Video Transcript
A copy center is printing leaflets for a local school. The time it takes for the job to be completed varies inversely with the number of copiers used. And if four copiers are used, the job will take three-quarters of an hour. How many copiers are required to complete the job in half an hour?
Since we’re told that the time it takes for the job to be completed varies inversely with the number of copiers used, let’s remind ourselves what this means. Suppose we have a pair of variables 𝑦 and 𝑥. If 𝑦 varies inversely with 𝑥, in other words, if 𝑦 is in inverse proportion to 𝑥, we can write that 𝑦 is proportional to one over 𝑥. The corresponding equation we can then write is 𝑦 equals 𝑘 over 𝑥 for some real constant 𝑘. And we call that the constant of variation or the constant of proportionality.
So let’s define the two variables in our question. We’ll let ℎ be the number of hours that it takes for a job to be completed. Then, we’ll let 𝑛 be the number of copiers used. This means ℎ is proportional to one over 𝑛, which gives us the equation ℎ equals 𝑘 over 𝑛. Our job will be to use the information in the question to find the value of 𝑘. Once we know the value of 𝑘, we can find the number of copiers required to complete the job in half an hour. First, we’re told that if four copiers are used, the job will take three-quarters of an hour. In other words, when 𝑛 is equal to four, ℎ is three-quarters.
By substituting 𝑛 is equal to four and ℎ is equal to three-quarters into our earlier equation, we find that three-quarters equals 𝑘 over four and either by comparing the numerators or multiplying through by four, we find 𝑘 is equal to three. So the equation that links the number of hours that it takes for a job to be completed and the number of copiers is ℎ equals three over 𝑛.
Now, the question wants us to find the number of copiers required to complete the job in half an hour. So we substitute ℎ equals one-half into our equation. When ℎ is a half, our equation is a half equals three over 𝑛. Multiplying both sides of this equation by 𝑛, and we get 𝑛 over two equals three. And then we multiply by two, so we find 𝑛 is equal to six. So if the job needs to be completed in half an hour, we know that we need to use six copiers.
Now, actually, this makes a lot of sense if we think about things logically. It takes four copiers 45 minutes to complete the job. Let’s imagine now we halve the number of copiers used. If we do that, it’s now going to take double the amount of time. The less copiers, the longer it will take. So two copiers are going to take 90 minutes. With this in mind, imagine we now have six copiers. We’ve tripled the number of copiers we have. So we can reduce the amount of time it will take by a factor of one third. Once again, we find six copiers take 30 minutes.
