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

madhavsrinivas

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Updated on: 01 Dec 2013, 06:06

Originally posted by madhavsrinivas on 30 Nov 2013, 07:30.
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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Difficulty:

95% (hard)

Question Stats:

51% (03:21) correct

49% (03:16) wrong

based on 127 sessions

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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

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tuanle

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14 Dec 2013, 04:31

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madhavsrinivas

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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28 Feb 2014, 14:43

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Is there any elegant approach to solve this one? Its really tough if you go step by step!

Cheers
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05 Mar 2014, 23:50

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tuanle

madhavsrinivas

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

Bunuel

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PareshGmat

tuanle

madhavsrinivas

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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madhavsrinivas

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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21 Jun 2014, 09:03

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cssmarimo

madhavsrinivas

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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faamir

cssmarimo

madhavsrinivas

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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madhavsrinivas

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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16 Feb 2017, 21:00

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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?

Thanks.
-Max

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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

Sent from my MotoG3 using GMAT Club Forum mobile app

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02 Aug 2017, 03:51

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madhavsrinivas

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.

Don't forget to appreciate if you like my solution. Thanks

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Jayeshvekariya

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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08 Oct 2017, 05:43

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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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PareshGmat

tuanle

madhavsrinivas

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

Can I ask how you got 3/2 and 2/1? Is it just the one third and a half being reversed?

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chick3nnugg3t

Can I ask how you got 3/2 and 2/1? Is it just the one third and a half being reversed?

Let me know if the following post helps:

https://gmatclub.com/forum/emma-got-a-c ... l#p1340521

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madhavsrinivas

Alternate approach:

Let x = the pocket money
Since $4 more than than 1/2 the pocket money is spent on the book, the amount spent on the book = $4 + \frac{1}{2}x$
Amount remaining = $x - (\frac{1}{2}x+4) = \frac{1}{2}x - 4$
Since $4 more than 1/3 the remaining amount is spent on the pencil, the amount spent on the pencil = $4 + \frac{1}{3}(\frac{1}{2}x-4) = 4 + \frac{1}{6}x - \frac{4}{3}$

Let A = the correct answer choice.
Since the difference betwen the two prices must be equal to the correct answer, we get:
$(4 + \frac{1}{2}x) - (4 + \frac{1}{6}x - \frac{4}{3}) = A$
$\frac{1}{3}x + \frac{4}{3} = A$
$x + 4 = 3A$
$x = 3A - 4$
Implication:
The pocket money is equal to 4 less than 3 times the correct answer choice.

We can PLUG IN THE ANSWERS, which represent the value of A.
Since the amount spent on the book = $4+\frac{1}{2}x$ , the correct option for A must be EVEN, so that the value of x is even.
Eliminate A and E.
When the correct answer is plugged in, the amount remaining for Emma to buy choicolates must be twice the amount spent to buy chocolates for the sister.

D: x = 3*38 - 4 = 110
Book = 4 more than 1/2 of 110 = 4 + 55 = 59
Remaining amount = 110-59 = 51
Pencil = 4 more than 1/3 of 51 = 4 + 17 = 21
Remaining amount = 51-21 = 30
Chocolates for the sister = 4 + 1/5 of 30 = 4 + 6 = 10
Amount remaining for Emma to buy chocolates = 30-10 = 20
Success!

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