One-third of all U. S. taxpayers may deduct charitable contributions

It's a pure ratio question, so we can invent a total number and use it to answer the question. 300 is convenient, since we want to be able to divide our total both by 3 and by 100. Then we know 45% of these, or 135 people, can make some kind of deduction; 100 can make a federal deduction; and 120 can make a state deduction. So we have a standard Venn diagram situation:

- x people can make both deductions, so are in the overlapping part of the Venn diagram
- 100-x people are in the federal deductions *only* region of the diagram
- 120-x people are in the state deductions *only* region of the diagram

Since these three things must add up to the total number of people, 135, in the two circles, we have

220 - x = 135
x = 85

and 85/300 = 17/60 is the answer.

Well I see that many people like me are choosing D. It took me almost 10 mins to realize the mistake

So we know that:
Number of taxpayers who deduct charitable contributions = CCd = 1/3
Number of taxpayers who deduct state income tax = SId = 40% = 2/5
Number of taxpayers who dont deduct either = 55/100
=> Number of taxpayers who deduct both = Td = 45/100
Let us assume the overlap between CCd and SId to be x

Applying simple set theory

We know:
CCd + SId - x = Td
1/3 + 2/5 -x = 45/100

If we take the set to be of 100 people,

33 + 40 - x = 45
=> x = 28
=> Portion that can claim both = 28/100 = 7/25 and we assume the correct answer is D

But we make one small mistake in considering 1/3 = 33 whereas it is 33.3333333

So now we need to relook at the below equation:
CCd + SId -x = Td
1/3 + 2/5 - x = 45/100

If we take the set to be of 100 people,

33.3333 + 40 - x = 45
=> x = 28.3333
=> Portion that can claim both = 28.3333/100
Multiple both numerator and denominator by 3
=> Portion = 85/300 = 17/60

which means the correct answer is E.