Could you please share a solution for this?

Given:

• Out of 2400 first year students, everyone must take either Calculus, or English Composition, or both
• Half of all the students take Calculus, half do not
• One-third of those who take Calculus also take English Composition

To find:

• The number of students take English Composition

Approach and Working:
As one-third of those who take Calculus also take English Composition, we can say

• Number of students take both Calculus and English Composition = \(\frac{1}{3} *(\frac{1}{2} * 2400) = 400\)

Also, those students who do not take Calculus, must take English Composition

• Number of students who take English Composition only = \(\frac{1}{2} * 2400 = 1200\)
• Therefore, the total number of students taking English Composition = 400 + 1200 = 1600

Hence, the correct answer is option D.

Answer: D

Here's a table view of the overlapping sets:



First, we know the total is 2400.

Then, given that they must take at least one of the subjects, we know the number of students who do not take calculus and do not take english is 0. This fills the "middle" square.

Next, half of the total number of students take calculus. So we know 1200 take calculus and 1200 do not. This fills the "bottom" row.

Of the 1200 who take calculus, a third of those students also take english. So, 400 students take english and calculus. This leaves 800 students who take calculus and do not take english. This fills the "left" column.

This is a question of overlapping sets and a classic RTFQ trap!

And here's a quick way to solve this problem:

Set (A) only + Set (B) only + Both + Neither = Total Number

Set(A) = English
Set(B) = Calculus
Both = Calculus and English
Neither = RTFQ trap!
Total = Given in question

Recognise that "all must take one" eliminates the possibility of students taking neither subject. Therefore, Neither = 0.

Set(A) = ?
Set(B) = 1200
Both = 400
Neither = 0
Total = 2400

Let's plug the values into the formula:

? + 1200 + 400 + 0 = 2400
? + 1600 = 2400
? = 800

You could draw a Venn diagram, or a table like the other suggested answer shows, but it would be quicker to recognise the detail in the question wording and simply plug the numbers into the formula. Drawing a table may take you 45-60 seconds. Using the formula would take 20 seconds.

Ultimate Tuition offers a GRE Quant Cheat Sheet that has a great chapter on overlapping sets.

Check out the Ultimate GRE Quant Cheat Sheet!