120 children are participating in a summer camp and all of them must

Question

Tough and Tricky questions: Overlapping Sets.

120 children are participating in a summer camp and all of them must choose at least one of the French or Spanish language workshops in which to participate. 32 of the children have selected to participate in both workshops. If 24 children participate exclusively in the French workshop, what percent of all the children exclusively participate in the Spanish workshop?

A. 40%
B. 48%
C. 50%
D. 53%
E. 64%

Kudos for a correct  solution .

omegacube · Answer

n(A Union B) = n(A - B) + n(B - A) + n(A intersection B)
Implies 120 = 24 + n(B - A) + 32
Thus n(B - A) = 64

Percentage = 64/120% = 53%

Hence D

g106 · Answer

Total children participating = 120 = Exclusively French + Both + exclusively Spanish
 Exclusively Spanish = 120 - (24 + 32)
 = 64
 Percentage = (64/120) * 100 = 53

ANS D

qw1981 · Answer

F= French F.N= Not French Same represenation for spanish.

F F.N

S 32 64=120-32-56

S.N 24 0 (as one language is mandatory)

Now 64/120 %= ? Note that we know 60 is half of 120 hence % would be just above 50. Only answer that meets this is D.

PareshGmat · Answer

Answer = D. 53% Refer diagram below: over.png \frac{64}{120} * 100 = 53%

Bunuel · Answer

OFFICIAL SOLUTION:

The best way to solve these kinds of problems is by creating a Venn diagram. Start by drawing two circles that overlap such that they share some fraction of their areas. The boundaries of the circles will encompass the number of students that participate in each language workshop. The section where they overlap represents students that take both languages. You can arbitrarily label the left lobe with an F for French and the Right lobe with an S for Spanish. The sum of the populations in each lobe must be equal to the sum of the children, or 120.

Since there are 24 children exclusively taking French the F lobe should have the number 24 in it. Since there are 32 children taking both languages, the middle lobe should have a 32 in it. We will solve for the number of children taking only Spanish, so label the S lobe with an x. The sum of the lobes, as we mentioned must equal 120, so we can solve for x by solving the equation: 24 + 32 + x = 120, or x = 64. So, there are exactly 64 children exclusively participating in the Spanish workshop.

Finally, to determine what percent of the student population the 64 exclusive Spanish taking students constitute, we set up the ratio: part/whole = 64/120 = 8/15, or 53 percent.

veergmat · Answer

A U B = nA+nB- n(A intersection B)

OR = n(A-B) + n(B-A) + n(A intersection B)

120 = n(S-F)+24+32
n(S-F) = 64

64/120*100 = 53.3 % ANSWER D is closest

msrivas2 · Answer

Hi Bunuel,

Can we solve this problem using formula : Total = S1 +S2 - Both +Neither?
Please explain if we can.

siv · Answer

Yes we can. Let, S1 be all French including both French + Spanish and S2 be all Spanish including French and Spanish.
Hence, S2= 32 + 24.
120 (total) = S1 + ( 32 + 24 ) - 32 (both) + 0
Therefore, S1 = 96.

Now, Exclusive French will be 96-32 = 64.

53% of 120 will be approx. 64.

Hope it helps.

bumpbot · Answer

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