A drink is made by mixing orange juice and pineapple juice in the ratio two to five. Part a) Draw a graph which can be used to work out the amount of pineapple juice needed given the amount of orange juice. Draw your graph to show up to 10 litres of orange juice. Part b) How much pineapple juice is needed to make a drink that contains seven litres of orange juice?
We have this ratio two to five and we know that this is the orange juice to the pineapple juice. For every two parts of orange juice, you need five parts of pineapple juice. Our 𝑦-axis on this graph will represent the amount of pineapple juice needed if we’re given the amount of orange juice. Before we make a scale for our 𝑦-axis, it would be helpful to know what the range would be.
On our graph, the maximum amount of orange juice is 10 litres. We can use this 10 litres to find out what the maximum value of pineapple juice on the 𝑦-axis should be. Two times five equals 10. We can multiply five times five to find the litres of pineapple juice we need. Five times five equals 25. And that means if we have 10 litres of orange juice, we’ll need 25 litres of pineapple juice.
This tells us that we’ll need our 𝑦-axis to go up to at least 25. If we count the bigger boxes, we see that there are 15 of them. Let’s imagine you saw that 25 and you thought, “Well, we can graph by 25.” And you started your graph at five, 10, 15, 20, and 25. And while there is nothing wrong with this necessarily, it’s going to make a very small graph and we’re not utilizing all of our space.
We could use a smaller scale. Let’s see if we could count by twos. We know that there are 15 marks. This way our maximum 𝑦-value is 25 and we space that out more evenly across the grid. Now we’ve labelled our 𝑦-axis, we’re ready to start plotting some points.
First, let’s imagine that there were zero litres of orange juice. If there’s no orange juice, then there’s no reason to have any pineapple juice. Our first point would be zero, zero at the origin. We also know that if there are two parts orange, there will be five parts pineapple. When we take two litres of orange juice, we need five litres of pineapple juice. The five is going to fall out halfway between four and six at the point two, five.
We also know the point 10, 25 that we already calculated: 10 on the 𝑥-axis and halfway between 24 and 26 on the 𝑦-axis. At this point, we could go ahead and draw a line that starts at the origin zero, zero and continues through the point 10, 25. Our graph would look like this.
Part b now wants to know how much pineapple juice is needed to make a drink that contains seven litres of orange juice. If we look at the graph where we have seven litres of orange juice, we have a point here. It’s between 16 and 18, but closer to 18. However, we want to find an exact answer, not an estimate. And that means we need to use the ratio two to five to solve for the pineapple juice if we have seven litres of orange juice.
The question is what can you multiply by two to give you seven. We know that two times three equals six and two times one-half equals one. If you multiply two by 3.5, you’ll get seven. And to keep these proportions balanced, we would need to multiply five by 3.5. Five times three equals 15, five times one-half equals 2.5, 15 plus 2.5 equals 17.5.
This confirms what we saw on the graph. It would take 17.5 litres of pineapple juice to make a drink that contains seven litres of orange juice.