600 students took the test on English and Maths. 35% students failed

Wow! That's a great question to test both on concepts of set theory, percentage and Applications on the said topics.

Lets check this out!
Instead of using the value of 600 at this stage and incresing the complexity of the calculations ,lets keep it simple and assume that there are 100 students who took the test.

Of these 100 students, 35 failed in English which implies that 100-35 or 65 students PASSED in English.

Also, of these 100 students,45% failed in Math which implies that 100-45 0r 55 students PASSED in Math.

40% of those who passed in Math also passed in English. This is where we have to use some quick calculations.

40% of 55 = 22 students PASSED in Math and also passed in English!!

We have assumed that Total is 100 and hence lets calculate the number of students who PASSED in both the subjects.

Here, 65+ 55 - 22 = 98 students have PASSED in both the subjects.
(This is because Passed in English + Passed in Math - the overlap of Passed in both the subject will give us the number of students passed in either of the subjects)

Since 98 students passed in either of the subjects, 100-98 or 2 students FAILED in both the subjects.

A Quick GMAT Tip-On GMAT, never lose the context and the information of the Q. stem.

Since out of 100,we have 2 students failing in both the subjects, multiply this value by 6 to find the number of students failing in both the subjects out of 600.

This would be 6 * 2 = 12 students

(option e)

Hope this helps !
Solve more of such questions, "think through" and increase the probability of improving on the areas of Sets and Percentage!

Devmitra Sen
GMAT Quant SME