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On a certain table is a collection of 45 pens of various col

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On a certain table is a collection of 45 pens of various col [#permalink] New post   10 May 2019, 08:27
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On a certain table is a collection of 45 pens of various colors of ink and various designs. Of these 45 pens, 60% are ballpoint pens. Also, 60% of the pens on the table have blue ink.

Quantity A
Quantity B
The number of pens that are ballpoint with blue ink
8

A) The quantity in Column A is greater.
B) The quantity in Column B is greater.
C) The two quantities are equal.
D) The relationship cannot be determined from the information given.
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A
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Re: On a certain table is a collection of 45 pens of various col [#permalink] New post   10 May 2019, 08:48
1
ballpoint pens =45*0.6=27
and blue ink pen =45*0.6=27
so ballpoint pens+blue ink pen=54
but there are actually 45 pen
so 54-45=9
so 9 ballpoint pens must have blue ink.
that is why A is the answer.
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On a certain table is a collection of 45 pens of various col [#permalink] New post   10 May 2019, 08:50
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GreenlightTestPrep wrote:
On a certain table is a collection of 45 pens of various colors of ink and various designs. Of these 45 pens, 60% are ballpoint pens. Also, 60% of the pens on the table have blue ink.

Quantity A
Quantity B
The number of pens that are ballpoint with blue ink
8



One approach is to use 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 (aka overlapping sets questions).

Here, we have a population of pens, and the two characteristics are:
- ballpoint or not ballpoint
- blue or not blue

So, we can set up our matrix as follows:
Image
NOTE: I placed a star in the box representing pens that are ballpoint AND have blue ink.

From here, let's determine the GREATEST value and the LEAST value that can be placed in the top-left box.

Since the squares in the top row adds to 27, and since the squares in the left-hand column must add to 27, the GREATEST number we can place in the top-left box is 27
We get:
Image

In order the MINIMIZE the value in the top-left box, we must MAXIMIZE the values in the top-right box and in the bottom-left box.
Since the two boxes in the bottom row must add to 18, we can see that 18 is the BIGGEST number to place in the bottom-left box
Since the two boxes in the right-hand column must add to 18, we can see that 18 is the BIGGEST number to place in the top-right box
We get:
Image

When we fill in the remaining boxes, we see that...
Image
...the SMALLEST number we can place in the top-left box is 9

We get:
QUANTITY A: Some value from 9 to 27 inclusive
QUANTITY B: 8

Answer: A

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Re: On a certain table is a collection of 45 pens of various col [#permalink] New post   19 Aug 2019, 13:38
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Here is another approach.
Attachments

pens overlapping.png
pens overlapping.png [ 281.86 KiB | Viewed 11610 times ]

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Re: On a certain table is a collection of 45 pens of various col [#permalink] New post   22 Jul 2021, 20:35
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To my mind, this would be the reasoning here; feel free to correct me if I'm wrong:
1. We know the total number of pens (45).
2. We know that exactly 27 pens are ballpoint and that 27 pens contain blue ink.
3. We know that they have to overlap for two reasons: (1) on an obvious/intuitive level, 60% is more than half, so it's impossible for the two not to overlap in that 45-pen system, and (2), we can see that if we add up 27 and 27, we get 54, which is more than the total number of pens on the table.

4. Since we know they have overlap, we could define that this way:
Total = Set1 + Set2 - Overlap
45 = 27 + 27 - O
45 = 54 - O
O = 9

5. Since 9 > 8, the answer is A.
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On a certain table is a collection of 45 pens of various col [#permalink] New post   18 Dec 2021, 06:19
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We want to know what is the min and max of the intersection of Ball Point and Blue.

The max is the case where all Ball Point pens are also blue, i.e. intersection would be 27

*Visualizing a two group Venn Diagram*
The underlying idea to get the min is: WE MINIMIZE THE INTERSECTION WHEN WE LET THE THE GROUP OUTSIDE THE UNION BE AT ITS SMALLEST.

Why? By making the group outside the union as small as possible, we "free up space" for the union to be as large as possible. Why is this important? Because with more free space we are allowed to "separate" the groups of Ball Point Pens and Blue pens further apart, hence making the intersection smaller.

Let x=QA.
We let the groups outside the union be 0.
Therefore
45=27+27-x
x=9

Therefore QA is between 9 and 27 inclusive

Final Answer: A
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Re: On a certain table is a collection of 45 pens of various col [#permalink] New post   17 Jun 2024, 05:41
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Re: On a certain table is a collection of 45 pens of various col [#permalink]
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