GMAT math question - please help!?

Question

Machine A produces pencils at a constant rate of 9,000 pencils per hour and machine B produces pencils at a rate of 7000 pencils per hour. If the two machines together must produce 100,000 pencils and if each machine can operate for at most 8 hours, what is the least amount of time in hours that B must operate?

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I have a pretty long convoluded way of solving this, but I'm hoping someone can show me something less time consuming. Thanks!

Answer 1

Well, start by working machine A to the max.
9,000/hr x 8 hours = 72,000

That leaves 100,000 - 72,000 = 28,000 for machine B to make.

At 7,000/hr that would be 4 hours.

Answer 2

Machine A produces pencils at a constant rate of 9,000 pencils per hour and machine B produces pencils at a rate of 7000 pencils per hour. If the two machines together must produce 100,000 pencils and if each machine can operate for at most 8 hours, what is the least amount of time in hours that B must operate?

Well together machines A and B produce 16000 pencils per hour

So in 6 hours they will produce 96,000 pencils
In another ¼ hour they will produce 4,000 more pencils combined (2,250 from machine A and 1,750 from machine B) So each machine operates simultaneously for 6¼ hours) Total machine time = 12.5 hours

If you seek the least amount of time for machine B then machine A must operate for 8 hours producing 72,000 and the other 28,000 will need to be produced on machine B and that will take 4 hours.

All up machine B will work for 4 hours and the complete task will take a minimum of 8 hours (machine A) plus 4 hours (machine B ie 12 hours compared to 12.5 hours by working the two machines in the 1st case

Least total time

Let machine A operate for a hours and machine B for b hours

Then total time T = (a + b) hours

And 9000a + 7000b = 100000
ie 9a + 7b = 100
So b = (100 - 9a)/7

So T = a + (100 - 9a)/7
= (100 - 2a)/7

So minimum total time occurs when a is maximum ie when a = 8
whence b = 4 and T = 12

So minimum total machine of 12 hours time occurs when machine A works 8 hours and machine B works 4 hours.

Answer 3

Machine A produces pencils at a constant rate of 9,000 pencils per hour and machine B produces pencils at a rate of 7000 pencils per hour. If the two machines together must produce 100,000 pencils and if each machine can operate for at most 8 hours, what is the least amount of time in hours that B must operate?

Well together machines A and B produce 16000 pencils per hour

So in 6 hours they will produce 96,000 pencils
In another ¼ hour they will produce 4,000 more pencils combined (2,250 from machine A and 1,750 from machine B) So each machine operates simultaneously for 6¼ hours) Total machine time = 12.5 hours

If you seek the least amount of time for machine B then machine A must operate for 8 hours producing 72,000 and the other 28,000 will need to be produced on machine B and that will take 4 hours.

All up machine B will work for 4 hours and the complete task will take a minimum of 8 hours (machine A) plus 4 hours (machine B ie 12 hours compared to 12.5 hours by working the two machines in the 1st case

Least total time

Let machine A operate for a hours and machine B for b hours

Then total time T = (a + b) hours

And 9000a + 7000b = 100000
ie 9a + 7b = 100
So b = (100 - 9a)/7

So T = a + (100 - 9a)/7
= (100 - 2a)/7

So minimum total time occurs when a is maximum ie when a = 8
whence b = 4 and T = 12

So minimum total machine of 12 hours time occurs when machine A works 8 hours and machine B works 4 hours.

Answer 4

Since machine A produces more pencils per hour, we should run machine A as long as possible, which is 8 hours.

If x stands for the least amount of time that B must operate, we can write a balanced equation,

9,000(8) + 7,000x = 100,000

Solving for x, x = 4 hours.

Actually, most of GMAT math questions can be done by mental math, like this one.

Subtract 72,000 from 100,000 to get 28,000, and then divide 28,000 by 7,000 to get 4 hours.

Answer 5

the longer "A" operates, the shorter "B" must operate. So if "A" can operate for 8 hrs that's 72,000 pencils. That leaves 28,000 pencils. It takes "B" 4 hrs to make that many pencils and that's the shortest amount of time "B" can operate for!

Answer 6

4 hours.

Machine A can make 72,000 in 8 hours (9,000 x 8). 100,000 less 72,000 is 28,000. Divided by 7,000 (machine B) is 4.

Answer 7

100000/9000=11 1/9
9000*8=72000
28000/7=4
so B must operate for 4 hours (least)

Answer 8

You probably have a bright future in management because you don't seem shy about asking people to do your work; even resorting to the bullshit about having solved it without giving any details. The more I'm around management types the more disgusted I get with them.