A foreign language club at Washington Middle School consists of \(n\) students, \(\dfrac25\) of whom are boys. All of

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