Three machines have equal constant work rates. It takes h + 3 hours to

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

MahmoudFawzy · Accepted Answer

let rate of 1 machine is 'r' ( = number of toys per hour)

rate of 3 machines together =  \frac{360}{h}  = 3r
rate of 2 machines together =  \frac{360}{h+3}  = 2r

rate of 3 machines / rate of 2 machines =  \frac{3r}{2r}  =  \frac{360}{h} * \frac{h+3}{360}  =  \frac{h+3}{h} 
3h = 2h + 6
h = 6

if  \frac{360}{h}  = 3r
then r = 20

D

Archit3110 · Answer

rate of 2 machines: 360/h+3
rate per machine when all 3 machines working together : 360/h

lets solve for h first
rate of 2 machines
360/h + 360/h = 360/h+3
h= 6

given rate per machine 360/h = toys per machine ; 360/6*3 = 20
IMO D

fskilnik · Answer

1\,\,{m{mach}}\,\,\, 	o \,\,\,{{?\,\,{m{toys}}} \over {1\,\,{m{hour}}}}

\left. \matrix{
 2\,\,{m{mach}}\,\,\, 	o \,\,\,{{360\,\,{m{toys}}} \over {h + 3\,\,{m{hours}}}}\,\,\,\,\,\,\, \Rightarrow \,\,\,\,\,\,1\,\,{m{mach}}\,\,\, 	o \,\,\,{{180\,\,{m{toys}}} \over {h + 3\,\,{m{hours}}}}\,\,\,\, \hfill \cr 
 3\,\,{m{mach}}\,\,\, 	o \,\,\,{{360\,\,{m{toys}}} \over {h\,\,{m{hours}}}}\,\,\,\,\,\,\, \Rightarrow \,\,\,\,\,\,1\,\,{m{mach}}\,\,\, 	o \,\,\,{{120\,\,{m{toys}}} \over {h\,\,{m{hours}}}} \hfill \cr} ight\}\,\,\,\,\,\,\, \Rightarrow \,\,\,\,\,\,{{180} \over {h + 3}} = {{120} \over h}\,\,\,\,\,\, \Rightarrow \,\,\,\,\, \ldots \,\,\,\,\,\,\, \Rightarrow \,\,\,\,\,\,h = 6

1\,\,{m{mach}}\,\,\, 	o \,\,\,{{120\,\,{m{toys}}} \over {h = 6\,\,{m{hours}}}} = {{20\,\,{m{toys}}} \over {1\,\,{m{hour}}}}\,\,\,\,\,\,\,\,\,\, \Rightarrow \,\,\,\,\,\,? = 20

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

Regards, 
Fabio.

BrentGMATPrepNow · Answer

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

Abhishek009 · Answer

(h + 3)*2 = 3h

2h + 6 = 3h

Or, h = 6

Thus we have ,

SO, Efficiency of each machine is  \frac{360}{9*2} =20 ; Answer must be (D) 20

ScottTargetTestPrep · Answer

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

MathRevolution · Answer

=>

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

Therefore, the answer is D.
Answer: D

Lipun · Answer

Work rate of each machine is constant.

2 machines working together for 'h+3' hours produce 360 toys. => 1 machine produces 180 toys in 'h+3' hours. ----(1)
3 machines working together for 'h' hours produce 360 toys. => 1 machine produces 120 toys in 'h' hours. ----(2)

From equations (1) & (2), the machine produces additional 60 toys in 3 hours. => Per hour it produces 20 toys.

vn049636 · Answer

W = R*T

Let r be rate of each machine

360 = 2/r *(h+3) and 360 = 3/r *(h)

So 2h + 6 = 3h => h = 6 => three machines can produce 60 toys per hour and each machine can produce 20 toys per hour

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