Machine A and Machine B can produce 1 widget in 3 hours work

A+B = 1/3

2A + B = 1/2

Hence A=1/6

ANS = 6

\(\frac{1}{A}+\frac{1}{B}=\frac{1}{3}\)
so \(\frac{AB}{A+B}=3\)
AB=3A+3B

if A rate were doubled then
\(\frac{2}{A}+\frac{1}{B}=\frac{1}{2}\)
\(\frac{2B+A}{AB}=\frac{1}{2}\)
AB=4B+2A

Then AB must be the same in both equations
3A+3B=4B+2A
B=A

Now substitute in first equation
\(\frac{1}{A}+\frac{1}{A}=\frac{1}{3}\)
\(\frac{2}{A}=\frac{1}{3}\)
A=6

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 machine A produce 1 widget in a hours, and machine B produce 1 widget in b hours. It follows that machine A's rate is 1/a widgets/hour, and machine B's rate is 1/b widgets/hour.

We are told that the two machines together can produce 1 widget in 3 hours, thus the combined rate of the two machines is 1/3 widgets/hour. We can write:

\(\Rightarrow\) 1/a + 1/b = 1/3

If machine A's speed were doubled, it would take half as much time for machine A to produce 1 widget, which means machine A would produce 1 widget in a/2 hours. Thus, machine A's rate would be 1/(a/2) = 2/a widgets/hour. We are told that under this assumption, it takes the two machines 2 hours to produce 1 widget, which is the same thing as saying the combined rate of the two machines is 1/2 widgets/hour. Thus:

\(\Rightarrow\) 2/a + 1/b = 1/2

Let's subtract the equation 1/a + 1/b = 1/3 from 2/a + 1/b = 1/2:

\(\Rightarrow\) 2/a + 1/b = 1/2
\(\Rightarrow\) -(1/a + 1/b = 1/3)

\(\Rightarrow\) 1/a = 1/2 - 1/3 = 1/6

\(\Rightarrow\) a = 6

Answer: E

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