# Video: Recognize and Generate Simple Equivalent Fractions

Tim Burnham

Equivalent fractions have equal value, and the numerator and the denominator of one are a common multiple of the numerator and the denominator of the other.

11:39

### Video Transcript

In this video, we’re gonna look at equivalent fractions. We’ll explain what we mean by the term equivalent fraction, we’ll see some real life examples involving a delicious chocolate cake, and we’ll learn how to recognize and generate equivalent fractions. But first, let’s just recall what we know about fractions. We call the upstairs number the numerator and we call the downstairs number the denominator. This little bar in the middle we call the divisor bar. Where in fact there’s a really fancy word, vinculum, that’s another name for a divisor bar.

This particular fraction we’ll pronounce as three-fifths. And if we look at what three-fifths represents, it means sort of three out of five. If I take a circle and I chop it into five equal sections and I colour in three of them, three-fifths of the circle has been coloured in. So the fraction represents three out of five things, or we could interpret it as three divided by five. The divisor bar at the middle means divided by, so three divided by five.

Okay then, let’s start off with a nice tasty-looking chocolate cake. If we would’ve split it down the middle into two equal parts, we would have half a cake and half a cake. So you can see if we put the two halves back together, we get a whole cake. Now if we cut each half exactly in half again, we’ve got quarters of a cake. And hopefully you can see that if I put two-quarters together, a quarter plus another quarter, I get the same as a half a cake. It’s an equivalent proportion of the whole cake. That means it’s the same proportion of the cake. So we say a half and two-quarters are equivalent fractions because they’re the same proportion. I’m enjoying thinking about this cake, so let’s consider some more equivalent fractions.

So now we’ve got each quarter in half and we’ve made eighths of a cake. So we already know that a half a cake is the same as two-quarters of a cake. Now if we think about the same proportion of the whole cake on the last example, that’s four-eights of a cake. So four-eights of the cake is equivalent to two-quarters and that’s equivalent to a half. All of these fractions are equivalent to each other because they represent the same proportion of the cake. Now, let’s start off with a new cake the same size and cut into six equal pieces.

It’s reasonably easy to say that if I take three of those six, that is also half a cake. So that is also equivalent to a half or two-quarters or four-eighths. So what we’ve been doing here is looking for fractions which are equivalent to a half. And in finding those fractions, we’ve also found fractions that are equivalent to each other. So two-quarters is equivalent to a half and four-eighths is equivalent to a half and three-sixths is equivalent to a half. But in doing that, we’ve also found that two-quarters is equivalent to four-eighths is equivalent to three-sixths as well as a half.

But there are equivalent fractions that are not equivalent to a half. For example, if we took two-eighths of a cake, that is the same as a quarter of our cake. So two-eighths and a quarter are equivalent fractions. Now if I wanted to try and find the number of sixths that are equivalent to a quarter, that becomes a bit more tricky. One-sixths would be less than a quarter and two-sixths would be bigger than a quarter. To get an equivalent fraction to a quarter in sixths, we would need one and a half sixths. But that’s not a proper fraction. We can’t have a fraction within a fraction, so we can’t write it like that.

So if we specify a particular denominator, a number of pieces that we’re gonna cut the cake into, we’re not necessarily going to be able to find a nice whole number for the numerator to find an equivalent fraction to some other fraction like a quarter or two-eighths. We’re not really covering this in this video. But look, if I cut all of those six’s into a half, I’d have twelfths. And a quarter is going to be equivalent to three of those twelfths. So we’re gonna always find an equivalent fraction, but it’s not necessarily going to have the particular denominator that you first thought of. But anyway, we’ll cover that in another video in more detail.

All this looking at cakes is actually make me a bit hungry now, so I’m gonna put them way and we’re just gonna deal with circles. So here’s a circle that we split into five equal parts, fifths. And we’ve coloured in three of the parts red. So the shaded part is three-fifths of the whole circle. Now we can cut each of those fifths in half again, so the circle will be split into ten equal parts. And the same region is shaded and it covers six out of the ten regions.

So the area that we described as being three-fifths of the circle can also be described as six-tenths of the circle. So three-fifths and six-tenths are equivalent fractions. So we can say that fractions are equivalent when they represent a same proportion of a whole. Now let’s do some examples without cakes or circles and we’ll just look at the numbers. So fractions are equivalent if you can multiply or divide the top and bottom numbers to get the same fraction. With a quarter and three twelfths, if we take the one and multiply that by three, we get three. If we take the four and multiply that by three, four times three is twelve. So we’ve multiplied the top and bottoms by the same number, by three, to get the second fraction. That means that those two fractions are equivalent.

Okay, let’s take three twelfths and compare it to twelve forty-eighths. Well three times four is twelve and twelve times four is forty-eight. So these are equivalent fraction. Okay, let’s consider twelve forty-eighths and six twenty-fourths. Well to go from twelve to six, I’ve got to divide by two. And to go from forty-eight to twenty-four, I’ve got to divide by two. So I’ve divided the top and the bottom by the same numbers to get that second fraction, so they are equivalent fractions.

So we’ve looked at a quarter, three twelfths, twelve forty-eighths, and six twenty-fourths, and we worked out pairs of equivalent fractions. But of course we could go directly from one quarter to twelve forty-eighths by multiplying by twelve. If I multiply the top by twelve, I’ve got twelve. If I multiply the bottom by twelve, I’ve got forty-eight. In fact, to go from any of those fractions to any of other those fractions, I can always find the same number that I multiply or divide the top and bottom by to get to the other fraction. Now I can always find an equivalent fraction just by multiplying the top and bottom by the same number, so six times a hundred is six hundred. And twenty-four times a hundred is twenty-four hundred or two thousand four hundred. So there’s another equivalent fraction in that list.

Right, now it’s time for you to test if you’ve learnt what we’ve been teaching you in this video. Are these fractions equivalent? Three-fifths and nine fifteenths. Number two: two-tenths and six twentieth. Number three: six twelfths and three-quarters. And number four: eight twentieths and two-fifths. So pause the video and we’ll be back in a minute just to check those answers. Okay, so how do I get from three to nine? Well I have to multiply by three. How do I get from five to fifteen? Well I have to multiply by three. So I had to multiply the numerator and the denominator by three to get nine fifteenths, so they are equivalent fractions.

Number two. How do I get from two to six? I have to multiply by three. How do I get from ten to twenty? I have to multiply by two. So I had to multiply the numerator and the denominators by different numbers to get the numbers in the second fraction, so they’re not equivalent fractions.

Okay, number three. How do I get from six to three? I have to divide by two. How do I get from twelve to four? I have to divide by three. So in this case, I had to divide the numerator and the denominator by different numbers in order to get the second fraction, so they’re not equivalent fractions. And in number four, to get from eight to two, I have to divide by four. To get from twenty to five, I also have to divide by four. So because I had to divide the numerator and the denominator by four, those two are equivalent fractions.

Let’s summarise what we’ve taught about in this video then. Fractions are equivalent when they represent the same proportion of a whole. Splitting each fifth in half like this meant that we had twice as many sections and twice as many of them were coloured in. But still, the same proportion of the whole circle was coloured in, so we can see that three-fifths is equivalent to six tenths. Also fractions are equivalent when you can multiply or divide the top and bottom numbers of one fraction by the same number to get the other fraction.

So for five-sixths and ten twelfths, if I multiply the top by two and the bottom by two, five becomes ten, six becomes twelve. They are equivalent fractions. Similarly, eight thirtieths and four fifteenths are equivalent because eight divided by two gives us four and thirty divided by two gives us fifteen.

But if you have to multiply the top and bottom by different numbers to get the second fraction, they’re not equivalent. So five-sixths and ten eighteenths, we have to multiply five by two to get ten, but we have to multiply six by three to get eighteen. So these are not equivalent fractions. And ten twentieths and two-fifths are not equivalent. Because to turn ten into two, we have to divide by five. But to turn twenty into five, we have to divide by four. So we divided the top and bottom by different numbers, not equivalent.

Oh and one final thing, can you remember the fancy name that we gave to this divisor bar in the middle? It was a vinculum. Well that fascinating fact just about wraps it up for recognising and generating simple equivalent fractions.