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

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by M7MBA » Thu Sep 17, 2020 12:55 am

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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

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Source: Beat The GMAT — Problem Solving | Thu Sep 17, 2020 12:55 am

deloitte247: Legendary Member; Posts: 2214; Joined: Fri Mar 02, 2018 2:22 pmFollowed by: 5 members

Re: A foreign language club at Washington Middle School consists of $n$ students, $\dfrac25$ of whom are boys. All o

by deloitte247 » Sun Sep 20, 2020 12:53 pm

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

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Scott@TargetTestPrep: GMAT Instructor; Posts: 8155; Joined: Sat Apr 25, 2015 10:56 amFollowed by: 29 members; Location: Los Angeles, CA; Thanked: 43 times

Re: A foreign language club at Washington Middle School consists of $n$ students, $\dfrac25$ of whom are boys. All o

by Scott@TargetTestPrep » Thu Sep 24, 2020 7:06 am

M7MBA wrote: ↑
Thu Sep 17, 2020 12:55 am
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

Solution:

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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