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Emma got a certain amount as pocket money from her father. S

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Emma got a certain amount as pocket money from her father. She went to a book store and bought a book for herself by paying $4 more than half her pocket money. She then went to a shop and bought a pencil by paying $ 4 more than one-third the amount left with her. After this, she bought some chocolates for her younger sister by paying $ 4 more than one-fifth the amount left with her. She could have bought twice the number of chocolates for herself with the amount left with her, but she preferred to save the money. How much more did she pay for the book than for the pencil?

A. $49
B. $48
C. $36
D. $38
E. $37
[Reveal] Spoiler: OA

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Last edited by Bunuel on 01 Dec 2013, 06:06, edited 1 time in total.
Renamed the topic and edited the question.

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Re: Emma got a certain amount as pocket money from her father. S [#permalink]

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madhavsrinivas wrote:
Emma got a certain amount as pocket money from her father. She went to a book store and bought a book for herself by paying $4 more than half her pocket money. She then went to a shop and bought a pencil by paying $ 4 more than one-third the amount left with her. After this, she bought some chocolates for her younger sister by paying $ 4 more than one-fifth the amount left with her. She could have bought twice the number of chocolates for herself with the amount left with her, but she preferred to save the money. How much more did she pay for the book than for the pencil?

A. $49
B. $48
C. $36
D. $38
E. $37

Let x be the money Emma had at the time she went into the chocolate shop. We have: \(2 (\frac{x}{5} +4) = \frac{4x}{5} -5\), i.e. \(x=30\).
Similarly, The money she had before buying pencil was \(\frac{3}{2} (30+4)=51\); and the money in her pocket at the beginning was \(\frac{2}{1}(51+4)= 110\). She paid $59 for the book and $21 for the pencil. The answer should be D ($38)

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Re: Emma got a certain amount as pocket money from her father. S [#permalink]

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New post 28 Feb 2014, 14:43
Is there any elegant approach to solve this one? Its really tough if you go step by step!

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Re: Emma got a certain amount as pocket money from her father. S [#permalink]

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New post 05 Mar 2014, 23:50
tuanle wrote:
madhavsrinivas wrote:
Emma got a certain amount as pocket money from her father. She went to a book store and bought a book for herself by paying $4 more than half her pocket money. She then went to a shop and bought a pencil by paying $ 4 more than one-third the amount left with her. After this, she bought some chocolates for her younger sister by paying $ 4 more than one-fifth the amount left with her. She could have bought twice the number of chocolates for herself with the amount left with her, but she preferred to save the money. How much more did she pay for the book than for the pencil?

A. $49
B. $48
C. $36
D. $38
E. $37

Let x be the money Emma had at the time she went into the chocolate shop. We have: \(2 (\frac{x}{5} +4) = \frac{4x}{5} -5\), i.e. \(x=30\).
Similarly, The money she had before buying pencil was \(\frac{3}{2} (30+4)=51\); and the money in her pocket at the beginning was \(\frac{2}{1}(51+4)= 110\). She paid $59 for the book and $21 for the pencil. The answer should be D ($38)



Can you please elaborate as to how the first equation was set up??

I took initial amount of x & started calculating per conditions given, only to end up large variable fractions :(
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Re: Emma got a certain amount as pocket money from her father. S [#permalink]

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PareshGmat wrote:
tuanle wrote:
madhavsrinivas wrote:
Emma got a certain amount as pocket money from her father. She went to a book store and bought a book for herself by paying $4 more than half her pocket money. She then went to a shop and bought a pencil by paying $ 4 more than one-third the amount left with her. After this, she bought some chocolates for her younger sister by paying $ 4 more than one-fifth the amount left with her. She could have bought twice the number of chocolates for herself with the amount left with her, but she preferred to save the money. How much more did she pay for the book than for the pencil?

A. $49
B. $48
C. $36
D. $38
E. $37

Let x be the money Emma had at the time she went into the chocolate shop. We have: \(2 (\frac{x}{5} +4) = \frac{4x}{5} -5\), i.e. \(x=30\).
Similarly, The money she had before buying pencil was \(\frac{3}{2} (30+4)=51\); and the money in her pocket at the beginning was \(\frac{2}{1}(51+4)= 110\). She paid $59 for the book and $21 for the pencil. The answer should be D ($38)



Can you please elaborate as to how the first equation was set up??

I took initial amount of x & started calculating per conditions given, only to end up large variable fractions :(


It should be \(2 (\frac{x}{5} +4) = \frac{4x}{5} -4\), instead of \(2 (\frac{x}{5} +4) = \frac{4x}{5} -5\).

x there is the amount of money Emma had at the time she went into the chocolate shop.

She bought some chocolates for her younger sister by paying $ 4 more than one-fifth the amount left with her --> the price of the chocolates = \(\frac{x}{5} +4\) --> the he amount of money left = \(x-(\frac{x}{5} +4)=\frac{4x}{5} -4\).

She could have bought twice the number of chocolates for herself with the amount left with her --> \(2 (\frac{x}{5} +4) = \frac{4x}{5} -4\) --> \(x=30\).

Hope it's clear.
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Re: Emma got a certain amount as pocket money from her father. S [#permalink]

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New post 08 Mar 2014, 00:53
madhavsrinivas wrote:
Emma got a certain amount as pocket money from her father. She went to a book store and bought a book for herself by paying $4 more than half her pocket money. She then went to a shop and bought a pencil by paying $ 4 more than one-third the amount left with her. After this, she bought some chocolates for her younger sister by paying $ 4 more than one-fifth the amount left with her. She could have bought twice the number of chocolates for herself with the amount left with her, but she preferred to save the money. How much more did she pay for the book than for the pencil?
A. $49
B. $48
C. $36
D. $38
E. $37


I also want to know a better way to do this than to do it via brute force.

Anyway, this is how I did it.
M = initial amount Emma got as pocket money
B = price of the book
P = price of the pen
y = the amount Emma had after she bought the Book
C = price of the chocolate
x = the amount Emma had after she bought the pen

\(B = \frac{(M+8)}{2}+4\)

\(y = M - B = M - \frac{(M+8)}{2} + 4\) (I replaced it with y because it is too long for me)

\(P = \frac{y}{3} + 4\)

Now, Emma has x dollars.

\(C = \frac{x}{5}+4\)

With the money she has now, she can buy two chocolate, so

\(x - C = 2C\)

\(x - \frac{x}{5}+4 = \frac{2x}{5}+8\)

\(x = 30\)

\(C = 10\)

Going back to y,

\(y = P + x = P + 30\)

\(y = \frac{y}{3} + 4 + 30\)

\(y = 51\)

\(51 = M - \frac{(M+8)}{2} + 4\)

\(M = 110\)

\(B = \frac{(110+8)}{2}+4\)

\(B = 59\)

\(P = \frac{51}{3}+4 = 21\)

The difference of B and P is 38, thus (D).

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Re: Emma got a certain amount as pocket money from her father. S [#permalink]

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New post 21 Jun 2014, 09:03
cssmarimo wrote:
madhavsrinivas wrote:
Emma got a certain amount as pocket money from her father. She went to a book store and bought a book for herself by paying $4 more than half her pocket money. She then went to a shop and bought a pencil by paying $ 4 more than one-third the amount left with her. After this, she bought some chocolates for her younger sister by paying $ 4 more than one-fifth the amount left with her. She could have bought twice the number of chocolates for herself with the amount left with her, but she preferred to save the money. How much more did she pay for the book than for the pencil?
A. $49
B. $48
C. $36
D. $38
E. $37


I also want to know a better way to do this than to do it via brute force.

Anyway, this is how I did it.
M = initial amount Emma got as pocket money
B = price of the book
P = price of the pen
y = the amount Emma had after she bought the Book
C = price of the chocolate
x = the amount Emma had after she bought the pen

\(B = \frac{(M+8)}{2}+4\)

\(y = M - B = M - \frac{(M+8)}{2} + 4\) (I replaced it with y because it is too long for me)

\(P = \frac{y}{3} + 4\)

Now, Emma has x dollars.

\(C = \frac{x}{5}+4\)

With the money she has now, she can buy two chocolate, so

\(x - C = 2C\)

\(x - \frac{x}{5}+4 = \frac{2x}{5}+8\)

\(x = 30\)

\(C = 10\)

Going back to y,

\(y = P + x = P + 30\)

\(y = \frac{y}{3} + 4 + 30\)

\(y = 51\)

\(51 = M - \frac{(M+8)}{2} + 4\)

\(M = 110\)

\(B = \frac{(110+8)}{2}+4\)

\(B = 59\)

\(P = \frac{51}{3}+4 = 21\)

The difference of B and P is 38, thus (D).





Hi,
Can you please tell how did you arrive at the first equation:
\(B = \frac{(M+8)}{2}+4\)

I thought it should be B = (M/2) + 4. What am I missing?

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Re: Emma got a certain amount as pocket money from her father. S [#permalink]

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New post 21 Jun 2014, 11:22
faamir wrote:
cssmarimo wrote:
madhavsrinivas wrote:
Emma got a certain amount as pocket money from her father. She went to a book store and bought a book for herself by paying $4 more than half her pocket money. She then went to a shop and bought a pencil by paying $ 4 more than one-third the amount left with her. After this, she bought some chocolates for her younger sister by paying $ 4 more than one-fifth the amount left with her. She could have bought twice the number of chocolates for herself with the amount left with her, but she preferred to save the money. How much more did she pay for the book than for the pencil?
A. $49
B. $48
C. $36
D. $38
E. $37


I also want to know a better way to do this than to do it via brute force.

Anyway, this is how I did it.
M = initial amount Emma got as pocket money
B = price of the book
P = price of the pen
y = the amount Emma had after she bought the Book
C = price of the chocolate
x = the amount Emma had after she bought the pen

\(B = \frac{(M+8)}{2}+4\)

\(y = M - B = M - \frac{(M+8)}{2} + 4\) (I replaced it with y because it is too long for me)

\(P = \frac{y}{3} + 4\)

Now, Emma has x dollars.

\(C = \frac{x}{5}+4\)

With the money she has now, she can buy two chocolate, so

\(x - C = 2C\)

\(x - \frac{x}{5}+4 = \frac{2x}{5}+8\)

\(x = 30\)

\(C = 10\)

Going back to y,

\(y = P + x = P + 30\)

\(y = \frac{y}{3} + 4 + 30\)

\(y = 51\)

\(51 = M - \frac{(M+8)}{2} + 4\)

\(M = 110\)

\(B = \frac{(110+8)}{2}+4\)

\(B = 59\)

\(P = \frac{51}{3}+4 = 21\)

The difference of B and P is 38, thus (D).





Hi,
Can you please tell how did you arrive at the first equation:
\(B = \frac{(M+8)}{2}+4\)

I thought it should be B = (M/2) + 4. What am I missing?


You are right. Refer to other solutions above.
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Re: Emma got a certain amount as pocket money from her father. S [#permalink]

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madhavsrinivas wrote:
Emma got a certain amount as pocket money from her father. She went to a book store and bought a book for herself by paying $4 more than half her pocket money. She then went to a shop and bought a pencil by paying $ 4 more than one-third the amount left with her. After this, she bought some chocolates for her younger sister by paying $ 4 more than one-fifth the amount left with her. She could have bought twice the number of chocolates for herself with the amount left with her, but she preferred to save the money. How much more did she pay for the book than for the pencil?

A. $49
B. $48
C. $36
D. $38
E. $37



We can let m = the amount of money Emma has left after she bought the book and pencil. Using the information in the problem, she spent (1/5)m + 4 dollars on the chocolate she bought for her sister. So she would have m – [(1/5)m + 4] = (4/5)m – 4 dollars left. With this money, she could have bought twice the amount of the chocolate she bought for her sister. Thus:

(4/5)m – 4 = 2[(1/5)m + 4]
(4/5)m – 4 = (2/5)m + 8
4m – 20 = 2m + 40
2m = 60
m = 30

Emma has 30 dollars left after she bought the book and pencil. We can also let x = the Emma’s original amount of money.

She spent (1/2)x + 4 dollars on the book and had x - [(1/2)x + 4] = (1/2)x – 4 dollars leftover.

She spent [(1/3)[(1/2)x - 4] + 4 = (1/6)x - 4/3 + 12/3 = (1/6)x + 8/3 dollars on the pencil.

Thus, the amount of money she has left now would be:

(1/2)x - 4 - [(1/6)x + 8/3] = (3/6)x - 12/3 - (1/6)x - 8/3 = (1/3)x - 20/3 dollars.

Since we know the amount left over is 30 dollars, we can create the following equation to determine x:

(1/3)x - 20/3 = 30
x – 20 = 90
x = 110

Thus, she originally had 110 dollars and spent (1/2)(110) + 4 = 59 dollars on the book and (1/6)(110) + 8/3 = 55/3 + 8/3 = 63/3 = 21 dollars on the pencil.

The book costs 59 - 21 = 38 dollars more than the pencil.

Answer: D
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Re: Emma got a certain amount as pocket money from her father. S [#permalink]

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New post 16 Feb 2017, 21:00
1) Is a question like this realistic on the GMAT?
2) Is there an easier or faster method to solve this question? Lots of math is required with lots of room for silly error along the way. Also, doing this out would certainly take more than 2 minutes?

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New post 16 Feb 2017, 22:44
I find this question more quantitative rather than more logical.
I think the difficulty level is not high but quant involved is too much. Is there a shorter way or i should make random guess to get a more time for other question

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Re: Emma got a certain amount as pocket money from her father. S [#permalink]

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madhavsrinivas wrote:
Emma got a certain amount as pocket money from her father. She went to a book store and bought a book for herself by paying $4 more than half her pocket money. She then went to a shop and bought a pencil by paying $ 4 more than one-third the amount left with her. After this, she bought some chocolates for her younger sister by paying $ 4 more than one-fifth the amount left with her. She could have bought twice the number of chocolates for herself with the amount left with her, but she preferred to save the money. How much more did she pay for the book than for the pencil?

A. $49
B. $48
C. $36
D. $38
E. $37



This question is lengthy rather than tough and the language of the question might confuse you. It can be solved by taking x as the value of pocket money or to solve it faster we can use reverse solving approach.

Reverse solving approach can come into practice only if we understand the problem quickly and then solve a lots of such problems.

So. Lets 1st solve by:
Lengthy process :
Let X be the amount of pocket money received by Emma.

Money used to buy Book = 4 + X/2
Money Left = X/2 -4

Money used to buy Pencil = 4 + 1/3*(X/2 -4 )
Money Left = 2/3*(X/2 -4 ) - 4

Money used to buy Chocolate = 4 + 1/5*(2/3*(X/2 -4 ) - 4)
Money Left = [4/5* (2/3*(X/2 -4 ) - 4)-4]

Now money left can buy twice the number of chocolates.
So, [4/5* (2/3*(X/2 -4 ) - 4)-4] = 2 * [ 4 + 1/5*(2/3*(X/2 -4 ) - 4)]
-> 12 = 2/5*(2/3*(X/2 -4 ) - 4)
->30+4 = 2/3*(X/2 -4 )
-> 55= X/2
-> X = 110

So, Money used to buy Book = 4 + X/2 = 59

Money used to buy Pencil = 4 + 1/3*(X/2 -4 ) = 21

Difference = 38

Answer D

Let shorten this process by a bit.

Medium Lengthy process or Shorter Approach
To solve this process, first check what fractions are used. So here, 1/3rd, 1/5th and 1/2 is used to purchase the items. So we will LCM of denominators of fraction to assume the value of pocket money received by Emma.
LCM of 3,5,2 = 30


Let 30X be the amount of pocket money received by Emma.

Money used to buy Book = 4 + 15X
Money Left = 15X -4

Money used to buy Pencil = 4 + 1/3*(15X-4 ) = 5X +8/3
Money Left = 2/3*(15X -4 ) - 4 = 10X -20/3

Money used to buy Chocolate = 4 + 1/5*(10X-20/3) = 2X + 4 -4/3 = 2X + 8/3
Money Left = [4/5* (10X-20/3)-4] = 8X-16/3 - 4 = 4X-28/3

Now money left can buy twice the number of chocolates.
So, [8X-28/3] = 2 * [2X + 8/3]
-> 4X = 28/3 + 16/3 = 44/3
-> X = 11/3
-> 30X = 110

Pocket Money received = 110

So, Money used to buy Book = 4 + 15X = 59

Money used to buy Pencil = 5X +8/3 = 55/3 +8/3 = 21

Difference = 38

Answer D

Lets try to solve it by relatively more shorter approach by solving the problem in reverse order.

Shortest Approach
Let the amount left after buying pencil be X
So, Money used to buy chocolate = X/5 + 4
Money left after buying chocolate = 4X/5 - 4

Money left after buying chocolate = 2 x Money used to buy chocolate
-> 4X/5 - 4 = 2 (X/5 + 4)
-> 2X/5 = 12
-> X = 30

So, Money left after buying Book = 3/2*(30+4) = 51
Money used to buy Pencil = 4 + 1/3 *51 = 21

Money received by Emma = 2*(51+4) = 110
Money used to buy book = 4+1/2*110 = 59

So, Difference = 59 - 21 = 38

Answer D

So, we can solve by these three methods.
Longer method will take approx 5 minutes.
Shorter Method will take approx 3.5 minutes.
SHortest method will take approx 2 minutes. But it need a lots of practice to start with this approach.

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Re: Emma got a certain amount as pocket money from her father. S [#permalink]

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New post 02 Aug 2017, 03:56
Jayeshvekariya wrote:
I find this question more quantitative rather than more logical.
I think the difficulty level is not high but quant involved is too much. Is there a shorter way or i should make random guess to get a more time for other question

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Jayeshvekariya, jlgdr
You can refer my solution in the penultimate post.

PareshGmat , I have elaborated all the solutions in the penultimate post.

mbah191, 1. Such question may be asked in GMAT as 700 level question. There might be 2 steps only instead of 3 steps in this question which makes it really tough. But with practice the time may be reduced easily.
2. Refer my solution to solve it within 2 minutes. Yes there is a chance of lots of silly mistakes in this problem.
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New post 08 Oct 2017, 05:43
Formation of equations using the given information itself takes so much time; I highly doubt that majority of students would be able to solve this sum within 2 minutes.

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Re: Emma got a certain amount as pocket money from her father. S   [#permalink] 08 Oct 2017, 05:43
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