Elena purchased Brand X pens for $4.00 apiece and brand Y

Question

Elena purchased Brand X pens for $4.00 apiece and brand Y for $2.80 apiece. If Elena purchased a total of 12 of these pens for $42.00, how many brand X pens did she purchase?

A) 4
B) 5
C) 6
D) 7
E) 8

fiendex · Accepted Answer

I took a different approach that is more intuitive to me:

4x + 2.80 (12 - x) = 42.00
 4x + 33.60 - 2.80x = 42.00
 1.20x = 8.40
 x = 7

kys123 · Answer

I have been using this one which I feel is pretty simple as well.

Maximum cost of 12 units ($4 *12 pens)= 48
Difference in price between pen and pencil = 4 -2.8 = 1.2.
48 - 42 = $6 
$6/1.2 = 5
You need to make 5 pen into pencil for the sum to be 42.

12 - 5 = 7 pens

Bunuel · Answer

4x+2.8y=42 --> multiply by 2.5 (to get the integers) --> 10x+7y=105 --> only one positive integers solutions x=7 and y=5 (how to solve: 7y must have the last digit of 5 in order the last digit of the sum to be 5).

Answer: D.

Answer: D.

Bunuel · Answer

Interestingly enough even if we were not told that Elena purchased a total of 12 items we could still solve the problem. Another non-negative solution of 10x+7y=105 is x=0 and y=15, but since x=0 is not listed then x=7 is correct answer anyway.

carcass · Answer

I agree with Bunuel.........even though a similar problem on MGMAT guide is not sub 600 but 650 up........

40x + 28y = 420

We have 12 pens looking at the answer choices pick D : 7*40 + 28*5 = 420.......> 420 = 420. D

EMPOWERgmatRichC · Answer

Hi All,

This question has a great Number Property 'shortcut' built into it than can help you avoid having to do too much math.

Since the total spent is exact $42 and Brand Y pens cost $2.80 each, we have to 'multiply away' all of those $0.80 increments and end up with a round number. The smallest positive integer that will do that is 5... 5($2.80) = $14.

Considering that there are only 12 total pens and the answers choices are 4-8 inclusive, there's only one possible number of Brand X pens that makes sense...

Final Answer:  D

GMAT assassins aren't born, they're made,
Rich

Shiv2016 · Answer

Bunuel  
I need a little help with this question.

x+y= 12...................1
4x+2.80y=42.............2

From 1, y= 12-x

Substitute in eq. 2

I approximated 2.80 to 3 as the multiplication would be a little long and I thought that as I am taking a higher value, the answer will be less than the approximated answer.

4x+36-3x= 42

x= 6

But the answer is more than 6. Where did I really make the mistake?

Bunuel · Answer

4x +  2.80y  = 42

When you change 2.8 with 3, you increase the value of the red summand. So, the value of 4x, and therefore of x itself, will be lower than they actually are.

Helium · Answer

Answer required is number of brand X pens. - An integer

Brand X pen costs $ 4.00

So total cost of required brand X pens will be integer (No of brand X pens * $ 4)

$42 - integer = integer

Hence, we are looking for a number when multiplied with $ 2.8 will give integer.

A quick look at options tells - 5 (12 - 7). So required answer brand X pens = 7

Probus · Answer

Hello,

Notice that

4* some integer x+2.80 * some integer y = 42.00
Clearly 2.80 must be multiplied with a number that yields integer. So it can be 5,10,15,20 ... So on

Now If y=15 x=0,and y =5 x= 7 satisfies our conditions that 4x+2.8y=42

Hence Choice D

SameeranKarpe · Answer

You can use alligation method where the ratio of weights is the reverse of its deviation from average.
Average price per pen is 42/12=3.5
Deviation from average for x = 0.5
Deviation from average for y= 0.7

x/y = 4/2.8 = 0.7/0.5 = 7/5

As the sum is 12, you can conclude from the ratio that x = 7, y =5

SameeranKarpe · Answer

Bunuel  will this be a correct approach -

You can use alligation method where the ratio of weights is the reverse of its deviation from average.
Average price per pen is 42/12=3.5
Deviation from average for x = 0.5
Deviation from average for y= 0.7

x/y = 4/2.8 = 0.7/0.5 = 7/5

As the sum is 12, you can conclude from the ratio that x = 7, y =5

AnishPassi · Answer

Approach 1

Had the number of X and Y brand pens purchased been equal, the average price/ pen would have been the simple average of 2.8 and 4 (which is 3.4).

The actual average price/ pen is 42/12 =3.5.
3.5 > 3.4, so Elena would have purchased more X pens than Y pens.
That eliminates A, B and C.

Next I could choose one option and work backward to check whether it fits.

e.g. (D)7.
Total revenue:
7 x 4 =28
5 x 2.8 =14
Total: 28 + 14 =42
This fits. So, must be correct.

Alternately if I try (E) 8. 
Total revenue:
8 x 4 =32
4 x 2.8 =11.2
Total: 32 + 11.2 > 42.
Too high. Rejected. (D) must be correct.

Approach 2

Say, all the pens purchased were the $4 ones. She purchased 12 pens. So that would bring the total expense to $48. 
(I considered $4 pens instead of $2.8 simply because 4 looks like an easier number to deal with.) 4 x12 = $48. That’s too much. The actual total expense is $42.

The only way to reduce the expense is to exchange a certain number of the more expensive pens with the same number of cheaper pens. Since the total number of pens has to remain 12.

We have to go from $48 to $42. So, we have to reduce the total revenue by $6.

What happens when I exchange one expensive pen for one less expensive pen? The revenue reduces by 4-2.8 = $1.2.

So in order to reduce the revenue by $6, I would need to make 6/1.2 = 5 such exchanges. So overall, the number of the more expensive pens would be 12 - 5 = 7..

Elena purchased Brand X pens for $4.00 apiece and brand Y

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Elena purchased Brand X pens for $4.00 apiece and brand Y

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