Bunuel wrote:
Machine A takes 2 more hours than machine B to make 20 widgets. If working together, the machines can make 25 widgets in 3 hours, how long will it take machine A to make 40 widgets?
(A) 5
(B) 6
(C) 8
(D) 10
(E) 12
Kudos for a correct solution.
VERITAS PREP OFFICIAL SOLUTION:We need to find the time taken by machine A to make 40 widgets. It will be best to take the time taken by machine A to make 40 widgets as the variable x. Then, when we get the value of x, we will not need to perform any other calculations on it and hence the scope of making an error will reduce. Also, value of x will be one of the options and hence plugging in to check will be easy.
Machine A takes x hrs to make 40 widgets.
Rate of work done by machine A = Work done/Time taken = \(\frac{40}{x}\)
Machine B take 2 hrs less than machine A to make 20 widgets hence it will take 4 hrs less than machine B to make 40 widgets. Think of it this way: Break down the 40 widgets job into two 20 widget jobs. For each job, machine B will take 2 hrs less than machine A so it will take 4 hrs less than machine A for both the jobs together.
Time taken by machine B to make 40 widgets = x – 4
Rate of work done by machine B = Work done/Time taken = \(\frac{40}{(x - 4)}\).
We know the combined rate of the machines is 25/3
So here is the equation:
\(\frac{40}{x} + \frac{40}{(x - 4)} = \frac{25}{3}\)
The steps till here are not complicated. Getting the value of x poses a bit of a problem.
Notice here that that the right hand side is not an integer. This will make the question a little harder for us, right? Wrong! Everything has its pros and cons. The 3 of the denominator gives us ideas for the values of x (as do the options). To get a 3 in the denominator, we need a 3 in the denominator on the left hand side too.
x cannot be 3 but it can be 6. If x = 6, \(\frac{40}{(6 - 4)} = 20\) i.e. the sum will certainly not be 20 or more since we have \(\frac{25}{3} = 8.33\) on the right hand side.
The only other option that makes sense is x = 12 since it has 3 in it.
\(\frac{40}{12} + \frac{40}{(12 - 4)} = \frac{10}{3} + 5 = \frac{25}{3}\)
Answer (E)If we did not have the options, we might have tried x = 9 too before landing on x = 12. Nevertheless, these calculations are not time consuming at all since you can get rid of the incorrect numbers orally. Making a quadratic and solving it is certainly much more time consuming.
Another method could be to bring 3 to the left hand side to get the following equation:
\(\frac{120}{x} + \frac{120}{(x - 4)} = 25\)
This step doesn’t change anything but it helps if you face a mental block while working with fractions. Try to practice such questions using these techniques – they will save you a lot of time.
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