Three machines have equal constant work rates. It takes h +

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Max@Math Revolution: Elite Legendary Member; Posts: 3991; Joined: Fri Jul 24, 2015 2:28 amFollowed by: 37 members; Location: Las Vegas, USA; Thanked: 19 times

by Max@Math Revolution » Sun Jan 13, 2019 11:53 pm

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[Math Revolution GMAT math practice question]

Three machines have equal constant work rates. It takes h + 3 hours to produce 360 toys when 2 machines work together, and it takes h hours to produce 360 toys when 3 machines work together. How many toys can each machine produce per hour when working on its own?

A. 12
B. 15
C. 18
D. 20
E. 24

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

by fskilnik@GMATH » Mon Jan 14, 2019 4:20 am

Max@Math Revolution wrote:[Math Revolution GMAT math practice question]

Three machines have equal constant work rates. It takes h + 3 hours to produce 360 toys when 2 machines work together, and it takes h hours to produce 360 toys when 3 machines work together. How many toys can each machine produce per hour when working on its own?

A. 12
B. 15
C. 18
D. 20
E. 24

$$1\,\,{\rm{mach}}\,\,\, \to \,\,\,{{?\,\,{\rm{toys}}} \over {1\,\,{\rm{hour}}}}$$
$$\left. \matrix{
2\,\,{\rm{mach}}\,\,\, \to \,\,\,{{360\,\,{\rm{toys}}} \over {h + 3\,\,{\rm{hours}}}}\,\,\,\,\,\,\, \Rightarrow \,\,\,\,\,\,1\,\,{\rm{mach}}\,\,\, \to \,\,\,{{180\,\,{\rm{toys}}} \over {h + 3\,\,{\rm{hours}}}}\,\,\,\, \hfill \cr
3\,\,{\rm{mach}}\,\,\, \to \,\,\,{{360\,\,{\rm{toys}}} \over {h\,\,{\rm{hours}}}}\,\,\,\,\,\,\, \Rightarrow \,\,\,\,\,\,1\,\,{\rm{mach}}\,\,\, \to \,\,\,{{120\,\,{\rm{toys}}} \over {h\,\,{\rm{hours}}}} \hfill \cr} \right\}\,\,\,\,\,\,\, \Rightarrow \,\,\,\,\,\,{{180} \over {h + 3}} = {{120} \over h}\,\,\,\,\,\, \Rightarrow \,\,\,\,\, \ldots \,\,\,\,\,\,\, \Rightarrow \,\,\,\,\,\,h = 6$$
$$1\,\,{\rm{mach}}\,\,\, \to \,\,\,{{120\,\,{\rm{toys}}} \over {h = 6\,\,{\rm{hours}}}} = {{20\,\,{\rm{toys}}} \over {1\,\,{\rm{hour}}}}\,\,\,\,\,\,\,\,\,\, \Rightarrow \,\,\,\,\,\,? = 20$$

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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GMATGuruNY: GMAT Instructor; Posts: 15539; Joined: Tue May 25, 2010 12:04 pmFollowed by: 1906 members; Location: New York, NY; Thanked: 13060 times; GMAT Score:790

by GMATGuruNY » Mon Jan 14, 2019 4:54 am

Max@Math Revolution wrote:[Math Revolution GMAT math practice question]

Three machines have equal constant work rates. It takes h + 3 hours to produce 360 toys when 2 machines work together, and it takes h hours to produce 360 toys when 3 machines work together. How many toys can each machine produce per hour when working on its own?

A. 12
B. 15
C. 18
D. 20
E. 24

(number of machines)(number of hours) = number of toys that are produced.

In the prompt above, the blue values are THE SAME.
Implication:
2 machines take h+3 hours to produce the same number of toys as 3 machines in 2 hours:
2(h+3) = 3h
2h + 6 = 3h
6 = h.

Since 3 machines produce 360 toys in h=6 hours, the rate for 3 machines = 360/6 = 60 toys per hour.
Since the rate for 3 machines = 60 toys per hour, the rate for each machine = 60/3 = 20 toys per hour.

The correct answer is D.

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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

by Brent@GMATPrepNow » Mon Jan 14, 2019 6:19 am

Max@Math Revolution wrote:Three machines have equal constant work rates. It takes h + 3 hours to produce 360 toys when 2 machines work together, and it takes h hours to produce 360 toys when 3 machines work together. How many toys can each machine produce per hour when working on its own?
A. 12
B. 15
C. 18
D. 20
E. 24

It takes h + 3 hours to produce 360 toys when 2 machines work together
So, during those h + 3 hours, EACH machine makes 180 toys
So, ONE machine can produce 180 toys in h + 3 hours
rate = output/time
So, we can write: rate of ONE machine = 180/(h + 3)

It takes h hours to produce 360 toys when 3 machines work together
So, during those h hours, EACH machine makes 120 toys
rate = output/time
So, we can write: rate of ONE machine = 120/h

How many toys can each machine produce per hour when working on its own?
Since we've written the rate of ONE machine in two ways, we can state the following:
180/(h + 3) = 120/h
Cross multiply to get: 180(h) = 120(h + 3)
Expand to get: to get: 180h = 120h + 360
Solve: h = 6

The first piece of information tells us that it takes h + 3 hours to produce 360 toys when 2 machines work together.
So, we now know that it takes 2 machines 6+3 hours to produce 360 toys
Simplify: it takes 2 machines 9 hours to produce 360 toys
This means it takes 1 machine 9 hours to produce 180 toys
So, 1 machine can produce 20 toys in 1 hour

Answer: D

Cheers,
Brent

Brent Hanneson - Creator of GMATPrepNow.com

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Scott@TargetTestPrep: GMAT Instructor; Posts: 8134; Joined: Sat Apr 25, 2015 10:56 amFollowed by: 29 members; Location: Los Angeles, CA; Thanked: 43 times

by Scott@TargetTestPrep » Mon Jan 14, 2019 5:43 pm

Max@Math Revolution wrote:[Math Revolution GMAT math practice question]

Three machines have equal constant work rates. It takes h + 3 hours to produce 360 toys when 2 machines work together, and it takes h hours to produce 360 toys when 3 machines work together. How many toys can each machine produce per hour when working on its own?

A. 12
B. 15
C. 18
D. 20
E. 24

Since it takes 2 machines h + 3 hours to make 360 toys, it takes 1 machine h + 3 hours to make 180 toys for a rate of 180/(h + 3).

Since it takes 3 machines h hours to produce 360 toys, it takes 1 machine h hours to make 120 toys for a rate of 120/h.

Setting our two rates equal we have:

180/(h + 3) = 120/h

180h = 120(h + 3)

180h = 120h + 360

60h = 360

h = 6

Thus, each machine can produce 120/6 = 20 toys per hour.

Answer: D

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Max@Math Revolution: Elite Legendary Member; Posts: 3991; Joined: Fri Jul 24, 2015 2:28 amFollowed by: 37 members; Location: Las Vegas, USA; Thanked: 19 times

by Max@Math Revolution » Wed Jan 16, 2019 12:02 am

=>

The work rate for each machine is given by 360 / {2(h+3)} = 180 / ( h + 3 ). Another expression for this work rate is 360 / (3h) = 120 / h.
Thus, 180 / ( h + 3 ) = 120 / h or 3 / ( h + 3 ) = 2 / h.
So, 3h = 2(h+3) and h = 6.
The sum of the work rates of the 3 machines is 360 / 6 = 60.
The work rate of each machine is 60/3 = 20 toys / hour.

Therefore, the answer is D.
Answer: D

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