Question

Ten people can dig five holes in three hours. If n people digging at the same rate dig m holes in d hours:

Answer 1

First. get the unit rate. That is, transform "Ten people can dig five holes in three hours" into "One person digs one holes in one hour."

We can say that

$\frac{5\,\mbox{holes}}{3\,\,\mbox{hours}\cdot\mbox{10\,\,\mbox{person}}}=\frac{\frac{5}{3}\,\,\mbox{holes}}{1\,\,\mbox{hour}\cdot\mbox{10\,\,\mbox{person}}}$  

(that's still for 10 people). For 1 person this rate will be 1/10th of the above, so:

$\frac{\frac{5}{3}\,\,\mbox{holes}}{1\,\,\mbox{hour}\cdot 10 \,\,\mbox{person}}=\frac{\frac{5}{30}\,\,\mbox{holes}}{1\,\,\mbox{hour}\cdot 1 \,\,\mbox{person}}$

So the unit rate is 5/30. We can now set up an equation expressing the number of holes "m" dug up by "n" people in "d" hours:

$m = \frac{5}{30}\cdot n\cdot d$

which is clearly linear in n and d.

Now we can answer the questions:

(1) Is n proportional to m when d=3? Answer: Yes

$m = \frac{5}{30}\cdot n \cdot 3 = \frac{1}{2}n\\\implies n = 2m$

which shows that n is proportional to m in this case.

(2) Is n proportional to d when m=5? Answer: No

$5 = \frac{5}{30}\cdot n \cdot d\\\implies n = \frac{30}{d}$

There is an inverse proportionality, therefore this is not proportional.

(3) Is m proportional to d when n=10? Answer: Yes

$m = \frac{5}{30}\cdot 10 \cdot d = \frac{5}{3}d$

There is a proportional relationship between m and d.