Overlapping problem

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Overlapping problem

by Mo2men » Wed Nov 09, 2016 9:46 am
A sporting goods store received a shipment of baseball gloves that included 5 brown gloves for every 6 black gloves. Did the store receive at least 250 gloves in the shipment?

(1) 44% of the left-handed gloves in the shipment were black.

(2) The shipment included 84 black, right-handed gloves.

Source: Veritas

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by [email protected] » Wed Nov 09, 2016 10:39 am
Hi Mo2men,

This DS question revolves around a ratio. In these situations, it's important to remember that ratio questions often come down to "multiples."

We're given a starting ratio of Brown gloves to Black gloves = 5/6. We're told that Left-handed and Right-handed gloves are ordered individually (which means that there's NO ratio governing the number of Left or Right). We're asked if the total number of gloves is at least 250? This is a YES/NO question.

From the prompt, we KNOW that the number of Brown gloves MUST be a multiple of 5 and the number of Black gloves MUST be a multiple of 6...

Fact 1: 44% of left-handed gloves are Black.

44/100 = 11/25 left-handed gloves are Black, so the minimum number of this type = 11

This means....

56/100 = 14/25 left-handed gloves are Brown, so the minimum number of this type = 14

We don't know the number (or color) of Right-handed gloves, which is a significant missing piece of information, so we could have ANY NUMBER of Right-handed Black and Right-handed Brown gloves (as long as those number properly complete the given ratio). For example:

Brown Left = 14
Black Left = 11
Brown Right = 1
Black Right = 7
Total = 33 and the answer to the question is NO

Brown Left = 14
Black Left = 11
Brown Right = 101
Black Right = 127
Total = 253 and the answer to the question is YES

Fact 1 is INSUFFICENT

Fact 2: 84 Black Right-handed Gloves

Using the original ratio, this info tells us that there are a MINIMUM of 70 Brown gloves, so the MINIMUM TOTAL = 154 (answer is NO), but the total could be higher (e.g. in the 1000s, so the answer is YES.
Fact 2 is INSUFFICIENT

Combined, we know:
Black Right-handed = 84
Brown Rightt-handed = ???
Black Left-handed = multiple of 11
Brown Left-handed = multiple of 14
Total Brown = multiple of 5
Total Black = multiple of 6

Here's where things get a bit more complex:

Since the Total Black must be a multiple of 6....84 + (multiple of 11) = (multiple of 6)

The MINIMUM multiple of 11 that "fits" this factoid would be 66, so we'd have 84 + 66 = 150 Black gloves AT THE MINIMUM

With those 150 Black gloves, we'd have to match the ratio in the prompt, so the Total number of Brown gloves would be 125 AT THE MINIMUM.

So, we end up with a MINIMUM of 275 gloves; the answer to the question is ALWAYS YES.

Final Answer: C

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by GMATGuruNY » Wed Nov 09, 2016 10:52 am
Mo2men wrote:A sporting goods store received a shipment of baseball gloves that included 5 brown gloves for every 6 black gloves. Did the store receive at least 250 gloves in the shipment?

(1) 44% of the left-handed gloves in the shipment were black.

(2) The shipment included 84 black, right-handed gloves.

Source: Veritas
Organize the information in a MATRIX.

A sporting goods store received a shipment of baseball gloves that included 5 brown gloves for every 6 black gloves.
Since black/brown = 6x/5x, the following matrix is yielded:
Image

Did the store receive at least 250 gloves in the shipment?
In the matrix above, the total number of gloves = 11x.
Question stem, rephrased:
Is 11x ≥ 250?

Statement 1: 44% of the left-handed gloves in the shipment were black.
Let the total number of left-handed gloves = y, implying that the number of black left-handed gloves = (44/100)y = (11/25)y.
The following matrix is yielded:
Image
Since (11/25)y must be an INTEGER VALUE, y must be a MULTIPLE OF 25.
If y=25, then it's possible that 11x < 250.
If y=2500, then 11x > 250.
INSUFFICIENT.

Statement 2: The shipment included 84 black, right-handed gloves.
The following matrix is yielded:
Image
If 6x = 84, then 11x < 250.
If 6x = 840, then 11x > 250.
INSUFFICIENT.

Statements combined:
Image
The leftmost column indicates the following equation:
84 + (11/25)y = 6x.
(11/25)y = 6x - 84 = (multiple of 6) - (multiple of 6) = multiple of 6.

Since (11/25)y must be a multiple of 6, and y must be a multiple of 25, the least possible value for y = 6*25 = 150.
Substituting y=150 into 84 + (11/25)y = 6x, we get:
84 + (11/25)(150) = 6x
150 = 6x
x = 25.
Since the least possible value for x is 25, 11x > 250.
SUFFICIENT.

The correct answer is C.
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by Brent@GMATPrepNow » Wed Nov 09, 2016 5:16 pm
I thought I'd point out that Mitch's approach is also known as the Double Matrix Method. This technique can be used for most questions featuring a population in which each member has two characteristics associated with it.
Here, we have a population of baseball gloves, and the two characteristics are:
- Black or brown
- left-handed or right-handed

This question type is VERY COMMON on the GMAT, so be sure to master the technique.

To learn more about the Double Matrix Method, watch this video: https://www.gmatprepnow.com/module/gmat- ... ems?id=919

Once you're familiar with this technique, you can attempt these additional practice questions:

Easy Problem Solving questions
- https://www.beatthegmat.com/finance-majo ... 67425.html

Medium Problem Solving questions
- https://www.gmatprepnow.com/module/gmat- ... /video/920
- https://www.beatthegmat.com/posted-speed ... 72374.html
- https://www.beatthegmat.com/motel-t271938.html
- https://www.beatthegmat.com/of-the-appli ... 70255.html
- https://www.beatthegmat.com/opening-nigh ... 64869.html
- https://www.beatthegmat.com/at-least-100 ... 74669.html
- https://www.beatthegmat.com/prblem-solving-t279424.html

Difficult Problem Solving questions
- https://www.gmatprepnow.com/module/gmat- ... /video/946
- https://www.beatthegmat.com/ratio-problem-t268339.html
- https://www.beatthegmat.com/overlapping- ... 65223.html
- https://www.beatthegmat.com/fractions-t264254.html
- https://www.beatthegmat.com/overlapping- ... 64092.html
- https://www.beatthegmat.com/mba/2011/05/ ... question-2

Easy Data Sufficiency questions
- https://www.gmatprepnow.com/module/gmat- ... /video/943
- https://www.beatthegmat.com/for-what-per ... 70596.html
- https://www.beatthegmat.com/ds-quest-t187706.html

Medium Data Sufficiency questions
- https://www.beatthegmat.com/sets-matrix-ds-t271914.html
- https://www.beatthegmat.com/each-of-peop ... 71375.html
- https://www.beatthegmat.com/a-manufacturer-t270331.html
- https://www.beatthegmat.com/in-costume-f ... 69355.html
- https://www.beatthegmat.com/mba/2011/05/ ... question-1

Difficult Data Sufficiency questions
- https://youtu.be/dsCeqF9Kbk8
- https://www.beatthegmat.com/double-set-m ... 71423.html
- https://youtu.be/dOZ9KM1m5Hs
- https://www.beatthegmat.com/sets-t269449.html
- https://www.beatthegmat.com/mba/2011/05/ ... question-3

Cheers,
Brent
Brent Hanneson - Creator of GMATPrepNow.com
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