Machine A and Machine B can produce 1 widget in 3 hours working together at their respective constant rates. If Machine A's speed were doubled, the two machines could produce 1 widget in 2 hours working together at their respective rates. How many hours does it currently take Machine A to produce 1 widget on its own?

A. 1/2
B. 2
C. 3
D. 5
E. 6

Let a be the number of hours it takes Machine A to produce 1 widget on its own. Let b be the number of hours it takes Machine B to produce 1 widget on its own.

The question tells us that Machines A and B together can produce 1 widget in 3 hours. Therefore, in 1 hour, the two machines can produce 1/3 of a widget. In 1 hour, Machine A can produce 1/a widgets and Machine B can produce 1/b widgets. Together in 1 hour, they produce 1/a + 1/b = 1/3 widgets.

If Machine A's speed were doubled it would take the two machines 2 hours to produce 1 widget. When one doubles the speed, one cuts the amount of time it takes in half. Therefore, the amount of time it would take Machine A to produce 1 widget would be a/2. Under these new conditions, in 1 hour Machine A and B could produce 1/(a/2) + 1/b = 1/2 widgets. We now have two unknowns and two different equations. We can solve for a.

The two equations:
2/a + 1/b = 1/2 (Remember, 1/(a/2) = 2/a)
1/a + 1/b = 1/3

Subtract the bottom equation from the top:
2/a – 1/a = 1/2 – 1/3
1/a = 3/6 – 2/6
1/a = 1/6
Therefore, a = 6.

The correct answer is E.

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My question is, how do you know that if Machine A's speed doubles, it would take both machines 2 hours to produce 1 widget? Help!

I think this is stated in the problem.

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Ben Ku
Instructor
ManhattanGMAT

Let a be the Machine A's speed and let b be the Machine B's speed.

Machine A and Machine B can produce 1 widget in 3 hours working together at their respective constant rates
---> 1/a + 1/b = 3

If Machine A's speed were doubled, the two machines could produce 1 widget in 2 hours working together at their respective rates
---> 1/2a + 1/b = 2

Solving these two equations, I got a = 1/2
So, the number of hours it takes Machine A to produce 1 widget is 2.

Could you pls explain why my approach is wrong? Thank you.

Another approach for this problem, basing on the same logic.

a and b are speeds of Machine A and B, respectively.

We have two following equations:
a + b = 1/3
2a + b = 1/2

a = 1/6.
The number of hours is 6.

I'm so confused between these two approaches, bcuz it seems to me that the logic is the same, basing on Rates to find the solution.

A's rate is doubled; that gives us 2/a for its rate rather than 1/(2a). Let me know if you need any further clarification..

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Tim Sanders
Manhattan GMAT Instructor

Hi Tim,

Please let me know how does it become to 2/a. I solved with 1/2a. I am always confused with these rate & work question.

Please help.

Thanks,
Abhishek

Abhishek,
Let's use some real numbers to illustrate where you went wrong.

Let's assume that I can clean my house in 4 hours. Thus, my rate is 1/4.

If I become twice as fast at cleaning my house, I can now clean it in 2 hours. My new rate is 1/2.

How do I go from 1/4 to 1/2? I multiply by 2 in the numerator. 1/4 * 2 = 2/4 or 1/2.

If I multiply by 2 in the denominator, I get 1/4 * 1/2 = 1/8. I am now saying that I clean my house in 8 hours, meaning that I became half as fast. I went the wrong way.

Hope this makes more sense!

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Jamie Nelson
ManhattanGMAT Instructor