ =\ \frac{5}{6}of\ remaining\ girls$$
$$Number\ of\ remaining\ girls=\ \frac{3n}{5}-\frac{1n}{5}=\frac{2n}{5}$$
$$Therefore,\ number\ of\ girls\ studying\ french=\ \frac{5}{6}\cdot\frac{2n}{5}=\frac{1n}{3}$$
$$if\ the\ remaining\ girls\ study\ german,\ how\ many\ girls\ study\ german$$
$$total\ number\ of\ girls\ =\ girls\ studying\ \operatorname{span}ish\ +\ french\ +\ german$$
$$\frac{3n}{5}=\frac{1n}{5}+\frac{1n}{3}+number\ of\ girls\ studying\ german$$
$$number\ of\ girls\ studying\ german=\frac{3n}{5}-\frac{1n}{5}-\frac{1n}{3}$$
$$=\frac{9n-3n-5n}{15}$$
$$=\frac{1n}{15}or\ \frac{n}{15}$$
$$Answer\ =\ E$$