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# A man purchased some pens, pencils and erasers. Can the number of eras

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Math Expert
Joined: 02 Sep 2009
Posts: 50042
A man purchased some pens, pencils and erasers. Can the number of eras  [#permalink]

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05 Jul 2017, 00:33
00:00

Difficulty:

75% (hard)

Question Stats:

58% (01:47) correct 42% (01:38) wrong based on 66 sessions

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A man purchased some pens, pencils and erasers. Can the number of erasers purchased be 3?

(1) The ratio of the number of pens to the number of pencils was the same as the ratio of the number of pencils to the number of erasers.
(2) Total number of items purchased was 21.

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A man purchased some pens, pencils and erasers. Can the number of eras  [#permalink]

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Updated on: 27 Sep 2017, 21:21
1
Bunuel wrote:
A man purchased some pens, pencils and erasers. Can the number of erasers purchased be 3?

(1) The ratio of the number of pens to the number of pencils was the same as the ratio of the number of pencils to the number of erasers.
(2) Total number of items purchased was 21.

(1) gives us

Pen : Pencil : Eraser
if the ratio is

3:3:3 then we can say Yes

but if the ratio is
1:3:9 then its No
Not sufficient

(2) Total is 21

Now
Pen : Pencil : Eraser
x:y:z
x+y+z = 21
Not sufficient

On combining
x/y=y/z
y^2 = xz

Lets take one example
y = 6 and x = 12 and y = 3
Pen : Pencil : Eraser

"Can the number of erasers purchased be 3" Yes possible

Hence C
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Luckisnoexcuse

Originally posted by Luckisnoexcuse on 05 Jul 2017, 00:46.
Last edited by Luckisnoexcuse on 27 Sep 2017, 21:21, edited 1 time in total.
Intern
Joined: 01 Jun 2016
Posts: 29
A man purchased some pens, pencils and erasers. Can the number of eras  [#permalink]

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27 Sep 2017, 21:02
Bunuel wrote:
A man purchased some pens, pencils and erasers. Can the number of erasers purchased be 3?

(1) The ratio of the number of pens to the number of pencils was the same as the ratio of the number of pencils to the number of erasers.
(2) Total number of items purchased was 21.

Solution:

Let the number of pen be X,
the number of pencil be Y,
the number of Erasers be Z.

Statement 1 - \frac{X}{Y} = \frac{Y}{Z}
=> XZ = $$Y^2$$ -------------------------------------------------Equation I

Y can be any number.

Statement 1 Not sufficient

Statement 2 - X + Y + Z = 21 -------------------------------------------------Equation II

Y can be any number.

Statement 2 Not sufficient

Together 1 & 2:

From equation 1
$$Y^2$$ = Can only be equal to 4,9,16, and 25.
so, Y can be 2, 3, 4, 5.

Only when X = 16, Z = 1 and Y = 4 will satisfy both of the above equations.

hence Y = 4 and not 3 so

Together both the statements are sufficient. C

Intern
Joined: 26 Feb 2017
Posts: 17
Re: A man purchased some pens, pencils and erasers. Can the number of eras  [#permalink]

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17 Oct 2017, 05:17
Can someone please give another simple technique to solve this?
Manager
Joined: 12 Feb 2017
Posts: 70
Re: A man purchased some pens, pencils and erasers. Can the number of eras  [#permalink]

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28 Oct 2017, 04:08
atomicmass wrote:
Bunuel wrote:
A man purchased some pens, pencils and erasers. Can the number of erasers purchased be 3?

(1) The ratio of the number of pens to the number of pencils was the same as the ratio of the number of pencils to the number of erasers.
(2) Total number of items purchased was 21.

Solution:

Let the number of pen be X,
the number of pencil be Y,
the number of Erasers be Z.

Statement 1 - \frac{X}{Y} = \frac{Y}{Z}
=> XZ = $$Y^2$$ -------------------------------------------------Equation I

Y can be any number.

Statement 1 Not sufficient

Statement 2 - X + Y + Z = 21 -------------------------------------------------Equation II

Y can be any number.

Statement 2 Not sufficient

Together 1 & 2:

From equation 1
$$Y^2$$ = Can only be equal to 4,9,16, and 25.
so, Y can be 2, 3, 4, 5.

Only when X = 16, Z = 1 and Y = 4 will satisfy both of the above equations.

hence Y = 4 and not 3 so

Together both the statements are sufficient. C

consider,
there are 6 pencils, 12 pens and 3 erasures.
then also pencil^2= pens*erasures holds.
6^2=36
12*3=36.
Answer would still remain C. Because we are asked to find "can the numbers of erasures purchased be 3?" So, yes in one scenario it can be.
Math Expert
Joined: 02 Aug 2009
Posts: 6978
Re: A man purchased some pens, pencils and erasers. Can the number of eras  [#permalink]

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28 Oct 2017, 05:07
Bunuel wrote:
A man purchased some pens, pencils and erasers. Can the number of erasers purchased be 3?

(1) The ratio of the number of pens to the number of pencils was the same as the ratio of the number of pencils to the number of erasers.
(2) Total number of items purchased was 21.

Hi Bunuel, may be you require to look into the Q ...

the wordings are CAN the erasers be 3...

(1) The ratio of the number of pens to the number of pencils was the same as the ratio of the number of pencils to the number of erasers.
let the ratio be 1:1:1...all three can be 3 each 3:3:3... YES erasers CAN be 3
also each can be 4:4:4 that is 4 each... ans is No

BUt we are looking for CAN, so ans is YES and should be sufficient

(2) Total number of items purchased was 21
x+y+z=21... we can easily find the match where erasers are 3 and remaining two add up to 18
so again should be suffficient..

But if the Q was "IS number of erasers purchased 3", each statement would be insuff
two solution that will fit in..
1) 12:6:3.....12+6+3=21....Also ratio is 12:6 or 2:1 and 6:3 or 2;1.... 6^2=12*3... so eraser can be 3
2) 16:4:1...16+4+1=21...Also ratio is 16:4=4:1 and 4:1....4^2=16*1...... so eraser will not be 3

and answer should be E for "is number of erasers 3?"..
and ans should be D for " can the number of erasers be 3?"

--== Message from the GMAT Club Team ==--

THERE IS LIKELY A BETTER DISCUSSION OF THIS EXACT QUESTION.
This discussion does not meet community quality standards. It has been retired.

If you would like to discuss this question please re-post it in the respective forum. Thank you!

To review the GMAT Club's Forums Posting Guidelines, please follow these links: Quantitative | Verbal Please note - we may remove posts that do not follow our posting guidelines. Thank you.

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3) effects of arithmetic operations : https://gmatclub.com/forum/effects-of-arithmetic-operations-on-fractions-269413.html

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Re: A man purchased some pens, pencils and erasers. Can the number of eras &nbs [#permalink] 28 Oct 2017, 05:07
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