A group of scouts receives a donation of 1,000 dollars to fund places on an international jamboree. The amount each scout receives for their trip varies inversely with the number of scouts from the group going to the jamboree. Write an equation for 𝑚, the amount each scout receives, in terms of 𝑛, the number of scouts from the group who are going to the jamboree. And if 25 scouts from the group are going to the jamboree, how much will each scout receive from the donation?
We’re told that the amount of the donation of 1,000 dollars that each scout receives varies inversely to the number of scouts. So, to write an equation for 𝑚 in terms of 𝑛, let’s recall what we mean when we talk about items varying inversely to one another. Suppose we have variables 𝑦 and 𝑥. If 𝑦 varies inversely to 𝑥, that means that when 𝑥 increases, 𝑦 decreases. We write this as 𝑦 is proportional to one over 𝑥, and the corresponding equation we use to describe this is 𝑦 equals 𝑘 over 𝑥. 𝑘 is a constant, and we call it the constant of variation or the constant of proportionality.
Now, in this question, we’re told that 𝑚 and 𝑛 are inversely proportional to one another. This means 𝑚 is directly proportional to one over 𝑛 or 𝑚 equals 𝑘 over 𝑛. In order to write an equation for 𝑚, we need to work out the value of 𝑘. Now, we can use a little bit of logic and what we know about sharing to answer this, or we can use some starting figures. Let’s think about what those starting figures could be.
For instance, imagine we had one scout. It follows that that one scout would receive all of the money, so we can substitute 𝑚 equals 1,000 and 𝑛 equals one into our equation. So, 1,000 equals 𝑘 over one, meaning 𝑘 is simply equal to 1,000. And so, our equation is 𝑚 equals 1,000 over 𝑛. But of course, this makes a lot of sense if we think slightly logically. If we’re sharing 1,000 dollars between 𝑛 scouts, we know that sharing means divide, so it would make sense that each scout gets 1,000 divided by 𝑛 dollars. Either way, the equation for 𝑚 in terms of 𝑛 is 𝑚 equals 1,000 over 𝑛.
The second part of this question asks us how much money each scout receives if 25 of them go on the trip. So, to answer this, we’re going to use the equation we calculated in the first part of this question. But we’re going to let 𝑛 be equal to 25. So, 𝑚 in dollars is going to be the amount each scout will receive. It’s 1,000 divided by 25 which, of course, is equal to 40. So, the amount each student will receive is 40 dollars.