Transcript

NARRATOR:

Let's start with a 1/4 plus a 1/4. To work this out, let's say we have a pie which we divide into four pieces. If our first 1/4 in our calculation was this quarter of the pie, and we were going to add it to another quarter of the pie, if I was to eat both of those quarters, so a quarter and then another quarter, how much would I have eaten? You could work this out by looking at the pie, and seeing that I've eaten two of the four pieces.

So 2/4 of the pie, which is the same as half of the pie. If we look at it mathematically, you can see that the denominators or the bottom numbers in the fractions stayed the same. This is because they represent the total number of pieces we had in this example. For the numerators, the top numbers in the fraction, we added them together. I had one of the four pieces of pie, and then I had another one of the four pieces of pie. So I ate two of the four pieces of pie, which is one half of the whole pie. It works the same way for subtraction.

If we had 3/7 take away 2/7, that would equal 1/7. As we just subtract the two from the 3 to get one, and we keep the denominator the same. Thinking again about this as pieces of pie, if you had three of the seven pieces of pie and you were to give away two of those seven pieces, you would be left with just one of the seven pieces of pie.

You can see that it is fairly straightforward to add and subtract fractions, when the denominator is the same. But what happens when we have different denominators? Let's say we have 1/4 plus 1/2. If you take the example of the pie again, for the first quarter of the calculation, let's colour this quarter of the pie in. And then if I was to eat another half of the pie, I would eat this much. So what does that equal in total?

Well, there's a couple of ways we can think about this. First of all, we could rewrite one half. One half of the pie is the same as 2 quarters. There's one quarter here, and another quarter here. So one half is the same as 2 quarters, 2 over 4. That means 1/4 plus 1/2 is the same as 1/4 plus 2 quarters. All we did here was change a half to two quarters, by multiplying the numerator and the denominator, the top and the bottom of the fraction by 2.

And you can do that to any fraction, as long as you multiply the numerator and the denominator by the same number. The reason I picked two here, is because I wanted to get the same denominator as a quarter. To finish off this calculation, we just add the numerators to give 3 and keep the denominator the same of course. So the answer is 1/4 plus 2/4 equals 3/4. If you look at the image of the pie, you can see 3/4 of it has been shaded in.

Now let's try 1/2 plus a 1/3. With this, it isn't possible to multiply the denominator 2 by anything to get 3, or to multiply the denominator 3 by anything to get 2. So instead, in order to find the common denominator, we need to find the least common multiple of 2 and 3. So what's the least common multiple of 2 and 3?

Well, that's the smallest number that's a multiple of 2 and a multiple of 3, or the smallest number that both 2 and 3 divide into exactly. The smallest number that's a multiple of both 2 and 3 is 6, so let's convert both these factions to something over 6. So 1/2 is equal to what over 6? Well, if I ate a half of a pizza which had been divided into six slices, I would have eaten three slices. And that makes sense, because one is a half of 2 and 3 is a half of 6.

Similarly, if I ate a third of that pizza with six slices, it would be the same as 2/6. So 1/2 plus a 1/3 is the same thing as 3/6 plus 2/6. All we've done is write the original calculation with different denominators. We've essentially just changed the number of pieces in the pie. And now at this point, the problem becomes easy. We add the numerators to give 5, 3/6 plus 2/6 gives us 5/6. And we've kept the denominator the same as always.

And it works the same for subtraction. 1/2 take away 1/3, is the same as 3/6 take away 2/6. And that is equal to 1/6.