Snatches of conversation about work planning in the office drift in my direction during my lightning visits to print teaching materials and make photocopies. Person A, working alone, can translate a document in 4 hours – I overhear – while person B, working alone, can do the same job in 6 hours. So, if we get them to share the job, we’ll have the job done in 5 hours (1/2 times 4 + 1/2 times 6).
That is not exactly true. Although the answer is much less, and although lots of other factors may distort the calculation significantly in real life, and although it does not really matter that much after all, here are a couple of examples how to get it exactly right.
Example 1
Person A can do a job in 4 hours, while person B can do the same job in 6 hours. How long would it take both, working together, to do the same job?
Problems of this kind often involve figuring with the principle of 1, where 1 represents the entire job (one job). It is clear that we must somehow add the times of A and B to get this entire job. But how? Let t be the number of hours it takes A and B to complete the job working together. Person A can do the job faster, or more of it in the same time, or, a fourth of the job against a sixth of the job which person B can do in the same time (1/4 is greater than 1/6). Our equation will thus be:
t(1/4) + t(1/6) = 1
By solving the equation for t, we get t = 2.4 hours
Problem 2
We can reverse the situation while at the same time running a check on the previous problem.
It takes persons B 2 hr longer to do a job than it takes person A. If they work together, they can do the job in 2.4 hours. How long would it take each, working alone, to do the job?
Let 2.4 be the number of hours it takes A and B to complete the job working together. Person A can do 1/t of the job while person B can do 1/(t + 2) of it in the same time. So:
2.4(1/t) + 2.4[1/(t + 2)] = 1
We will get two answers, one of which is negative, hence meaningless, while the other is t = 4. This is the time it takes A to do the job. B will do it in 4 + 2 hours.
That is not exactly true. Although the answer is much less, and although lots of other factors may distort the calculation significantly in real life, and although it does not really matter that much after all, here are a couple of examples how to get it exactly right.
Example 1
Person A can do a job in 4 hours, while person B can do the same job in 6 hours. How long would it take both, working together, to do the same job?
Problems of this kind often involve figuring with the principle of 1, where 1 represents the entire job (one job). It is clear that we must somehow add the times of A and B to get this entire job. But how? Let t be the number of hours it takes A and B to complete the job working together. Person A can do the job faster, or more of it in the same time, or, a fourth of the job against a sixth of the job which person B can do in the same time (1/4 is greater than 1/6). Our equation will thus be:
t(1/4) + t(1/6) = 1
By solving the equation for t, we get t = 2.4 hours
Problem 2
We can reverse the situation while at the same time running a check on the previous problem.
It takes persons B 2 hr longer to do a job than it takes person A. If they work together, they can do the job in 2.4 hours. How long would it take each, working alone, to do the job?
Let 2.4 be the number of hours it takes A and B to complete the job working together. Person A can do 1/t of the job while person B can do 1/(t + 2) of it in the same time. So:
2.4(1/t) + 2.4[1/(t + 2)] = 1
We will get two answers, one of which is negative, hence meaningless, while the other is t = 4. This is the time it takes A to do the job. B will do it in 4 + 2 hours.
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