In a town of 8,000 residents, 65 percent of all residents

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BTGmoderatorDC: Moderator; Posts: 7187; Joined: Thu Sep 07, 2017 4:43 pmFollowed by: 23 members

by BTGmoderatorDC » Tue Oct 16, 2018 5:56 pm

In a town of 8,000 residents, 65 percent of all residents own a car, 55 percent own a motorcycle, and 25 percent own neither a car nor a motorcycle. How many residents own a car but not a motorcycle?

A. 800
B. 1,600
C. 2,000
D. 3,600
E. 4,400

OA B

Source: Manhattan Prep

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Source: Beat The GMAT — Problem Solving | Tue Oct 16, 2018 5:56 pm

fskilnik@GMATH: GMAT Instructor; Posts: 1449; Joined: Sat Oct 09, 2010 2:16 pmFollowed by: 33 members; Thanked: 59 times

by fskilnik@GMATH » Wed Oct 17, 2018 5:33 am

BTGmoderatorDC wrote:In a town of 8,000 residents, 65 percent of all residents own a car, 55 percent own a motorcycle, and 25 percent own neither a car nor a motorcycle. How many residents own a car but not a motorcycle?

A. 800
B. 1,600
C. 2,000
D. 3,600
E. 4,400
Source: Manhattan Prep

$$?\,\, = \,x = 65\% \left( {{\rm{Tot}}} \right) - {\rm{both}}$$
$$75\% \left( {{\rm{Tot}}} \right) = 65\% \left( {{\rm{Tot}}} \right) + 55\% \left( {{\rm{Tot}}} \right) - {\rm{both}}\,\,\,\,\,\,\left[ {\,{\rm{simplifier}}\,} \right]\,\,\,\,\,\,\,\,\,\, \Rightarrow \,\,\,\,\,\,\,{\rm{both}} = 45\% \left( {{\rm{Tot}}} \right)$$
$$? = 20\% \left( {{\rm{Tot}}} \right) = {1 \over 5}\left( {800 \cdot 10} \right) = 2 \cdot 800 = 1600$$

This solution follows the notations and rationale taught in the GMATH method.

Regards,
Fabio.

Fabio Skilnik :: GMATH method creator ( Math for the GMAT)
English-speakers :: https://www.gmath.net
Portuguese-speakers :: https://www.gmath.com.br

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Brent@GMATPrepNow: GMAT Instructor; Posts: 16207; Joined: Mon Dec 08, 2008 6:26 pmFollowed by: 1268 members; Location: Vancouver, BC; Thanked: 5254 times; GMAT Score:770

In a town of 8,000 residents, 65 percent of all residents

by Brent@GMATPrepNow » Wed Oct 17, 2018 6:50 am

BTGmoderatorDC wrote:In a town of 8,000 residents, 65 percent of all residents own a car, 55 percent own a motorcycle, and 25 percent own neither a car nor a motorcycle. How many residents own a car but not a motorcycle?

A. 800
B. 1,600
C. 2,000
D. 3,600
E. 4,400

Another approach is to use the Double Matrix Method.
This technique can be used for most questions featuring a population in which each member has two characteristics associated with it (aka overlapping sets questions).
Here, we have a population of residents, and the two characteristics are:
- own a car or do not own a car
- own a motorcycle or do not own a motorcycle

65% of 8000 = 5200
55% of 8000 = 4400
25% of 8000 = 2000

So, our matrix looks like this so far:

Since there are 8000 resident altogether, we know that, if 5200 own a car, then 2800 do NOT own a car
And, if 4400 own a motorcycle, then 3600 do NOT own a motorcycle
We get:

When we fill in the rest of the matrix we get:

Brent Hanneson - Creator of GMATPrepNow.com

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swerve: Legendary Member; Posts: 2535; Joined: Sun Oct 29, 2017 2:04 pmFollowed by: 6 members

by swerve » Wed Oct 17, 2018 3:22 pm

Car = 65%
Motor = 55%
Both Car and Motor = x%

Neither Car or Motor = 25%
Car = A + x
Motor = B + x

A + B - x = 75
x = 45%

A + x = 65%
A = 20%

20% of 8,000 = 1,600.

Regards!

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Scott@TargetTestPrep: GMAT Instructor; Posts: 8138; Joined: Sat Apr 25, 2015 10:56 amFollowed by: 29 members; Location: Los Angeles, CA; Thanked: 43 times

by Scott@TargetTestPrep » Thu Oct 18, 2018 5:15 pm

BTGmoderatorDC wrote:In a town of 8,000 residents, 65 percent of all residents own a car, 55 percent own a motorcycle, and 25 percent own neither a car nor a motorcycle. How many residents own a car but not a motorcycle?

A. 800
B. 1,600
C. 2,000
D. 3,600
E. 4,400

Let's start with making the total 100 percent; thus:

Total = Car owners + Motorcycle owners - both + neither

100 = 65 + 55 - both + 25

100 = 145 - both

Both = 45

So 65 - 45 = 20 percent of the 8,000 residents own a car but not a motorcycle, and 20% of 8,000 is 0.2 x 8,000 = 1,600.

Answer: B

Scott Woodbury-Stewart
Founder and CEO
[email protected]

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