Every student at Darcy School is in at least one of three clubs: horse

Question

Every student at Darcy School is in at least one of three clubs: horseback riding, embroidery, and country dancing, which are the only clubs in existence at the school. The ratio of the number of students in exactly two clubs to the number of students in exactly one club is 4:3, while the ratio of the number of students in exactly two clubs to the number of students in at least two clubs is 5:7. Which of the following could be the total number of students at Darcy School?

A. 63
B. 69
C. 74
D. 82
E. 86

Kudos for a correct solution.

A. 63
B. 69
C. 74
D. 82
E. 86

Bunuel · Accepted Answer

MANHATTAN GMAT OFFICIAL SOLUTION: Although the setup of this problem seems to call for a Venn Diagram with three circles, this question is actually solely about ratios. We need to figure out the ratio of students who are in only one club to the ratio of students in exactly two clubs to the ratio of students in all three clubs. One : Two : Three Clubs = ? We can use the unknown multiplier to help us represent the two ratios. The ratio of the number of students in exactly two clubs to the number of students in exactly one club is 4:3. https://gmatclub.com/forum/download/file.php?id=27877 And the ratio of the number of students in exactly two clubs to the number of students in at least two clubs is 5:7. Make sure to use a different variable for this ratio. Also, note that this ratio is students in 2 clubs to students in at least two clubs. That means that there are really only 2 students in three clubs for every 5 people in two clubs. https://gmatclub.com/forum/download/file.php?id=27878 Now we have two different ways of expressing the number of students in exactly two clubs. We know that the number of students in two clubs is a multiple of both 4 and 5. Therefore, the number of students in two clubs must be a multiple of 20. Multiply the first ratio by 5 and the second ratio by 4. https://gmatclub.com/forum/download/file.php?id=27879 We can now combine the two ratios into one ratio, which will be 15z : 20z : 8z. The number of students in the school must be a multiple of 15 + 20 + 8 = 43. The only answer that is a multiple of 43 is 86. The correct answer is E. Screen-shot-2011-10-08-at-1.16.41-PM.png Screen-shot-2011-10-08-at-1.15.56-PM.png Screen-shot-2011-10-08-at-1.14.59-PM.png

GMATinsight · Answer

Look at the explanation in image attached

Answer: option E

vb27 · Answer

First ratio is 4/3, second ratio is 5/7. The 4 and 5 represent exactly two clubs. By simplifying, that is multiplying the first by 5 and the second by 4, we get the ratio of the 3 as 15:20:28 which adds up to 63.

So the answer is A?

So the answer is A?

GMATinsight · Answer

You are right till the calculation of 15:20:28

but please understand that 20 (the number of students in exactly two clubs) is subset of 28 (Number of students in at least two clubs) only. SO you don't have to add them.

So total will be only sum of 15 and 28 = 43

I hope this helps!

Aves · Answer

This is a good question to test the concept of fractions and ratios.

It took me a while to solve this but once you know the trick, it is quite easy.

From the question, we know that  \frac{Exactly 2 Clubs}{Exactly 1 Club}= \frac{4}{3}  and  \frac{Exactly 2 Clubs}{At least 2 Clubs}   = \frac{5}{7}

The number of students in exactly 2 clubs is our link between 2 information here, so we have to make them equal by multiplying  \frac{4}{3}  by  \frac{5}{5}  and  \frac{5}{7}  by  \frac{4}{4}

You will get the ratios of Exactly 1 Club : Exactly 2 Clubs : At least 2 Clubs (or Exactly 2 clubs + Exactly 3 clubs)  = 15 : 20 : 28

You now know that the ratio of students in exactly 2 clubs is  20 , therefore the ratio of the students in exactly 3 clubs is  8 .

You will get the final ratio of Exactly 1 Club : Exactly 2 Clubs : Exactly 3 clubs  = 15 : 20 : 8

So, total student must be multiple of  15+20+8 = 43  and the only answer that is multiple of  43  is  86

mvictor · Answer

I used venn diagram to get to the correct answer choice...
suppose A=all 3
B, C, D - only in 2
E, F, G - in 1 only

we are told:
(B+C+D)/(F+G+E)=4/3
F+G+E = 3(B+C+D)/4

and we know that:
(B+C+D)/(A+B+C+D)=5/7
or that A=2(B+C+D)/5

now..
all together is A+B+C+D+E+F+G
2(B+C+D)/5 + B+C+D+3(B+C+D)/4
we have 2 fractions..we can add those two to get 23(B+C+D)/20
so B+C+D must be a factor of 20.
suppose B+C+D=20
then A=40/5 = 8
and E+G+F=15
or total = 43.
since we do not have 43 in our answer choices, we need to find an answer that is a multiple of 43.
so E.

sahilvijay · Answer

Answer is Clearly E

see attached image .

hellosanthosh2k2 · Answer

Let number of students in exactly one club = x
Let number of students in exactly two clubs = y
Let number of students in exactly three clubs = d
so number of students in atleast two clubs = y + d
Given :  y/x = 4/3  =>  x = (3/4)y 
  y/(y+d) = 5/7 
 =>  2y = 5d  =>  d = 2y/5 
 =>  y + d  =  y + 2y/5  =  7y/5

Total : x + y + d =  (3/4)y + 7y/5 = (43/20)y

=> y multiple of 20, if y = 20, total = 43 (no options)
=> y = 40, total = 86 => (E)

hadimadi · Answer

Hi,

no big calculations needed, since we can compare the two ratios because both have a common comparator, namely the number of students in exactly two classes. We we have 4:3, 5:7, and 4,5 respectively represent the people in exactly two classes for both, expanding to 20 we get.

28:20:15

Total Students=Students in exactly 1 class + students in at least 2 -> 28+15=43 -> M43 is the answer -> (E)

Regor60 · Answer

a= one class
b= two classes
c= at least two classes
d= three classes

So, enrollment is equal to

a+b+d

From the problem statement:

b/a=4/3 so b=(4/3)a

b/c=b/(b+d)=5/7, so

5(b+d)=7b, so d=b(2/5)=

(2/5)(4/3)a= a(8/15)

So a+b+d = a+(4/3)a +(8/15)a

which equals 43a, meaning the total has to be a multiple of 43.

The only answer that fits is 86.

Posted from my mobile device

Purnank · Answer

Its very easy actually, 2 Things are given that are sufficient to calculate Total number of students.
Assuming E1, E2 and E3 being the count of students in exactly 1, 2 and 3 clubs.
1st - The ratio of the number of students in exactly two clubs to the number of students in exactly one club is 4:3
 \frac{E2}{E1} = \frac{4}{3} 
2nd - the ratio of the number of students in exactly two clubs to the number of students in at least two clubs is 5:7
 \frac{E2}{E2+E3} = \frac{5}{7} 
 \frac{E2}{E3} = \frac{5}{2} 
Now from both of the fractions of E1/E2 and E2/E3
we get,
E1 : E2 : E3 = 15 : 20 : 8 
means E1 + E2 + E3 = 43*Constant
Means the Total count should be multiple of 43 and from given options we can see that E fits perfectly.
  Answer is Option E.

bumpbot · Answer

Automated notice from GMAT Club BumpBot: 

A member just gave Kudos to this thread, showing it’s still useful. I’ve bumped it to the top so more people can benefit. Feel free to add your own questions or solutions.

 This post was generated automatically.

Every student at Darcy School is in at least one of three clubs: horse

Prep Toolkit

Top 5 GMAT Debrief Videos

615 to 715 in 15 Days (Ria's Strategies)

More than Quant/Verbal Abilities: Speed vs. Accuracy

745 with Only Self-Prep and GMAT Club

From 555 to 765: 210-point Improvement

Perfect 805 Score Debrief by Julia

My Rewards

GMAT Problem Solving (PS) Questions

Every student at Darcy School is in at least one of three clubs: horse

Prep Toolkit

615 to 715 in 15 Days (Ria's Strategies)

More than Quant/Verbal Abilities: Speed vs. Accuracy

745 with Only Self-Prep and GMAT Club

From 555 to 765: 210-point Improvement

Perfect 805 Score Debrief by Julia

My Rewards