### Video Transcript

There is a directly proportional relationship between the distance a truck travels and the fuel it needs to cover the distance. If a truck needs 26 litres of fuel to cover a distance of 156 kilometres, what is the amount of fuel needed for the truck to cover a distance of 360 kilometres?

Now the first important thing in this question is the fact that we’re told it’s a directly proportional relationship between the distance travelled and the fuel needed to cover the distance. And we’re also told that we need 26 litres of fuel to cover a distance of 156 kilometres.

Let’s define a couple of variables then. Let 𝑑 be the distance that we’re covering in the truck in kilometres. And let 𝑓 be the amount of fuel that we need to cover that distance in litres. Now there’s a special symbol that we can use to represent a directly proportional relationship. And we can say that 𝑑 is directly proportional to 𝑓.

In a directly proportional relationship, the value of one variable is a simple multiple of the value of the other variable. And we can represent that in an equation like this. 𝑑 is equal to some constant number times 𝑓.

And the question told us that when we use 26 litres of fuel, in other words, when 𝑓 is equal to 26, then we can travel a distance of 156 kilometres. So 𝑑 is 156. So we can substitute those values in for 𝑓 and 𝑑. And we see that the 156 is equal to 𝑘 times 26.

Now I can divide both sides of that equation by 26 to work out what 𝑘 is equal to. If I divide the left-hand side, I’ve got 156 over 26. And if I divide the right-hand side, 𝑘 times 26 divided by 26 is just 𝑘.

Now we can simplify 156 over 26. For example, both the numerator and the denominator are divisible by two. So it’s equivalent to 78 over 13. But 78 is divisible by 13 as well. So if I divide the numerator and the denominator by 13, I get six over one. In other words, 𝑘 is equal to six.

Another way simplifying that would be to recognise that 26 is 25 plus one. And we know that four times 25 is 100. So six times 25 is 150. Six times one is six, so six times 26 is 150 plus six, 156. So if six times 26 is 156, then 156 divided by 26 is six. So whichever way you simplify, we know that 𝑘 is equal to six.

Now we know that 𝑘 is equal to six. We can replace 𝑘 in this equation to get this equation. 𝑑 is equal to six times 𝑓. And dividing both sides of that by six, we know that 𝑓 is equal to 𝑑 divided by six.

Now that second form is gonna be particularly useful because our question asked us to find out the amount of fuel, 𝑓, that we need in order to cover a distance of 360 kilometres. So when 𝑑, the distance, is 360, then 𝑓, the amount of fuel we need, is 360 divided by six. And sixes go into 36 six times. So they go into 360 60 times. So our answer is that we need 60 litres of fuel to cover a distance of 360 kilometres.

Now just before we go, let’s look at a completely different way of tackling this question. We’re told that there’s a directly proportional relationship between distance and fuel. And we’re told that we can achieve a distance of 156 kilometres with 26 litres of fuel. Now we want to know how many litres of fuel we’re gonna need to cover 360 kilometres.

The directly proportional relationship tells us that if we travelled twice as far, we’re going to use twice as much fuel, for example. So if we can work out what we need to multiply 156 by to get 360, then we can use the same multiplier on the litres to work out how much fuel we’ll need. But 360 isn’t a nice easy multiple of 156. So we’re gonna break it down into a two-stage process.

156 can be turned into one by dividing by itself, 156. And one can be turned into 360 by multiplying by 360. Now we can apply the same logic to the fuel. If we’re only travelling one hundred and fifty sixths of the distance, we’re only gonna use a hundred and fifty sixths the amount of fuel. So one kilometre requires 26 over 156 litres of fuel. Now I’m not gonna simplify that number just yet.

Now we can multiply it by 360 to see how much fuel we’ll need to travel 360 times as far as one kilometre, 360 kilometres. Now we can set about simplifying this expression. Now 26 goes into 26 one time, and it goes into 156 six times. So this simplifies to one over six times 360. And sixes go into six once and into 36 six times, so into 360 60 times.

Now 60 is equivalent to 60 over one. So we’ve got one times 60, which makes 60 on the numerator, and one times one, which makes one on the denominator. As we said, 60 over one is the same as 60. So either way you work it out, you get an answer of 60 litres of fuel are needed to cover a distance of 360 kilometres.