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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 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.
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Re: A group of Republicans and Democrats was surveyed and each [#permalink]

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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]

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New post 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]

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New post 26 Feb 2013, 21:26
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]

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New post 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 :oops:
could u harp on the fault in my approach??
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Re: A group of Republicans and Democrats was surveyed and each [#permalink]

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New post 02 Mar 2013, 01:55
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 :oops:
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]

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New post 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 :oops:
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 :evil:
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Re: A group of Republicans and Democrats was surveyed and each [#permalink]

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New post 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]

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New post 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
--> Answer is E
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Re: A group of Republicans and Democrats was surveyed and each [#permalink]

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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
repub.jpg [ 42.15 KiB | Viewed 3718 times ]

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Re: A group of Republicans and Democrats was surveyed and each [#permalink]

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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?

A. 81
B. 88
C. 160
D. 550
E. 710


Little shortcut to this task

Republicans who like both products: \(80\%*20\% = 16\%\) and this number equal to Democrats who like eclairs \(=55\%\) so

\(0.16R=0.55D\) --> \(16R=55D\) --> \(\frac{16}{55}=\frac{D}{R}\)

After this part we don't need any further calculations because from this ratio we can infer that common number of Democrats + Republicans will be multiple of 71 (16+55=71)
and only answer E is multiple of 71.
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Re: A group of Republicans and Democrats was surveyed and each [#permalink]

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Harley1980 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?

A. 81
B. 88
C. 160
D. 550
E. 710


Little shortcut to this task

Republicans who like both products: \(80\%*20\% = 16\%\) and this number equal to Democrats who like eclairs \(=55\%\) so

\(0.16R=0.55D\) --> \(16R=55D\) --> \(\frac{16}{55}=\frac{D}{R}\)

After this part we don't need any further calculations because from this ratio we can infer that common number of Democrats + Republicans will be multiple of 71 (16+55=71)
and only answer E is multiple of 71.



Although saying that the total number of Democrats and Republicans must be divisible by 71 is not incorrect, the total number of Democrats and Republicans must be actually divisible by 355. This is because saying total number of Republicans will be divisible by 55 doesn't ensure that the Republicans who like apple pies and eclairs is an integer. Similarly, saying that the total number of Democrats will be divisible by 16 doesn't ensure that the Democrats who like eclairs is an integer. Since Republicans who like both products is 16% (80%* 20%) of total Republicans, this implies that the number of Republicans must also be a multiple of 25, for the Republicans who like both products to be an integer. Since number of Republicans is divisible by both 25 and 55, the total number of Republicans must be divisible by 275 (LCM of 25 and 55). Since Democrats who like eclairs is 55% of total Democrats, this implies that the number of Democrats must also be a multiple of 20, for the Democrats who like eclairs to be an integer. Since number of Democrats is divisible by both 16 and 20, the total number of Democrats must be divisible by 80 (LCM of 16 and 20). Summing up 275 and 80 yields 355. I just wanted to point this out because this can make a difference if the multiple choices were different.
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Re: A group of Republicans and Democrats was surveyed and each [#permalink]

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New post 24 Oct 2015, 07:53
I'm slightly confused. The question says of those 80 Republicans who like Apple Pies, 16 also like eclairs. But what about the 20 who like Eclairs, are there any who like Apple Pies? That could be a possibility right?
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dina98 wrote:
I'm slightly confused. The question says of those 80 Republicans who like Apple Pies, 16 also like eclairs. But what about the 20 who like Eclairs, are there any who like Apple Pies? That could be a possibility right?


Hi Dina98,

This question is very different from most overlapping questions you will probably encounter on the GMAT. This question will require either 2 separate venn diagrams or 2 separate matrices, one for Democrats and one for Republicans. Since the question is asking for one possible value for the number of democrats and number of republicans, you shouldn't pick your own numbers. You can let the number of Republicans be R and the number of Democrats be D. This question is actually testing divisibility because the number of democrats and republicans must each be integers.

What do you mean by the 20 who like Eclairs?
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Re: A group of Republicans and Democrats was surveyed and each [#permalink]

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Re: A group of Republicans and Democrats was surveyed and each [#permalink]

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New post 28 Nov 2016, 16:22
(80/100 * 20/100)R = (55/100)D

We can use this information to find the ratio of Democrats to Republicans:
D/R = 16/55
So D:R = 16:55
Dems are 16/71 and Republicans are 55/71 of the total population. From this we know the total population must be an integer so the answer must be a multiple of 71. 710 fits the bill pretty cleanly.
Re: A group of Republicans and Democrats was surveyed and each   [#permalink] 28 Nov 2016, 16:22
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