Problem Solving

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

Detailed explanations would be appreciated

OA is E

Exactly one club = A
Exactly two clubs = B
Atleast two clubs (2 or 3 clubs) = C

A/B = 4/3 ; B/C = 5/7. Multiply,divide ratio 1 by 5 and ratio 2 by 4

A/B = 20/15 ; B/C = 20/28

Number of students in 3 clubs = 28x-20x = 8x

Total students 15x+8x = Total. This should be a multiple of 23x. Only 69 works

A=exactly one club
B=exactly two clubs
C=exactly three clubs
T=total number of students

T=A+B+C

B/A=4/3 B/(B+C)=5/7

Cross multiply:
3B=4A and 7B=5(B+C)
A=3B/4 and C=2B/5

Plugging in to the equation:

3B/4+B+2B/5=T

Multiply by 20 to eliminate fractions:

15B+20B+8B=20T

43B=20T

B=20T/43

B is an integer, so T must be a multiple of 43. Only one that works is E

Answer is E

Students in Exactly one club : E1
Students in Exactly two club : E2
Students in Exactly three club : E3

Students in Atleast two club : E2 + E3

From condition 1 => E2/E1 = 4/3
From condition 2 => E2/(E2 + E3 ) = 5/7 => E2/E3 = 5/2

So E1 : E2 : E3 = 15: 20 : 8
also Total students = E1 + E2 + E3

so Total students = 43x

knight247 wrote: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

Detailed explanations would be appreciated

OA is E

To combine ratios with a common element, the common element must be represented by the same value in each ratio.

Exactly 2 : Exactly 1 = 4:3 = 20:15.
Exactly 2 : At least 2 = 5:7 = 20:28.

Combining the ratios:
Exactly 1 : Exactly 2 : At least 2 = 15:20:28.

If there are 28 students in AT LEAST 2 clubs and 20 students in EXACTLY 2 clubs, then the number of students in ALL 3 clubs = 28-20 = 8.
Thus:
Exactly 1 : Exactly 2 : All 3 = 15:20:8.

Since 15+20+8 = 43, the total number of students must be a multiple of 43.

The correct answer is E.

Mitch,

How is this equation made?
If there are 28 students in AT LEAST 2 clubs and 20 students in EXACTLY 2 clubs, then the number of students in ALL 3 clubs = 28-20 = 8.

abhishek karumuri wrote:Mitch,

How is this equation made?
If there are 28 students in AT LEAST 2 clubs and 20 students in EXACTLY 2 clubs, then the number of students in ALL 3 clubs = 28-20 = 8.

AT LEAST 2 clubs includes those in EXACTLY 2 and those in ALL 3:

Thus:
At least 2 = (exactly 2) + (all 3)
28 = 20 + (all 3)
(All 3) = 28-20 = 8.

knight247 wrote: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

Detailed explanations would be appreciated

OA is E

Kindly check out the below link for video solution:

https://www.youtube.com/watch?v=mj8x_IV5Kzw

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the total students will be 43n, n being any positive integer.

Only E fits the bill.

I think the biggest point to get out of the question is that included in the at least 2 # there is an exactly 2 #. Look for the multiple of 43 ... E is the only choice that does so. Choose E