The number of hours 𝑛 needed for carrying out a certain task varies inversely with the number of workers who carry out the task. If the task is carried out by 23 workers in 35 hours, what is the time needed for 115 workers to carry out the task?
Okay. We can now look at how to solve this problem. Well it’s actually a proportionality problem. And there’re actually kind of three areas that shows that is proportionality. The first, it talks about something varying either directly or inversely. So in this case, it’s inversely. The next, is that it actually has a pair of values that we can use to find the proportionality constant which we’ll talk about in a bit. So in this case, 23 workers and 35 hours. And then finally, they will always ask you to find one of the variables. So in this case, we need to find what the time is if we have a 115 workers.
I think the first thing we’d have a look at is actually “inversely”. What does it mean? So inversely proportional, and I think the best way to identify this is by showing you using graphs. First thing, we have directly proportional. And as you can see, this means that as one thing rises, so does the other. So it could be, as the amount of product that you buy, the amount of money spent increases. So that’s when this is directly proportional. But on the right, we can see an inversely proportional. An idea of that in our graph. And it shows actually as one thing increases, the other decreases. So this one here could be, for instance, you could look at something like the value of the car as the older it gets. So in general, unless we get a classic car, a car’s price will decrease the older it gets. So that’s what inversely proportional is. And we’re gonna have a look at solving this problem now.
So we can look at this. We can say that 𝑛 is inversely proportional to 𝑤. So this is how we’d write it using the proportionality sign, where 𝑛 is the number of hours and 𝑤 is the number of workers. But this again is no use to us. So we need to put it into an equation. And to do that, we’ll introduce the proportionality constant. So now we’ve introduced the proportionality constant or the constant of proportionality, which is 𝑘. So we’ve got that 𝑛 is equal to 𝑘 over 𝑤. Right now, the first step with these kinds of problems, after we’ve made the equation, is to actually put in the values that we have to find 𝑘. So always remember to find 𝑘. Most values we’re gonna substitute in are the 23 workers and the 35 hours which gives us the equation 35 is equal to 𝑘 over 23. Great, so now we can actually find that 𝑘 because we got two values which can help us to find 𝑘. To do that, we’re gonna multiply each side by 23. And this gives us 805 is equal to 𝑘. Or how I’ve written it, 𝑘 is equal to 805.
Fantastic! We found 𝑘. We found the proportionality constant. So now we can look at solving the problem and finding the missing value. Now in order to solve the problem and find the missing time, we need to use the value of 𝑘 that we found which is 805 and put that back into our original equation to form a new equation which can be really very useful, which gives us the equation 𝑛 equals 805 divided by 𝑤. Great, so now all we need to do is substitute the value we have for the number of workers into this equation. And then we can actually find out what the time needed is. This gives us 𝑛 is equal to 805 divided by 115. So then we work that out. And it gives us a final value of 𝑛 of seven hours.
Okay, fantastic! We can now solve the problem. So we found out that the time needed for 115 workers to carry out the task is seven hours. And now I’m just gonna quickly recap, just to kind of give us an idea of the key points of this question. First of all, we identified that it’s a proportionality question. And we did that. And we saw that it was inversely proportional which means that one thing would decrease as the other increases. We next set up an equation which was 𝑛 equals 𝑘 over 𝑤 because it’s inversely proportional. So that one over, it’s inversely proportional. Then we find 𝑘 which is the proportionality constant. So we need to find 𝑘. And we do that by substituting in the pair of values we have. And then that finds 𝑘. And once we have 𝑘, we set up our new equation and substitute in the value that we have for the number of workers. And that’ll find us our final answer.