# Video: AQA GCSE Mathematics Higher Tier Pack 1 • Paper 2 • Question 5

AQA GCSE Mathematics Higher Tier Pack 1 • Paper 2 • Question 5

03:12

### Video Transcript

In a stationary box, the ratio of black pens to blue pens is three to five. The ratio of blue pens to pencils in the box is four to seven. Show that there are more pencils than pens in the box.

To answer this question, we’re going to need to find a relationship between the number of pencils in the box and the total number of black and blue pens in the box. Let’s consider what the information in the question is telling us.

The ratio of black pens to blue pens is three to five. For every three black pens there are in the box, there are five blue pens. The ratio of blue pens to pencils is four to seven. For every four blue pens in the box, there are seven pencils.

Notice how each ratio has a number that represents a proportion of blue pens. We need to find a way to scale both of these ratios up so that the number of parts of blue pens is the same in each ratio. We can scale these ratios up by finding the lowest common multiple of five and four. That’s the smallest number in both of five and the four times tables. The smallest number in both of five and the four times tables is 20. So the lowest common multiple of five and four is 20. So we’re going to scale both of these ratios up so that the number of parts representing blue pens is 20.

In the ratio of black pens to blue pens, we can scale this up by multiplying by four. Remember, to show that our ratio is equivalent, we have to do the same to both sides. Three multiplied by four is 12. So another way of writing the ratio of black pens to blue pens is 12 to 20.

To scale up the ratio representing blue pens and pencils, we multiply it by five. Remember, we do the same to both sides of our ratio. And seven multiplied by five is 35. So we can see that the ratio of blue pens to pencils is 20 to 35.

Now that the number of parts representing blue pens is the same, we can combine these ratios. And we can say that the ratio of black pens to blue pens to pencils is 12 to 20 to 35.

We said we need to find a way to compare the proportion of pencils and the total number of pens in the box. To do this, we can combine the number of parts representing black pens and blue pens. And that will give us the relationship between the number of parts representing pens and the number representing pencils. 12 plus 20 is 32. So for every 32 pens in the box, there are 35 pencils. 35 is a larger number than 32, so there must be more pencils than pens in the box.

Now it doesn’t actually matter that we don’t know the exact number of pens and pencils that are in the box. Yes, there might be 32 pens and 35 pencils. However, the ratio represents the relationship. For every 32 pens, there are 35 pencils. We could double this and say that, for every 64 pens, there are 70 pencils. No matter what we do though, the number of pens will always be smaller than the number of pencils. And our answer still stands. There are more pencils than pens.