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
A foreign language club at Washington Middle School consists of \(n\) students, \(\dfrac25\) of whom are boys. All of
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$$Total\ number\ of\ \ students=n$$ $
$$Total\ number\ of\ boys\ =\ \frac{2}{5}\ n$$
$$Number\ of\ girls=total\ studentsboys$$
$$=\frac{1}{1}n\frac{2}{5}n=\frac{5n2n}{5}=\frac{3n}{5}$$
$$Number\ of\ girls\ studying\ \operatorname{span}ish\ =\ \frac{1}{3}\cdot\frac{3n}{5}=\frac{1n}{5}$$
$$Number\ of\ girls\ studying\ french\ =\ \frac{5}{6}of\ remaining\ girls$$
$$Number\ of\ remaining\ girls=\ \frac{3n}{5}\frac{1n}{5}=\frac{2n}{5}$$
$$Therefore,\ number\ of\ girls\ studying\ french=\ \frac{5}{6}\cdot\frac{2n}{5}=\frac{1n}{3}$$
$$if\ the\ remaining\ girls\ study\ german,\ how\ many\ girls\ study\ german$$
$$total\ number\ of\ girls\ =\ girls\ studying\ \operatorname{span}ish\ +\ french\ +\ german$$
$$\frac{3n}{5}=\frac{1n}{5}+\frac{1n}{3}+number\ of\ girls\ studying\ german$$
$$number\ of\ girls\ studying\ german=\frac{3n}{5}\frac{1n}{5}\frac{1n}{3}$$
$$=\frac{9n3n5n}{15}$$
$$=\frac{1n}{15}or\ \frac{n}{15}$$
$$Answer\ =\ E$$
$$Total\ number\ of\ boys\ =\ \frac{2}{5}\ n$$
$$Number\ of\ girls=total\ studentsboys$$
$$=\frac{1}{1}n\frac{2}{5}n=\frac{5n2n}{5}=\frac{3n}{5}$$
$$Number\ of\ girls\ studying\ \operatorname{span}ish\ =\ \frac{1}{3}\cdot\frac{3n}{5}=\frac{1n}{5}$$
$$Number\ of\ girls\ studying\ french\ =\ \frac{5}{6}of\ remaining\ girls$$
$$Number\ of\ remaining\ girls=\ \frac{3n}{5}\frac{1n}{5}=\frac{2n}{5}$$
$$Therefore,\ number\ of\ girls\ studying\ french=\ \frac{5}{6}\cdot\frac{2n}{5}=\frac{1n}{3}$$
$$if\ the\ remaining\ girls\ study\ german,\ how\ many\ girls\ study\ german$$
$$total\ number\ of\ girls\ =\ girls\ studying\ \operatorname{span}ish\ +\ french\ +\ german$$
$$\frac{3n}{5}=\frac{1n}{5}+\frac{1n}{3}+number\ of\ girls\ studying\ german$$
$$number\ of\ girls\ studying\ german=\frac{3n}{5}\frac{1n}{5}\frac{1n}{3}$$
$$=\frac{9n3n5n}{15}$$
$$=\frac{1n}{15}or\ \frac{n}{15}$$
$$Answer\ =\ E$$
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Solution:M7MBA wrote: ↑Thu Sep 17, 2020 12:55 amA 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
We can let n be 60 (notice that it’s a multiple of the denominators of 5, 3 and 6). So, 60 x 2/5 = 24 are boys and 60  24 = 36 are girls. Furthermore, 36 x 1/3 = 12 girls study Spanish and (36  12) x 5/6 = 24 x 5/6 = 20 girls study French. Therefore, 36  12  20 = 4 girls study German. Since n = 60, 4/60 = 1/15 of the students study German.
Answer: E
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