If a can do the work in x days and b in y days, how long will they finish the job working together

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(Last Updated On: January 20, 2020)

Problem Statement: ME Board April 1995

If A can do the work in “x” days and B in “y” days, how long will they finish the job working together?

Problem Answer:

Together, they will finish the job in n = xy/(x+y).

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A does the work in x days. B does the work in y days. the basic formula is R*T = 1 where 1 represents the job that has to be done. T for worker A is equal to x days. T for worker B is equal to y days. formula for worker A is: R.1*x = 1 formula for worker B is: R.1*y = 1 For worker 1, R.1 is equal to 1/x. For worker 2, R.2 is equal to 1/y When they work together their rates are additive. This means that the basic formula becomes: (R.1 + R.2)*T = 1 R.1 is the rate for worker 1. R.2 is the rate for worker 2. T is the overall time it takes to do the job in days. Since R.1 = 1/x and R.2 = 1/y, then the formula becomes: (1/x + 1/y) * T = 1 if we solve for T, we get: T = 1 / (1/x + 1/y) That's the general formula. to see if the general formula works, then let's try some real numbers. Worker 1 can do the in 5 days. Worker 2 can do the job in 10 days. The time required to do the job together would then be equal to 1 / (1/5 + 1/10). simplify this equation to get: T = 1 / (3/10) solve for T to get: T = 10/3 days. let's see if that's true. worker 1 can do 1/5 of the job in 1 day because it takes him 5 days to do the job alone. worker 2 can do 1/10 of the job in 10 days because it takes him 10 days to do the job alone. in 10/3 days, worker 1 can do 1/5 * 10/3 = 10/15 of the job. in 10/3 days, worker 2 can do 1/10 * 10/3 = 1/3 of the job. since 1/3 of the job is equivalent to 5/15 of the job, then we get: worker 1 finishes 10/15 of the job in 10/3 days and worker 2 finishes 5/15 of the job in 10/3 days. together they have completed 10/15 + 5/15 of the job which is equal to the whole job. the formula works.