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In Motor City 90 percent of the population own a car, 15 percent own

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Joined: 02 Sep 2009
Posts: 50583
In Motor City 90 percent of the population own a car, 15 percent own  [#permalink]

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05 Nov 2017, 00:38
00:00

Difficulty:

55% (hard)

Question Stats:

49% (01:28) correct 51% (01:25) wrong based on 98 sessions

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In Motor City 90 percent of the population own a car, 15 percent own a motorcycle, and everybody owns one or the other or both. What is the percentage of motorcycle owners who own cars?

(A) 5%
(B) 15%
(C) 33 1/3%
(D) 50%
(E) 90%

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Posts: 2092
In Motor City 90 percent of the population own a car, 15 percent own  [#permalink]

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05 Nov 2017, 11:27
Bunuel wrote:
In Motor City 90 percent of the population own a car, 15 percent own a motorcycle, and everybody owns one or the other or both. What is the percentage of motorcycle owners who own cars?

(A) 5%
(B) 15%
(C) 33 1/3%
(D) 50%
(E) 90%

Attachment:

matrix.png [ 8.03 KiB | Viewed 2503 times ]

I - Double Matrix (see Diagram)

Total number of people = 100. Fill in:
Cars = 90
Motorcycles = 15
No Car, No Motorcycle = 0

Then:

1) Number who do not own a car: 100 - 90 = 10 (No Car)

2) Number of motorcycle owners who do not own a car MUST = 10 because 0 own neither, and the column must add up to 10 (so 10 = Motorcycle and No Car)

3) Number of car AND motorcycle owners: 15 - 10 = 5 - because row MUST add up to 15 (so 5 = Car and Motorcycle)

What is the percentage of motorcycle owners who own cars?
Motorcycle owners = 15
Motorcycle owners who own cars = 5

$$\frac{5}{15} * 100 =$$ (.33 * 100) = 33 1/3 %

II - Formula for Overlapping Sets

A + B - Both + Neither = Total number

Because we are given percentages, let the total number of people = 100

Let A = people who own cars = 90 percent = 90
Let B = people who own motorcycles = 15 percent = 15
Both?
Everyone owns a car or a motorcycle or both: "Neither" = 0

A + B - Both + Neither = Total number
90 + 15 - Both + 0 = 100
- Both = -5
Both = 5

5 people own both a motorcycle and a car.

What is the percentage of motorcycle owners who own cars?
Motorcycle owners = B, so: what percentage of B is "both"?
Or: "both" = what percentage of B?
Both = 5, B = 15

$$\frac{5}{15} * 100 =$$ (.33 * 100) = 33 1/3%

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Joined: 28 Dec 2010
Posts: 49
Re: In Motor City 90 percent of the population own a car, 15 percent own  [#permalink]

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06 Nov 2017, 00:02
It is a simple set theory question.
AUB = A + B - A∩B

AUB is overall: 100 (as everyone owns one or other)
A = 90 (cars)
B = 15 (bike)
A∩B is who owns both.

100 = 90 + 5 - A∩B
A∩B = 5

A.
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Posts: 2830
Re: In Motor City 90 percent of the population own a car, 15 percent own  [#permalink]

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08 Nov 2017, 16:27
Bunuel wrote:
In Motor City 90 percent of the population own a car, 15 percent own a motorcycle, and everybody owns one or the other or both. What is the percentage of motorcycle owners who own cars?

(A) 5%
(B) 15%
(C) 33 1/3%
(D) 50%
(E) 90%

We can use the following formula:

Total = total cars + total motorcycles - both + neither

We can let the total = 100 and b = both and create the equation, noting that since everyone owns at least one vehicle, the value of neither is 0:

100 = 90 + 15 - b + 0

100 = 105 - b

b = 5

Thus, 5/15 = 1/3 = 33⅓% of the motorcycle owners also own a car.

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Re: In Motor City 90 percent of the population own a car, 15 percent own &nbs [#permalink] 08 Nov 2017, 16:27
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