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# Alice has 3 times the number of stamps t that Doris does an

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Alice has 3 times the number of stamps t that Doris does an  [#permalink]

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21 Oct 2012, 08:10
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Difficulty:

55% (hard)

Question Stats:

67% (02:14) correct 33% (02:24) wrong based on 183 sessions

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Alice has 3 times the number of stamps that Doris does and Jane has 7 stamps more than Doris does. If Alice, Doris, and Jane each increase their number of stamps by 4, which of the following must be true after each person increases her number of stamps?

Alice has more stamps than Jane.
Jane has 3 more stamps than Doris.
The sum of the numbers of stamps that Alice and Doris have is a multiple of 4.

A.None
B.III only
C.I and III only
D.II and III only
E.I, II, and III

Spoiler: :: OE
Before each person increased her number of stamps, let's let the number of stamps that Alice has be A, let's let the number of stamps that Doris has be D, and let's let the number of stamps that Jane has be J.

Since Alice has 3 times the number of stamps that Doris does, . Since Jane has 7 stamps more than Doris does, . We have more variables than equations, so these two equations cannot be solved for all the variables. We should keep this in mind while looking at the statements.

Let's consider statement I. We know that and . Let's see if it must be true that after each person increases her number of stamps by 4, Alice has more stamps than Jane. If , then , and . After each person increases her number of stamps by 4, Alice has stamps, Doris has stamps, and Jane has stamps. In this case, where Alice has 7 stamps and Jane has 12 stamps, it is not true that Alice has more stamps than Jane. Eliminate choices (C) and (E) which contain statement I.

Next, consider statement II. We saw in statement I that if before each person increases her number of stamps Doris has 1 stamp, then after each person increases her number of stamps by 4, Alice has 7 stamps, Doris has 5 stamps, and Jane has 12 stamps. So Jane has stamps more than Doris, and statement II, which says that Jane has 3 more stamps than Doris, is not true. Eliminate choice (D) which contains statement II.

We can see that since before each person increases her number of stamps by 4, Jane has 7 stamps more than Doris, after each of Jane and Doris increase her number of stamps by 4, Jane will still have 7 more stamps than Doris. Again, statement II is not true.

Now let's consider statement III. We know that and . If , then and . After each person increases her number of stamps by 4, Alice will have stamps and Doris will have stamps. After each person increases her number of stamps by 4, the total number of stamps that Alice and Doris will have is 7 + 5 = 12, which is a multiple of 4, since 12 = 3 4. If we let , then and . After each person increases her number of stamps by 4, Alice will have 6 + 4 = 10 stamps and Doris will have 2 + 4 = 6 stamps. After each person increases her number of stamps by 4, the total number of stamps that Alice and Doris will have is 10 + 6 = 16, which is a multiple of 4, since 16 = 4 4. If we select any other set of values for A, D, and J that are consistent with the question stem, we will find that the sum of the numbers of stamps that Alice and Doris have after each person increases her number of stamps by 4 is a multiple of 4. Statement III must be true. Choice (B), III only, is correct.

It can be shown algebraically that statement III must be true. Before each person increases her number of stamps by 4, Alice has 3D stamps and Doris has D stamps. After Alice increases her number of stamps by 4, she will have 3D + 4 stamps. After Doris increases her number of stamps by 4, she will have D + 4 stamps. The total number of stamps that Alice and Doris have after each person increases her number of stamps by 4 is . Since D is an integer, 4D is a multiple of 4. The integer 8 is a multiple of 4, since 8 = 2 4. Since 4D and 8 are both multiples of 4, 4D + 8 must be a multiple of 4 because the sum of two multiples of 4 must be a multiple of 4.
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Re: Alice has 3 times the number of stamps t that Doris does an  [#permalink]

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21 Oct 2012, 08:49
1
thevenus wrote:
Alice has 3 times the number of stamps that Doris does and Jane has 7 stamps more than Doris does. If Alice, Doris, and Jane each increase their number of stamps by 4, which of the following must be true after each person increases her number of stamps?

Alice has more stamps than Jane.
Jane has 3 more stamps than Doris.
The sum of the numbers of stamps that Alice and Doris have is a multiple of 4.

A.None
B.III only
C.I and III only
D.II and III only
E.I, II, and III

Let Alice, Doris and jane have A, D and J stamps initially.
We know that
A =3D
and
J=d+7

After addition of 4 stamps they'll have A+4, D+4, and J+4 stamps respectively.

Lets analyze each statements:
Alice has more stamps than Jane.
this can be written as
A+4 > J+4
=> 3D+4 > D+7+4
=>D >7/3
It may not be true for many values of D.. D could very well be equal to 1,2 etc for which D<7/3. Not correct.

Jane has 3 more stamps than Doris.
this can be written as
J+4 = (D+4)+3
=> D+7+4 = D+4+3
Clealy not correct for any value of D.

The sum of the numbers of stamps that Alice and Doris have is a multiple of 4.
(A+4)+(D+4) = 4k, where k is any integer
=> 3D+4+D+4 =4k
=> 4D+8 = 4K
=> 4(D+2) = 4K
or K = D+2 .. clearly possible for any value of D. Hence Correct.

Hence Ans B. Only iii is correct.

Hope it helps.
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Joined: 02 Sep 2009
Posts: 60646
Re: Alice has 3 times the number of stamps t that Doris does an  [#permalink]

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12 Jun 2013, 09:16
1
skamal7 wrote:
Bunnel,
I have a little confusion in understanding your explaination .The question asks after the increase in number by 4 by all people .So doris will be having a minimum for 4 stamps right?
So we know that D is >4 right? Then how come we can eliminate the first one? Please explain

Doris will have more than or equal 4 stamps AFTER the increase but D is the number of stamps BEFORE the increase.

Hope it's clear.
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Re: Alice has 3 times the number of stamps t that Doris does an  [#permalink]

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22 Oct 2012, 07:39
Alice has 3 times the number of stamps that Doris does and Jane has 7 stamps more than Doris does. If Alice, Doris, and Jane each increase their number of stamps by 4, which of the following must be true after each person increases her number of stamps?

Alice has more stamps than Jane.
Jane has 3 more stamps than Doris.
The sum of the numbers of stamps that Alice and Doris have is a multiple of 4.

A.None
B.III only
C.I and III only
D.II and III only
E.I, II, and III

Given that:
A=3D;
J=D+7;

After increase:
Alice: A=3D+4;
Jane: J=D+11;
Doris: D+4.

Analyze each option:
Alice has more stamps than Jane.
3D+4>D+11 --> D>7/3=3.5. Thus this statement will be true if D is more than or equal to 4. Since we don't know that, then this statement is not always true.

Jane has 3 more stamps than Doris.
D+11=(D+4)+3. This is not true for ANY value of D.

The sum of the numbers of stamps that Alice and Doris have is a multiple of 4.
The sum of the numbers of stamps that Alice and Doris have is (3D+4)+(D+4)=4D+8=4(D+2). So, this sum is a multiple of 4 for ANY value of D, which smeans that this statement is always true.

Hope it's clear.
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Re: Alice has 3 times the number of stamps t that Doris does an  [#permalink]

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12 Jun 2013, 07:46
Bunnel,
I have a little confusion in understanding your explaination .The question asks after the increase in number by 4 by all people .So doris will be having a minimum for 4 stamps right?
So we know that D is >4 right? Then how come we can eliminate the first one? Please explain
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Alice has 3 times the number of stamps t that Doris does an  [#permalink]

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12 Oct 2018, 22:01
KAPLAN OE :

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Everything will fall into place…

There is perfect timing for
everything and everyone.
Never doubt, But Work on
improving yourself,
Keep the faith and
It will all make sense.
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Re: Alice has 3 times the number of stamps t that Doris does an  [#permalink]

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28 Nov 2019, 02:47
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Re: Alice has 3 times the number of stamps t that Doris does an   [#permalink] 28 Nov 2019, 02:47
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