Lesson Video: Equivalent Fractions Using Multiplication | Nagwa Lesson Video: Equivalent Fractions Using Multiplication | Nagwa

Lesson Video: Equivalent Fractions Using Multiplication Mathematics • Third Year of Primary School

In this video, we will learn how to multiply the numerator and denominator of a proper fraction by the same number to find equivalent fractions.

16:13

Video Transcript

Equivalent Fractions Using Multiplication

In this video, we’re going to learn how to multiply the numerator and the denominator of a fraction by the same number to help us find equivalent fractions.

Let’s begin with a blank canvas, and let’s imagine that we split it into two equal parts and we paint one of them. We’ve painted half the canvas, haven’t we? We’ve painted one part, but the whole shape was divided into two parts altogether. And of course, we know this is how we write one-half as a fraction. But what if we start with a similar canvas, but this time we divide it into twice as many parts? So instead of dividing it into two equal parts, we’ll split it into four. And what if we paint twice as many parts too? So instead of painting one part, let’s paint two.

Let’s write what we’ve done using fractions. So if you remember, we said we’re going to split the canvas into twice as many parts. And in a fraction, it’s the bottom number or the denominator that shows us the total number of equal parts. In the fraction one-half, we have two equal parts altogether. And so, if we have double this amount, we need to multiply the denominator by two. Two times two is four. And we can see our second canvas has been split into four equal parts, hasn’t it?

If you remember, the second thing that we said was that we were going to paint twice as many parts too. And in a fraction, it’s the numerator or the top number that shows us the number of selected parts, or the number of parts that we’re talking about, in this example, the number of parts that we’ve painted. We’ve only painted one part, and that’s why in one-half the numerator is one. But if we were to double this, just like we did with the denominator, we’d get a numerator of two. In other words, we need to paint two out of four parts.

Now, these are different fractions, one-half and two-quarters. But although they’re different, they worth the same. They’re what we call equivalent. So it’s a little bit like the times tables facts four times five and 10 times two. They’re not the same fact. But if we work them out, they worth the same; they’re equivalent. They’re both equal 20, don’t they? So although these fractions are different, they’re worth the same. And can you see? The same area of the canvas has been painted. One-half is equal to two-quarters.

But you know? The most important part of this example is not the canvases; it would all get very messy if we kept having to paint canvases to find equivalent fractions. The most important part of what we’ve done here is how we transformed the fractions. We multiplied the numerator and the denominator by the same number. And this last part is really important. If we’d multiplied by different numbers, these fractions won’t be equivalent at all. We chose to multiply them by two, and we came up with two-quarters.

Let’s try an example where we multiply by a different number. What fraction should we begin with? Should we have a number of fifths this time? How about three-fifths? And remember, to find an equivalent fraction, we need to multiply both the numerator and the denominator — here is the important bit — by the same number. This bit is so important; we’re going to keep repeating it. Oh, looks like our denominator has already been multiplied. Three-fifths is the same as how many fifteenths? Let’s use what we’ve learned already to help us.

We know that one way we can find an equivalent fraction to three-fifths is by multiplying both the numerator and the denominator by the same number. And because the denominator has already been multiplied, it’s gone from five to 15. What do we multiply five by to get 15? Five, 10, 15. Five times three is 15, isn’t it? And as we keep saying for our second fraction to be equivalent, we need to multiply the numerator by the same number too. So, we need to multiply this by three. And we know three threes are nine. And so, we can say three-fifths is equal to nine fifteenths.

Now again, you could look at these fractions and think, “Well, they look so different. How could they be worth the same?” Well, if you wanted to — and you don’t have to do this for every example because it would take forever — but you could draw a diagram to prove this. Oh dear! It looks like things are going to get messy again. So to start with, we’ll show three-fifths. That means dividing our canvas into five equal parts and painting three of them. So, this is three-fifths of the whole amount.

And then if you remember what we did with our fractions, we multiplied both the numerator and denominator by three. So instead of our canvas being divided into five equal parts, it now needs to be divided into 15. And the quickest way to do this on the diagram we already have is to draw a line here and here. We now have three times as many parts as we had before. And if we count them, we’ll see that there are 15.

And if we count the number of parts that we’ve painted, it’s now three times as many too. There are nine of these smaller parts that had been painted, and there’s little example to prove how we know that although these fractions do look different, they worth the same. Three-fifths equals nine fifteenths. And it doesn’t matter what number we multiply by as long as it’s the same.

Let’s have a go at answering some questions now where we’re on the hunt for some equivalent fractions. And we’re going to make sure that each time we multiply the numerator and the denominator by the same number.

Complete the following. Two-sevenths equals what fourteenths.

In this question, we’ve got two fractions that look a bit different. We’ve got a number of sevenths, two-sevenths, and a number of fourteenths. Now we don’t know how many fourteenths we have; this is a bit we need to complete. But although these two fractions look different, they are actually linked. And we know this because in between them is this equal sign. These fractions are equivalent; they’re worth the same.

But how are we going to complete the second fraction? How do we know what this missing numerator is? It could be any number, couldn’t it? Well, one way to find an equivalent fraction is to multiply both the numerator and the denominator by the same number. And if we look carefully at our diagram, we can see that this is what’s happened. We need to multiply them both by two. In fact, the denominator has already been multiplied by two, hasn’t it? Seven times two equals 14. So to find out how many fourteenths are the same as two-sevenths, we just need to multiply two by two as well. And two times two equals four.

Although these fractions look different, two-sevenths is actually worth the same as four fourteenths. And we could quickly check that we’ve got the right answer just by sketching a diagram. To show two-sevenths, we need to split up the whole amount into seven equal parts and select two of them. Then by multiplying the denominator by two, what we’re really saying is instead of seven equal parts, what we need is 14 equal parts. So if we draw a line across here, we can show 14 equal parts. And so, we can see that our numerator has doubled too. Instead of two parts being colored in, we now have four parts, but the two fractions have the same value.

We know that to find an equivalent fraction, we can multiply both the numerator and the denominator by the same amount. In this diagram, we could see that the denominator have been multiplied by two. So, we just had to do the same to the numerator. Two-sevenths equals four fourteenths. Our missing number is four.

Suppose you multiply the numerator and the denominator of the fraction one-half by two. What fraction do we get after this multiplication? Does this mean that the new fraction is equivalent to one-half? Yes or no.

In this question, we need to start off with the fraction one-half. And of course, we know that this represents one out of two equal parts. Now, we’re told to do something to this fraction. We need to multiply the numerator and the denominator by two. And we’re asked, what fraction do we get after this multiplication? Let’s find out.

The numerator in one-half of course is one. So if we’ve multiplied this by two, we get the answer two. And the denominator is two, so if we double this, we get four. The fraction that we get after this multiplication is two-quarters. We’re then asked, does this mean that the new fraction, which is the two-quarters that we’ve just made, is equivalent to one-half?

Let’s remind ourselves what the word “equivalent” means. If two things in maths are equivalent, and in this case we’re talking about two fractions, they may not look exactly the same, but they worth exactly the same; they have the same value. So, our question is asking us, is two-quarters worth the same as one-half? We could draw a little diagram to help us find this answer.

Here’s one whole. And instead of dividing it into two equal parts as we do when we show one-half, we need to double this amount. So we could draw a line here to divide it into four equal parts. And in the fraction one-half, one of our equal parts is shaded, but we need to double this too. So we need to shade in two equal parts. And we can see just by looking at the picture that two-quarters is worth the same as one-half. We could finish our diagram by putting in some equal signs, couldn’t we?

There are different ways that we can find equivalent fractions. And one of them is by multiplying the numerator and the denominator of a fraction by the same amount. In this question, we multiply both of them by two. We started with one-half. And by multiplying the numerator and the denominator by two, we got the fraction two-quarters. And because both numbers in the fraction were multiplied by the same amount, we know that the new fraction is equivalent to half. The answer to the second part of the question is yes.

Two-fifths equals eight what.

In this question, we’re shown two fractions that are worth the same. And we can tell this because of the equal sign in between them. Well, we say fractions, but really we’ve got one fraction and one part of a fraction because the denominator is missing in this fraction. Two-fifths is equal to eight what? To find the answer, we need to look at how our fractions have changed. And there’s only one number that we know in the second fraction, so we need to look at how our numerator has changed.

In the fraction two-fifths is the number two. And in our second fraction, it becomes the number eight. And you could say, “Well, we’ve added six.” Two plus six equals eight. But when we’re working with fractions, we need to think about how a number is multiplied and sometimes also divided, but in this case multiplied. What do we multiply two by to get eight? Two, four, six, eight. Two times four is eight, isn’t it?

Now this equal sign in between our two fractions is really important because if we want to find a fraction that’s equivalent to two-fifths, we’re going to have to do the same to the bottom number as we do to the top. We’re gonna have to multiply both the numerator and the denominator by the same number. And because we know the numerator has been multiplied by four, we’re going to have to multiply the denominator by four too. And we know that five times four equals 20.

Let’s read our statement. Two-fifths is equal to eight twentieths. So if we take two out of a possible five equal parts and then we multiply the denominator by four — so in other words, we have four times as many equal parts, and we can do this by drawing some lines here; there we go — we then have four times as many parts shaded. But we can see just by looking at the diagrams that the fractions are worth exactly the same.

We worked out that the numerator had been multiplied by four to go from two to eight. And so, to find the equivalent fraction we just needed to do the same to the denominator. Two-fifths equals eight twentieths. The missing number in this question is 20.

One-third equals what twelfths.

In this question, we’re given two fractions that are worth the same. We’ve got one-third. And then our second fraction is a number of twelfths, but we don’t know how many twelfths there are. This is the bit we need to find. Now we can see that our second fraction has been written in words. So, perhaps the first thing we could do is to write out our question again. And this time, write the fractions using numbers. It’s gonna be a little bit easier to see the numerator and the denominator if we’ve written them out using numbers.

So, we’ve got one-third and then a number of twelfths. And we haven’t mentioned this, but in between them is an equal sign, and this is really important. It shows us that both fractions are worth the same. They might be written differently, but they have the same value. But remember, the denominator in the fraction shows us the total number of equal parts that there are. And if we start off with three equal parts and we end up with 12 equal parts, what happened to those parts? Well, there are four times as many, aren’t there? Three times four is 12. So instead of three equal parts, the whole amount has been divided into 12 equal parts.

And to find the fraction that’s equivalent to one-third, we need to do the same to the numerator as we’ve done to the denominator. We’ve multiplied three by four to get 12. So, we also need to multiply one by four to give us four. And that’s how we know what our missing number is, and we’ll take care to write it using words too. We know that we can find equivalent fractions by multiplying the numerator and the denominator of a fraction by the same amount.

To get from a number of thirds to a number of twelfths, we know we had to multiply three by four. And to keep the fraction equivalent, we also had to do the same to the numerator. And as we can see from the diagram, one-third is equal to four twelfths. Our missing number, which we’ve written as a word, is four.

What have we learned in this video? We have learned how to multiply the numerator and the denominator of a fraction by the same amount to help find equivalent fractions.

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