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# A group of Republicans and Democrats was surveyed and each

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A group of Republicans and Democrats was surveyed and each [#permalink]  25 Feb 2013, 01:14
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A group of Republicans and Democrats was surveyed and each member was asked whether they liked apple pies or éclairs. 80% of the Republicans liked apple pie, 55% of the Democrats liked eclairs, and 20% of the Republicans who liked apple pie also liked eclairs. If the number of Republicans who liked both desserts is equal to the number of Democrats who liked éclairs, what is one possible value for the number of members of the group?

A. 81
B. 88
C. 160
D. 550
E. 710
[Reveal] Spoiler: OA

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Last edited by Bunuel on 26 Feb 2013, 01:32, edited 1 time in total.
Edited the question.
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Re: A group of Republicans and Democrats was surveyed and each [#permalink]  25 Feb 2013, 05:10
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Expert's post
emmak wrote:
A group of Republicans and Democrats was surveyed and each member was asked whether they liked apple pies or éclairs. 80% of the Republicans liked apple pie, 55% of the Democrats liked eclairs, and 20% of the Republicans who liked apple pie also liked eclairs. If the number of Republicans who liked both desserts is equal to the number of Democrats who liked éclairs, what is one possible value for the number of members of the group?

81

88

160

550

710

Let the number of republicans be 100x and that of the democrats be 100y. Now, we are given that 80x like apple pies. Again, 16x like both apple pie and eclairs amongst the republicans.

Also, 55y like eclairs amongst the democrats.

Also given that 16x = 55y

Thus, the total number of people in the group =$$100x+100y = 100(x+y) = 100*y*(\frac{55}{16}+1) = 100*y*(\frac{71}{16})$$

Now, the number of people has to be an integer.Also, as 71 and 16 are co-primes, thus 100y has to be a multiple of 16. Thus, 100y = 16k, where k is a non negative integer.

Thus, the total number of people = 100(x+y) = $$16k*\frac{71}{16}$$ = 71k. From the given options, for k=10, option E matches.

E.
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Re: A group of Republicans and Democrats was surveyed and each [#permalink]  26 Feb 2013, 10:55
vinaymimani wrote:
emmak wrote:
A group of Republicans and Democrats was surveyed and each member was asked whether they liked apple pies or éclairs. 80% of the Republicans liked apple pie, 55% of the Democrats liked eclairs, and 20% of the Republicans who liked apple pie also liked eclairs. If the number of Republicans who liked both desserts is equal to the number of Democrats who liked éclairs, what is one possible value for the number of members of the group?

81

88

160

550

710

Let the number of republicans be 100x and that of the democrats be 100y. Now, we are given that 80x like apple pies. Again, 16x like both apple pie and eclairs amongst the republicans.

Also, 55y like eclairs amongst the democrats.

Also given that 16x = 55y

Thus, the total number of people in the group =$$100x+100y = 100(x+y) = 100*y*(\frac{55}{16}+1) = 100*y*(\frac{71}{16})$$

Now, the number of people has to be an integer.Also, as 71 and 16 are co-primes, thus 100y has to be a multiple of 16. Thus, 100y = 16k, where k is a non negative integer.

Thus, the total number of people = 100(x+y) = $$16k*\frac{71}{16}$$ = 71k. From the given options, for k=10, option E matches.

E.

The question stated 20% of the Republicans who liked apple pie also liked eclairs, not 16% as you did. I think the question has a wrong number because the way you did is right.
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Re: A group of Republicans and Democrats was surveyed and each [#permalink]  26 Feb 2013, 21:26
Expert's post
nnk12391 wrote:
vinaymimani wrote:
emmak wrote:
A group of Republicans and Democrats was surveyed and each member was asked whether they liked apple pies or éclairs. 80% of the Republicans liked apple pie, 55% of the Democrats liked eclairs, and 20% of the Republicans who liked apple pie also liked eclairs. If the number of Republicans who liked both desserts is equal to the number of Democrats who liked éclairs, what is one possible value for the number of members of the group?

81

88

160

550

710

The question stated 20% of the Republicans who liked apple pie also liked eclairs, not 16% as you did. I think the question has a wrong number because the way you did is right.

Suppose 100 republicans are there in total. 80% like apple pie. Thus 80 republicans include all those who like ONLY apple pie & those who like BOTH apple pie and eclairs. Now, 20% of those who like apple pie, like eclairs also. Thus, out of 80 people, 16 like both. You can see that 16 is 20% of 80. Maybe the confusion is from the fact that 80 people already include the people who like both and that number is again manifested as a percentage.
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Re: A group of Republicans and Democrats was surveyed and each [#permalink]  02 Mar 2013, 01:38
vinaymimani wrote:
emmak wrote:
A group of Republicans and Democrats was surveyed and each member was asked whether they liked apple pies or éclairs. 80% of the Republicans liked apple pie, 55% of the Democrats liked eclairs, and 20% of the Republicans who liked apple pie also liked eclairs. If the number of Republicans who liked both desserts is equal to the number of Democrats who liked éclairs, what is one possible value for the number of members of the group?

81

88

160

550

710

Let the number of republicans be 100x and that of the democrats be 100y. Now, we are given that 80x like apple pies. Again, 16x like both apple pie and eclairs amongst the republicans.

Also, 55y like eclairs amongst the democrats.

Also given that 16x = 55y

Thus, the total number of people in the group =$$100x+100y = 100(x+y) = 100*y*(\frac{55}{16}+1) = 100*y*(\frac{71}{16})$$

Now, the number of people has to be an integer.Also, as 71 and 16 are co-primes, thus 100y has to be a multiple of 16. Thus, 100y = 16k, where k is a non negative integer.

Thus, the total number of people = 100(x+y) = $$16k*\frac{71}{16}$$ = 71k. From the given options, for k=10, option E matches.

E.

My approach was as follows:
0.16 R = 0.55 D (initial method same as yours where R= Republicans and D=Democrats)
$$\frac{4R}{25}$$ = $$\frac{11D}{10}$$
R can be = 25*11 and D = 25*4
which means R= 275 and D= 100
Alas the answer comes out to be 375 and none of the options match
could u harp on the fault in my approach??
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Kudos [?]: 713 [0], given: 135

Re: A group of Republicans and Democrats was surveyed and each [#permalink]  02 Mar 2013, 01:55
Expert's post
Quote:

My approach was as follows:
0.16 R = 0.55 D (initial method same as yours where R= Republicans and D=Democrats)
$$\frac{4R}{25}$$ = $$\frac{11D}{10}$$
R can be = 25*11 and D = 25*4
which means R= 275 and D= 100
Alas the answer comes out to be 375 and none of the options match
could u harp on the fault in my approach??

0.16 R = 0.55 D

or 16/100 R = 55/100 D

or 4/25 R = 11/20 D

Further, when you take R = 25*11, you get D as 80 . This adds upto 275+80 = 355. As yo can see that this is 710/2 , thus you can choose R = 25*11*2. This will give D as 160. The sum is 550+160 = 710.

Thanks.
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Re: A group of Republicans and Democrats was surveyed and each [#permalink]  02 Mar 2013, 02:22
vinaymimani wrote:
Quote:

My approach was as follows:
0.16 R = 0.55 D (initial method same as yours where R= Republicans and D=Democrats)
$$\frac{4R}{25}$$ = $$\frac{11D}{10}$$
R can be = 25*11 and D = 25*4
which means R= 275 and D= 100
Alas the answer comes out to be 375 and none of the options match
could u harp on the fault in my approach??

0.16 R = 0.55 D

or 16/100 R = 55/100 D

or 4/25 R = 11/20 D

Further, when you take R = 25*11, you get D as 80 . This adds upto 275+80 = 355. As yo can see that this is 710/2 , thus you can choose R = 25*11*2. This will give D as 160. The sum is 550+160 = 710.

Thanks.

Thanx....Silly mistake YET AGAIN
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Re: A group of Republicans and Democrats was surveyed and each [#permalink]  02 Mar 2013, 04:28
I gave up half way through till 100Y (55/16 + 1) :assumed I was going the wrong way and lost it....damn it pays to complete our thought process!

Just to check quickly, though sum does not mention, are we not taking into considerations people who liked neither?
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Re: A group of Republicans and Democrats was surveyed and each [#permalink]  02 Mar 2013, 13:05
80% R --> like a
55% D --> like e
20% R --> like a + e
80% x 20% R = 55% D --> No. of R like a + e = No. of D like e
=> R = 55/16 D
However, R + D should be one of the above answers => R + D = 71/16 D = B (the answers) B must divisible by 15/4, D is an integer and positive
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Re: A group of Republicans and Democrats was surveyed and each [#permalink]  15 Aug 2013, 12:10
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As per given data 16% of the republican comes into category of both who likes eclairs and apple pie, this number is equal to 55% of democrats who likes eclairs.
Following diagram is explanatory.

Attachment:

repub.jpg [ 42.15 KiB | Viewed 880 times ]

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Re: A group of Republicans and Democrats was surveyed and each   [#permalink] 15 Aug 2013, 12:10
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