In a group of 100 people, 40 study English and 30 study French. The number of people in the group who study neither language is twice the number of people who study both. How many people in the group study both languages? A. 10 B. 20 C. 26 D. 30 E. 34

(D) Here we will use the formula for Set Overlap. The general formula when there are two groups is: Total Number of Different People = (# in Group 1) + (# in Group 2) + (# in Neither Group) – (# in Both Groups).

The reason we subtract out the number of people in both groups is that they have been counted twice unless we subtract them out.

Let’s define some variables here: T = Total number of people. E = Number of people who study English. F = Number of people who study French. N = Number of people who study neither language. B = Number of people who study both languages. Using these variables we can now write the equation: T = E + F + N – B.

We are given numbers for T, E, and F, and we are told that N = 2B.

Let’s substitute these into our equation and solve: T = E + F + N – B 100 = 40 + 30 + 2B – B B = 30. The correct answer is choice (D). ---------- Shouldn't the correct answer for this questions be A, 10 people who study both languages. If there is a 100 total people: 40 study English and 30 study French which is 70 leaving 30 left over. Of that 30 the # of people who DO NOT study either is twice as many as those that study both; which to satisfy that statement would mean 20 study neither and 10 study both. Is that not correct?

The would have been correct if the question statement has been saying " ... 40 study ONLY English and 30 study ONLY French ...".

In the above question some of the 40 students, who study English, study French as well. The same goes for the 30 students, who study French. Those are students, who study the both languages.

The set of such students is the intersection of the "study English" and "study French" sets. You can easily see it on the Venn diagram for this question:

When we plug in the numbers, we get the following:

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