At a family summer party, each of the x members of the family

(1/3) * (1/7)
= 1/21 = both

anything other than both
= 1 - (1/21)
20/21

thanks
cheers through the kudos button if this helps

9x/10

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The questions asks for neither x nor y. I believe that some info is missing from this problem.

We have information for both x and y and x and not y.

chetan2u , could you please help with this

I get how it's figured out, but what about the people who chose to have a hotdog and not a hamburger? Do we not need to know this?

X is all the people - let's call this 21
Therefore, 7 chose to have a hamburger and 14 chose not to have a hamburger.
Of this 7, 1 chose to also have a hotdog. This leaves 6 people who only had a hamburger.
Of the 14 who chose not to have a hamburger, how many chose to have a hotdog and not a hamburger? - We don't know this piece of information, so surely it's a proportion of the 20x/21?

Say, Hamburger = H & Hotdog = D
Given n(H) =X/3 , Total = X
and n(D) = X- X/3 = 2X/3
Both = X/3 * 1/7 = X/21
Only H = X/3-X/21 = 6X/21
Only D = 2X/3-X/21 =13X/21
We know that, Total = n(H) + n(D) - n(H&B) + Neither
X = X/3 + 2X/3 - X/21 + Neither
Neither = X/21
So, who eat none = only H+ only D + Neither
= 6X/21+13X/21+X/21
= 20X/21.

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!

Answer: E

How to get the missing variable? I think the question is missing some data. HI Bunuel can you please check this?

Since x * 1/3 * 1/7 = x/21 of the family members have both a hamburger and a hotdog, 20x/21 of the family members do not have both.

Answer: E

No data is missing..Que asks how many people did not have both which includes people who had one (either hamburger or hotdog) + people who had neither (No ham no hotdog)

Hope that helps.

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