Ten workers perform one job in five days

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Question from Kenneth:

Hello:

Here is my question:

If ten workers perform one job in five days, one person performs one job in how many days?

Here is the calculation that I used:
(10 workers X 1 job X 5 days)/(1 person X 1 job X ? days)

The above equals 50/1, and the answer is 50 days because 50/1 = 50. In this calculation I can determine any number of workers or days if the number of jobs remains the same as that in the group of factors from the numerator (10 workers X 1 job X 5 days), that is 1 job. Here is another example to help clarify: (10 workers X 1 job X 5 days)/(? workers X 1 job X 10 days) This equals 50/10. The answer is 5 workers. So, if 10 workers can perform 1 job in 5 days, 5 workers can perform 1 job in 10 days.

Now, if I replace "1 job" from (10 workers X 1 job X 5 days) with a different number, for example, 4 jobs, this amount will prevent the calculation from producing the correct answer.

Here is an example: (10 workers X 1 job X 5 days)/(10 workers X 4 jobs X ? days) Mathematically, the calculation works, but the answer, 1.25 days, is not correct, if I'm not mistaken. If 10 workers can perform 1 job in 5 days, they cannot, by working at the same rate, perform 4 jobs in 1.25 days.

Can you explain, with a simple explanation, why the number representing the jobs in this calculation needs to be the same in the group of factors in both the numerator and in the denominator in order to provide the correct answer?

I thank you sincerely for your help and assistance!

Hi Kenneth,

In the example you have you look at the fraction

(10 workers × 1 job × 5 days)/(1 person × 1 job × ? days)

and conclude that ? = 50 days. You conclude ? = 50 in order that the numerator and denominator have the same value, that is

10 workers × 1 job × 5 days = 1 person × 1 job × ? days

The units "workers × job × days" don't make sense to me but in the equation above "1 job" appears on both sides so I can cancel it and get

10 workers × 5 days = 1 person × ? days

Now the units make sense to me. The statement you started with "ten workers perform one job in five days" can be restated as "one job takes 10 × 5 = 50 worker-days" where a worker-day is the amount of work that one worker can accomplish in one day. Now it's clear that one worker will require 50 days to complete the job. At the same rate if you have 4 jobs to complete that will take 4 × 50 = 200 worker-days and thus with 10 workers will require 20 days.

I hope this helps,
Harley

Kenneth replied

I have a follow-up question to ask.

In you reply you indicated the following: 'The units "workers × job × days" don't make sense to me'
Can you explain why the units from the calculation do not make sense to you?
(10 workers X 1 job X 5 days)/(1 person X 1 job X ? days)

I thank you for your follow-up reply, and for your first reply. I did find it helpful!

Kenneth,

What I meant was that I didn't see how the product workers × job × days had any relevance to the problem. I didn't immediately see why you were getting the correct answer form this calculation. The key was your question "..why the number representing the jobs in this calculation needs to be the same in the group of factors in both the numerator and in the denominator in order to provide the correct answer?" If the number of jobs is the same in both the numerator and denominator then you can cancel the job number and then the product is workers × days. This I saw does have relevance to the problem as I explained in my earlier response.

I hope this helps,
Harley

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