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

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Math Expert
Joined: 02 Sep 2009
Posts: 58422
One-third of all U. S. taxpayers may deduct charitable contributions  [#permalink]

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11 Feb 2019, 02:44
00:00

Difficulty:

65% (hard)

Question Stats:

52% (03:00) correct 48% (02:32) wrong based on 61 sessions

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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

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One-third of all U. S. taxpayers may deduct charitable contributions  [#permalink]

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11 Feb 2019, 03:23
Bunuel wrote:
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

solved it using 2x2 grid

IMO E 17/160
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File comment: solution attached
apples.xlsx [30.57 KiB]

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Re: One-third of all U. S. taxpayers may deduct charitable contributions  [#permalink]

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15 Feb 2019, 23:38
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
$$\frac{y}{x}=\frac{17}{60}$$.
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Re: One-third of all U. S. taxpayers may deduct charitable contributions  [#permalink]

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16 Feb 2019, 16:45
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.
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Re: One-third of all U. S. taxpayers may deduct charitable contributions  [#permalink]

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17 Feb 2019, 00:03
1
Bunuel wrote:
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

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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Re: One-third of all U. S. taxpayers may deduct charitable contributions   [#permalink] 17 Feb 2019, 00:03
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