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# Word Problems/Unit Rate

Question
Hello:

Have I determined the unit rate for the following correctly?

If 4 pipes can drain a tank in 70 minutes, how long will it take 1 pipe to drain the same tank?

Since 1 pipe will take longer than 4 pipes, is the following solution correct?

(4/1) X 70 minutes/(1/4) X 4 pipes will equal 280 minutes/1 pipe.

I think that the quantities are inversely proportional to each other.
The inverse of 4 is 1/4.

Usually to find the unit rate, divide the numerator and denominator by the denominator. In this situation, this method will not produce the correct answer because of the inverse relationship.

Since the number of pipes times the amount of time is a constant, multiplying the 1st constant by a factor means we need to divide the 2nd by the same factor.

If we have ab = C, then we would have (2a)(b/2) = C, (3a)(b/3) = C, (na)/(b/n) = C, etc.

Problems like this all come down to the units involved.  This can be learned really well by looking at physics problems.

The tank has some volume measured in gallons, referred to as g.
The drainage rate of a pipe would be in gallons / minute, or r = g/m.

Since the result desired is in minutes, this can be seen to be gotten as g / r
since the units of r are g/m.  Putting this in gives g/(g/m) = 1/(1/m) = m,
and that is what is trying to be found.
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#### Scott A Wilson

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