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

Question

One-third of all U. S. taxpayers may deduct charitable contributions on their federal income tax returns. Forty percent of all taxpayers may deduct state income tax payments from their federal returns. If 55 percent of all taxpayers may not deduct either charitable contributions or state income tax, what portion of all taxpayers may claim both types of deductions?

(A) 3/20
(B) 9/50
(C) 1/5
(D) 7/25
(E) 17/60

(A) 3/20
(B) 9/50
(C) 1/5
(D) 7/25
(E) 17/60

Archit3110 · Answer

solved it using 2x2 grid

IMO E 17/160

globaldesi · Answer

Solved using venn diagram approach and simplifying the equation
let the total pop be x
thus 
 \frac{x}{3}+\frac{4x}{10}- y = \frac{45x}{100} , where y = who can claim both
solve the equation and thus 
 \frac{y}{x}=\frac{17}{60} .
Thus answer E

IanStewart · Answer

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.

eabhgoy · 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.

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

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