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

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

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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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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.

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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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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

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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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=>

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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