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### Ratio of Rates and Work

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Date: 06/06/98 at 20:38:49
From: Eugene Ni
Subject: Maths Problem: Ratio and Proportion

Dear Dr. Math,

Here's the question:

Four skilled workers can do a job in 5 days. Five semi-skilled workers
can do the same job in 6 days. How long does it take 1 skilled and
2 semi-skilled workers to do the job together?

I've tried this question several times, using inverse proportion, but
it doesn't work out.

Thanks,
Eugene
```

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Date: 06/06/98 at 20:54:04
From: Doctor Gary
Subject: Re: Maths Problem: Ratio and Proportion

Let's look at each type of worker separately, with the goal of
determining what fraction of a job one such worker will do in one day:

4 skilled workers can do a job in 5 days

Since it takes them 5 days, they do 1/5 of the job in each day. Since
there are 4 of them, each does 1/20 of the job in one day.

5 semi-skilled workers can do the same job in 6 days

Since it takes them 6 days, they do 1/6 of the job in each day. Since
there are 5 of them, each does 1/30 of the job in each day.

So the rate of work for a skilled worker is 1/20 of the job per day,
and the rate of work for a semi-skilled worker is 1/30 of the job per
day. Since we have 1 skilled worker and 2 semi-skilled workers, the
combined rate of work is:

1/20 + 1/30 + 1/30 = (3+2+2)/60 = 7/60 of the job per day

Since rate of work is defined as the quotient of work divided by time:

r  =  w/t

time must be the quotient of work divided by rate:

t  =  w/r

When we divide 1 job by 7/60 of the job per day, the answer is 60/7 of
a day, or 8 and 4/7 days.

- In 60/7 days, the skilled worker will do (60/7)(1/20), or 3/7,
of the job.

- In 60/7 days, each semi-skilled worker will do (60/7)(1/30),
or 2/7, of the job. So two semi-skilled workers will do 4/7 of
the job.

Together, the whole team will do 7/7 of the job in 60/7 days.

-Doctor Gary,  The Math Forum
Check out our web site! http://mathforum.org/dr.math/
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Associated Topics:
Middle School Ratio and Proportion
Middle School Word Problems

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