Manhattan Challenge Problem

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knight247: Legendary Member; Posts: 504; Joined: Tue Apr 19, 2011 1:40 pmFollowed by: 11 members; Thanked: 114 times

by knight247 » Mon Oct 03, 2011 5:24 am

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

Last edited by knight247 on Mon Oct 03, 2011 5:51 am, edited 1 time in total.

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Source: Beat The GMAT — Problem Solving | Mon Oct 03, 2011 5:24 am

sl750: Master | Next Rank: 500 Posts; Posts: 496; Joined: Tue Jun 07, 2011 5:34 amFollowed by: 1 member; Thanked: 38 times

by sl750 » Mon Oct 03, 2011 5:48 am

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

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GmatMathPro: GMAT Instructor; Posts: 349; Joined: Wed Sep 28, 2011 3:38 pmFollowed by: 54 members; Location: Austin, TX; Thanked: 236 times; GMAT Score:770

by GmatMathPro » Mon Oct 03, 2011 6:56 am

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

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ankur1986: Newbie | Next Rank: 10 Posts; Posts: 1; Joined: Thu Oct 06, 2011 9:01 am

by ankur1986 » Thu Oct 06, 2011 9:07 am

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

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GMATGuruNY: GMAT Instructor; Posts: 15539; Joined: Tue May 25, 2010 12:04 pmFollowed by: 1906 members; Location: New York, NY; Thanked: 13060 times; GMAT Score:790

by GMATGuruNY » Thu Oct 06, 2011 10:11 am

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.

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abhishek karumuri: Newbie | Next Rank: 10 Posts; Posts: 7; Joined: Wed Dec 22, 2010 3:55 am

by abhishek karumuri » Fri Oct 07, 2011 6:16 am

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.

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GMATGuruNY: GMAT Instructor; Posts: 15539; Joined: Tue May 25, 2010 12:04 pmFollowed by: 1906 members; Location: New York, NY; Thanked: 13060 times; GMAT Score:790

by GMATGuruNY » Fri Oct 07, 2011 6:55 am

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.

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abhi0697: Senior | Next Rank: 100 Posts; Posts: 57; Joined: Tue Apr 19, 2011 7:39 pmFollowed by: 2 members; Location: Singapore; Thanked: 6 times

by abhi0697 » Sat Oct 08, 2011 7:57 pm

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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parul9: Master | Next Rank: 500 Posts; Posts: 197; Joined: Thu Sep 15, 2011 10:22 amFollowed by: 2 members; Thanked: 6 times

by parul9 » Sat Oct 08, 2011 8:29 pm

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

Only E fits the bill.

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factor26: Senior | Next Rank: 100 Posts; Posts: 99; Joined: Sun Mar 13, 2011 12:42 pm

by factor26 » Sat Oct 08, 2011 8:58 pm

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

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