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A group of 20 people are drinking coffee. The total number t

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A group of 20 people are drinking coffee. The total number t [#permalink] New post 30 Jun 2020, 09:12
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A group of 20 people are drinking coffee. The total number taking cream in their coffee is 7 less than twice the total number taking sugar. The number taking both cream and sugar is the same as the number taking neither. How many people in the group take cream?

A. 9
B. 10
C. 11
D. 12
E. 13
[Reveal] Spoiler: OA

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Re: A group of 20 people are drinking coffee. The total number t [#permalink] New post 30 Jun 2020, 09:13
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Re: A group of 20 people are drinking coffee. The total number t [#permalink] New post 30 Jun 2020, 09:39
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Carcass wrote:
A group of 20 people are drinking coffee. The total number taking cream in their coffee is 7 less than twice the total number taking sugar. The number taking both cream and sugar is the same as the number taking neither. How many people in the group take cream?

A. 9
B. 10
C. 11
D. 12
E. 13


One approach is to use the Double Matrix Method.
This technique can be used for questions featuring a population in which each member has two characteristics associated with it (aka overlapping sets questions).

Here, we have a population of coffee drinkers, and the two characteristics are:
- takes cream or does not take cream
- takes sugar or does not take sugar

Aside: We can also use Venn diagrams and formulae to solve overlapping sets questions. However, as difficulty levels increase, it becomes harder to apply those other approaches, whereas the Double Matrix Method works every time.

20 people are drinking coffee. The total number taking cream in their coffee is 7 less than twice the total number taking sugar. The number taking both cream and sugar is the same as the number taking neither.
Let x = the number of people taking sugar
This means 2x - 7 = the number of people taking cream

Let k = the number of people taking both cream and sugar
So, k = the number of people taking neither cream nor sugar

We get the following:
Image


There are 20 people in total.
So, if x of them are taking sugar, then 20 - x of them are NOT taking sugar

Likewise, if (2x - 7) of them are taking cream, then 20 - (2x - 7) of them are NOT taking cream
Simplify to get: 27 - 2x of them are NOT taking cream

Add this to our diagram to get:
Image


Now let's focus on the value in the TOP-RIGHT box
Since the two boxes in the RIGHT-HAND column add to 20 - x, we can conclude that (20 - x) - k is the value in the TOP-RIGHT box
Image



Likewise, since the two boxes in the TOP row add to 2x - 7, we can conclude that (2x - 7) - k is the value in the TOP-RIGHT box
Image

Notice that we have two different ways to express the value in the TOP-RIGHT box
So, it must be the case that those quantities are equal
In other words: (20 - x) - k = (2x - 7) - k
Simplify each side: 20 - x - k = 2x - 7 - k
Add k to both sides: 20 - x = 2x - 7
Add x to both sides: 20 = 3x - 7
Add 7 to both sides: 27 = 3x
Solve: x = 9

How many people in the group take cream?
We already know that 2x - 7 people take cream
Since x = 9, the number of people who take cream = 2(9) - 7 = 11

Answer: C

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Re: A group of 20 people are drinking coffee. The total number t   [#permalink] 30 Jun 2020, 09:39
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