Recently, I came across this problem:
5 moles dig 4 holes in 3 minutes. How many minutes will it take 9 moles to dig 6 holes?
I decided that I should just use proportions and calculate it.
$\implies 5$ moles dig 1 hole in $\frac{3}{4}$ minutes.
$\implies 9$ moles dig $\frac{9}{5}$ holes in $\frac{27}{20}$ minutes.
I quickly realize that this won't work, because the proportions asked are not the same.
However, I can't think of any other way to calculate this.
I figured that this question can be in the form
If $n$ objects do $m$ things in $o$ time units, how many time units can $x$ objects do $y$ things?
How can I solve this type of problem?
Thanks for your help! Your help is appreciated!
Max0815
$\endgroup$ 24 Answers
$\begingroup$You are on the right track. You are expected to compute how many mole-minutes it takes to dig one hole. Then given any number of holes, you know how many mole-minutes are required, so divide by the number of moles.
$\endgroup$ 4 $\begingroup$Just start with $x=3 \ (\text{minutes})\cdot \ldots$, where 3 is the number you start from. And $x$ is the time to dig 6 holes by 9 moles. Then build a fraction of the following pairs of numbers: $(5,9);(6,4)$
The more moles are digging the less time is needed. Thus the fraction has to be smaller than 1. That means the factor is $\frac59$.
The more holes supposed to be dug the more time is needed. Thus the fraction has to be greater than 1. That means the factor is $\frac64$.
Thus $x=3\cdot \frac59\cdot \frac64=2.5 \ (\text{minutes})$
$\endgroup$ $\begingroup$The general approach to questions of this sort is to assume that a particular animal (or person) performs a certain task at a constant rate. In this problem we suppose all moles are the same, so doubling the number of moles will double the number of holes in a given length of time; doubling the time without changing the number of moles also will double the number of holes; but doubling the time and doubling the number of moles will cause them to dig four times as many holes.
You made a mistake by multiplying all three numbers by $\frac95.$ As shown in the examples above, the three numbers do not all go up or down in proportion.
The things that do go up or down proportionally are the number of holes and the product of the number of moles and the time:
$$ holes = constant \times moles \times minutes . $$
$\endgroup$ $\begingroup$Here is an easy way to see why your logic does not work:
Suppose 1 mole digs 1 hole in 1 minute
Then by your logic, it would follow that:
2 moles dig 2 holes in 2 minutes
But that's absurd: The 2 moles get twice as much done as 1 mole in the same time period, and so the 2 moles will dig 2 holes still in that same 1 minute.
$\endgroup$
