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Re: Machine A, B, and C, working together at their respective constant
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12 May 2021, 10:02
In this question on rates, the solution becomes easier if we take some simple variables to represent the respective rates of the machines A, B and C. This way, an equation can be formed using these variables which can then be solved to determine the values of the variables and hence the answers.
Let Machine A work at the rate of a units/per hour, machine B at b units/hour and machine C at c units/hour.
Let the total work representing the job be 6x units. Since this work is being completed in 6 hours by all three machines working together,
Combined rate = \(\frac{Total work }{ Total time}\) = \(\frac{6x }{ 6}\) = x units per hour.
The cardinal equation to solve questions on Rates (Time & Work) is Work = Time x Rate. Remember, if nothing else works, this equation will always work for you.
Back to the question now!
Since the combined rate of the three machines is a+b+c, we can say a+b+c = x. We also know,
a = 2b and c = (1/2) b.
Substituting these values in the equation, we have, 2b + b + \(\frac{b}{2}\) = x. On solving, we get, b = \(\frac{2x }{7}\) and therefore c = \(\frac{x}{7}\).
Since we want B and C to complete the job, combined rate of B and C = \(\frac{2x}{7}\) + \(\frac{x}{7}\) = \(\frac{3x}{7}\) units per hour.
Therefore, time taken to complete the job (of 6x units) = \(\frac{Work }{ Rate}\) = 6x / (3x/7) = 14 hours.
The correct answer option is A.
Notice how I took the total work as 6x units (instead of the uber popular 1 unit) so that I didn’t have to deal with a lot of fractions in my calculations.
Hope that helps!
Aravind BT