## At a family summer party, each of the $$x$$ members of the family chose whether or not to have a hamburger and whether

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### At a family summer party, each of the $$x$$ members of the family chose whether or not to have a hamburger and whether

by Vincen » Fri Feb 11, 2022 3:15 am

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At a family summer party, each of the $$x$$ members of the family chose whether or not to have a hamburger and whether or not to have a hotdog. If $$\dfrac13$$ chose to have a hamburger, and of those $$\dfrac17$$ chose to also have a hotdog, then how many family members chose NOT to have both?

A. $$\dfrac{x}{21}$$

B. $$\dfrac{x}{10}$$

C. $$\dfrac{9x}{10}$$

D. $$\dfrac{10x}{21}$$

E. $$\dfrac{20x}{21}$$

Source: EMPOWERgmat

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### Re: At a family summer party, each of the $$x$$ members of the family chose whether or not to have a hamburger and wheth

by Brent@GMATPrepNow » Fri Feb 11, 2022 6:38 am
Vincen wrote:
Fri Feb 11, 2022 3:15 am
At a family summer party, each of the $$x$$ members of the family chose whether or not to have a hamburger and whether or not to have a hotdog. If $$\dfrac13$$ chose to have a hamburger, and of those $$\dfrac17$$ chose to also have a hotdog, then how many family members chose NOT to have both?

A. $$\dfrac{x}{21}$$

B. $$\dfrac{x}{10}$$

C. $$\dfrac{9x}{10}$$

D. $$\dfrac{10x}{21}$$

E. $$\dfrac{20x}{21}$$

Source: EMPOWERgmat
These kinds of questions (Variables in the Answer Choices - VIACs) can be answered algebraically or using the INPUT-OUTPUT approach.
Here's the INPUT-OUTPUT approach.

It might be useful to choose a number that works well with the fractions given in the question (1/3 and 1/7).
So, let's say there are 21 family members at the party.
In other words, we're saying that x = 21

1/3 chose to have a hamburger, and of those 1/7 chose to also have a hotdog.
1/3 of 21 is 7, so 7 people chose to have a hamburger.
1/7 of 7 = 1, so 1 person had BOTH a hamburger AND a hotdog.

How many family members chose NOT to have both?
If 1 person had BOTH a hamburger AND a hotdog, then the remaining 20 people did not have BOTH a hamburger AND a hotdog.

So, when we INPUT x = 21, the answer to the question is "20 people did not have BOTH a hamburger AND a hotdog"

Now we'll INPUT x = 21 into each answer choice and see which one yields the correct OUTPUT of 20

A. 21/21 = 1. We want an output of 20. ELIMINATE A.
B. 21/10 = 2.1. We want an output of 20. ELIMINATE B.
C. (9)(21)/10 = 18.9. We want an output of 20. ELIMINATE C.
D. (10)(21)/21 = 10. We want an output of 20. ELIMINATE D.
E. (20)(21)/21 = = 20. PERFECT!