# Video: KS2-M19 • Paper 3 • Question 23

The distance from point 𝑃 to point 𝑅 is 800 meters. The distance from point 𝑃 to point 𝑄 is 4 times the distance from point 𝑄 to point 𝑅. Olivia says, it’s 600 meters from point 𝑃 to point 𝑄. Explain why Olivia is not correct.

06:36

### Video Transcript

The distance from point 𝑃 to point 𝑅 is 800 meters. The distance from point 𝑃 to point 𝑄 is four times the distance from point 𝑄 to point 𝑅. Olivia says, it’s 600 meters from point 𝑃 to point 𝑄. Explain why Olivia is not correct.

In this problem, we’re given a line. And the points on the line we can see are labelled 𝑃, 𝑄, and 𝑅. And the first piece of information that we’re given is that the whole distance from one end to the other — in other words, from point 𝑃 to point 𝑅 — is 800 meters. The next piece of information is really important to understand. We’re told the distance from point 𝑃 to point 𝑄, that’s the distance we’ve labelled in pink, is four times the distance from point 𝑄 to point 𝑅, which is the distance we’ve labelled in blue.

To help us understand problems like this, when we’ve got one amount that’s several times another amount, we could sketch a bar model. They’re really helpful. We can use this bar to represent the distance from point 𝑄 to point 𝑅. And then, we need to draw another four of these to represent the distance 𝑃 to 𝑄, because we know that that’s four times the distance from point 𝑄 to 𝑅. So what have we got? We’ve got a bar model that’s split up into five equal pieces. And, of course, we know that the length of the bar model represents the total length, which is 800 meters.

Now Olivia gives a wrong statement. And we know she’s wrong because we’re told to explain why she’s not correct. Olivia thinks that the distance from 𝑃 to 𝑄 — in other words, these four parts of the bar model — are worth 600 meters. There are two ways to show that she’s wrong. Firstly, we can use Olivia’s idea to help us think about what the length should be. We know that each of these bars between 𝑃 and 𝑄 are the same as the bar between 𝑄 and 𝑅. So if Olivia is correct and the distance from 𝑃 to 𝑄 is 600, we can divide 600 by four to find out the length of one bar. 600 divided by two, or split in half, is 300. And if we split 300 in half again, that’s the same as dividing by four. The answer is 150. So Olivia must think that each bar is worth 150 meters.

If the distance between 𝑄 and 𝑅 is 150 meters, then we just need to multiply this by five to find out the total length, which should be 800 meters. Zero times five equals zero. Five fives are 25, and one five is five plus the two that we’ve exchanged equals seven. If Olivia was correct, then the total distance should be 750 not 800. So this is one reason why we know Olivia is not correct. Let’s write this in a sentence. If the distance from 𝑃 to 𝑄 was 600 meters long, then the distance from 𝑄 to 𝑅 would be 600 divided by four, which is 150 meters. 150 times five equals 750 meters. But the total distance should actually equal 800 meters.

We did say there was a second way of explaining why Olivia is not correct. And to do that, we need to think about the total distance of 800 meters. If the distance between 𝑃 and 𝑅 is 800 meters, which it is, then we can work out the distance from 𝑃 to 𝑄 and prove that Olivia is not correct. As our bar model shows the distance from 𝑃 to 𝑄 is four-fifths of 800 meters, first, we can divide 800 by five to find the value of one-fifth. A fact that we can use to help us is that if we split 800 into 10, each part is worth 80. So 800 divided by half of 10 gives us double the size. In other words, the distance of each of our parts of the bar model is 160 meters.

One-fifth of 800 is 160. But we need to find four-fifths, because we’re looking for the distance between 𝑃 and 𝑄. To help us multiply by four, to multiply by four, we can just double and then double again. 160 doubled is 320 and 320 doubled equals 640. So the actual distance between 𝑃 and 𝑄 is 640 meters not 600 meters. Let’s write this in sentence form as another possible answer.

So we’ve found two possible ways to show Olivia that she’s not correct. The first possible answer, we look at Olivia’s answer, and we show that the total distance doesn’t equal 800 meters. If the distance from 𝑃 to 𝑄 was 600 meters long, then the distance from 𝑄 to 𝑅 would be 600 divided by four, which equals 150 meters. 150 meters multiplied by five equals 750 meters. But the total distance should actually equal 800 meters. In other words, Olivia’s number doesn’t add up.

In our second possible explanation, we work out the actual distance from 𝑃 to 𝑄. And we show that this is not the same as what Olivia says. The total distance is 800 meters. The distance from 𝑃 to 𝑄 is four-fifths of this amount. 800 divided by five equals 160. And 160 multiplied by four equals 640. So the distance from 𝑃 to 𝑄 is actually 640 meters long. So there we have two different possible answers to help explain why Olivia is not correct.