Lesson Video: Sharing a Quantity into a Ratio Mathematics

Video Transcript

In this video, we’ll learn how to share quantities into two- and three-part ratios. By this stage, you should feel fairly confident in identifying and using ratio notation as a way of comparing amounts. You should also know that simplifying a ratio is much like working with fractions. We look for common factors and divide through until the numbers are coprime. That is, they share no factors other than one.

Now, we’re going to look to extend these ideas and see how we can share given amounts into ratios with two or more parts. And we’ll consider two equally valid methods. Before we get into the nitty-gritty, let’s look at how we could solve a question pictorially.

Bart and Lisa share 20 sweets in the ratio two to three. How many sweets does each child get?

We’ll begin by recalling what we actually mean by this ratio two to three. This means that for every two sweets Bart gets, Lisa will get three. It follows that we expect that if we share these sweets out in this manner, Lisa will get more sweets than Bart. So let’s begin by giving Bart two sweets, and then we’ll give Lisa three. There are still quite a few sweets left. So let’s do that again. We give Bart two further sweets, and we give Lisa three.

Notice at this stage Bart has four sweets and Lisa has six. If we put these in a ratio that is four to six, we can simplify this back down to our original ratio two to three. That’s a really good way to check our answer. But there are still more sweets left. Let’s keep going until we’ve used them all up. Bart gets two more, and Lisa gets three. Bart gets another two, and Lisa gets the three that are left.

So we’ve used all of these sweets up. And for every two sweets we’ve given Bart, we’ve given Lisa three. We now see that Bart has eight sweets and Lisa has 12. Well, that’s all fine and well. But we really don’t want to do this every time. So we’re going to need to find a quicker method.

This method involves thinking about the ratio in terms of its individual parts. We go back to our original ratio. It consists of two parts here and three parts here. Two plus three is equal to five. So our ratio consists of a total of five parts. Let’s draw a bar model to represent this. Our bar model needs to have five parts, just like our ratio. And, of course, we need to fill this bar with our 20 sweets. So how do we share 20 sweets into five parts or five boxes? What mathematical symbol means share?

Well, the symbol divide means to share. So we divide 20 by five, and that gives us four. In doing this, we’re working out the value of one part. One part is worth four sweets. And now we know this, we go back to the start and remind ourselves how many parts each child gets. Well, Bart gets two parts — I’ve shaded his parts yellow — whereas Lisa gets three parts — I’ve shaded that in pink.

We now know that each individual box contains four sweets. One part is worth four. And so to work out the number of sweets Bart gets, we multiply four by two. That gives us eight. Similarly, we can multiply four by three to work out the number of sweets Lisa gets; that’s 12. And a really quick way we can double check whether we’ve performed the correct calculation is to check that our final values add up to the original. Well, eight plus 12 is indeed 20 as required. So we’re done.

And hopefully, we see that this is a much more efficient method than repeatedly giving two sweets and three sweets and so on, especially when it comes to sharing much larger amounts. Note that these processes can be used with integers as we have done — that’s whole numbers — but also with noninteger numbers.

In general, the steps are as follows. We begin by adding the numbers we have in our ratio. This tells us the number of parts that we’re sharing into. So we divide our original amount by this value. That then tells us what one part is worth. So we take that value and we multiply each bit of our ratio by that number. This is sometimes called the add-divide-multiply method for fairly obvious reasons.

It’s also important to notice that when we’re not actually sharing an amount into a ratio, we use similar sets, although our first step won’t necessarily be adding the numbers. Let’s look at an example of how this works.

Divide 48 kilograms in the ratio two to one.

One way we have to answer this question is sometimes called the add-divide-multiply method. Let’s recall the steps that we take. The first thing we do is establish how many parts we have by adding the numbers in our ratio. Well, we’re sharing in a ratio of two to one. So we add two and one to give us three. That tells us that we’re dividing our original amount into three parts.

Next, we do that division. We divide the original amount by this total. In this case, that’s 48 divided by three, which is equal to 16. That tells us what one part is worth. So here we see that one part is worth 16 kilograms.

Finally, we take this number and we multiply it by each bit in our ratio. Well, the first bit of our ratio is two. Two times 16 is 32. So when we share 48 kilograms in the ratio of two to one, our first bit is 32 kilograms.

Now, of course, the second bit is simply one part. So we don’t really need to do this multiplication. But we will for completion. One times 16 is 16, remembering of course that order matters. So we need to list the largest value first. We see that when we share 48 kilograms in the ratio two to one, we get 32 kilograms and 16 kilograms.

Now, using a step-by-step method is all fine and well. But it’s really important to understand what we’re doing at each step. Let’s look at an example where we might be inclined to blindly follow these steps without really thinking about what we’re doing.

Chloe and Michael shared their lottery winning in the ratio of three to two. Michael received 360 pounds. How much did Chloe receive?

When we answer this question, we’re going to need to be really careful. We’re told that Chloe and Michael share their lottery winning. And at first glance, we might see the ratio of three to two in the value of 360 pounds and think we’re going to share this value in the ratio three to two. In fact, this is the amount that Michael received. So in essence, this has already been shared for us. And our job is then to work out how much Chloe receives.

So let’s think about the steps we take when we’re sharing into a ratio and what they actually mean. The first thing we do is add the numbers in our ratio. This tells us the total number of parts we have. Then we divide our amount by this number to find the value of one part. Once we know the value of one part, we can go on to the next step. We multiply each bit of our ratio by this number. And so, in this case, we can disregard the first step.

But what we are going to do is divide to find the value of one part. Chloe and Michael share their winning in the ratio of three to two. So the second part of this ratio represents the number of parts that Michael gets. He gets two parts. So 360 pounds must be worth two parts. We’re going to divide this number by two to find the value of one part. 360 divided by two is 180. So we see that one part is worth 180 pounds.

And so, once we’ve done this, what’s our final step? Well, we’re going to multiply 180 pounds by the number of parts that Chloe receives. Chloe receives three parts, so we multiply 180 by three. This gives us a value of 540 pounds. Chloe receives 540 pounds. And, in fact, we could check our answer.

We’ve said that Chloe receives 540 pounds and Michael 360. So let’s put this in a ratio and simplify it. We could do this in a few steps. But in fact, both of these numbers are divisible by 180. Since the units are the same, we no longer need them. And we see that to simplify 540 to 360, we get three to two. This is the same as the ratio given to us in our question. So we know we’ve done it correctly. Chloe gets 540 pounds.

In our next example, we’ll consider a slightly different method for sharing a number into a given ratio.

David and Isabella share 30 cupcakes in the ratio two to three. How many do they each get?

One way we have to answer this question is the add-divide-multiply method. But there is another method which is equally valid. And that involves thinking about bar modeling and fractions. The first step is the same as the add-divide-multiply method. We look at our ratio. That’s the ratio two to three. We then add the numbers in our ratio. Two plus three is equal to five, meaning we’re looking at a total of five parts in this question. And so a bar model will be made up of five parts, as shown.

Remember, order is important. So David gets two of these parts. That’s represented by the yellow bits. And Isabella gets three. That’s represented by the pink parts. Now, we can think about this in terms of fractions. And we can see that the portion of the bar that I’ve shaded yellow represents two-fifths of the whole, whereas the portion that I’ve shaded pink represents three-fifths of the total.

We can extend this into the number of cupcakes and say that, well, if there are 30 cupcakes, David gets two-fifths of these. Similarly, Isabella will get three-fifths of these. By recalling that “of” can quite regularly be interchanged with the multiplication symbol, we see that we can answer this question either by finding two-fifths of 30 and three-fifths of 30 or by multiplying two-fifths by 30 and three-fifths by 30.

Let’s multiply two-fifths by 30. We’ll write 30 as a fraction. It’s 30 over one. Next, we cross cancel. We divide the denominator of our first fraction by five and the numerator of our second by five. Two times six gives us 12, and one times one gives us one. But 12 ones is simply 12. And so we see that David gets 12 of the cupcakes.

To find the number that Isabella gets, we could subtract this from 30. But let’s multiply again. We do three-fifths times 30 over one and, once again, divide through by five. This gives us three times six, which is 18 over one, or simply 18. David gets 12 cupcakes, and Isabella gets 18.

Note that, of course, we can check our answer by checking that 12 and 18 do add up to 30, which they do. Note also that at this stage, we could have found the value of one-fifth by dividing 30 by five. Then whatever value we get, we double to find the value of two-fifths. Either method is perfectly valid.

In our next example, we’ll go back to the add-divide-multiply method and consider how it will work when sharing using a three-part ratio.

Divide 72 centimeters in the ratio two to three to one.

Let’s recall one of the methods we have for sharing into a ratio. It’s called the add-divide-multiply method. In this method, the first thing we do is we add the numbers in our ratio. This will tell us the total number of parts we’re sharing into. Our ratio is two to three to one. And two plus three plus one is six. So we have a total of six parts.

Step two is to divide the amount we’ve been given by this number. This will tell us what one part is worth. We’re dividing 72 then by six, and that gives us 12. Of course, we’re actually working in centimeters. So we can say that one part is worth 12 centimeters.

Our final step is to multiply each bit of the ratio by this number. Essentially, if we know the value of one part, we can find the value of two by timesing it by two, we can find the value of three by timesing it by three, and so on. Two multiplied by 12 is 24, so the first bit we get is 24 centimeters. Three times 12 is 36, so our second bit is 36 centimeters. Then one multiplied by 12, which we don’t really need to do but we’ll do for completion, is 12. And the third bit is 12 centimeters.

When we divide 72 centimeters in the ratio two to three to one, we get 24 centimeters, 36 centimeters, and 12 centimeters. In fact, 24 plus 36 plus 12 gives us the original 72 as we required. So we know that we’ve probably done this correctly.

In our final example, we’ll go back to the fraction method and consider how it works when using a three-part ratio.

Three friends win 6000 pounds. They decide to share the money in the ratio three to five to four. Calculate the amount each one receives.

To answer this question, we’re going to use a fractional method. By considering the ratio three to five to four, we want to work out what fraction of the total amount each friend wins. Now, for ease, we’re going to name these friends 𝐴, 𝐵, and 𝐶. And the first thing we’re going to do is add the numbers in our ratio. What this does is tells us the total number of parts we’re sharing the winnings into. Three plus five plus four is 12. And so we’re sharing this 6000 pounds into 12 parts.

Remember, order matters. So 𝐴 gets three of these parts, 𝐵 gets five, and 𝐶 gets four. In terms of fractions then, we can say that if there are a total of 12 parts and 𝐴 gets three of these, 𝐴 must get three twelfths of the winning amount. In fact, we can simplify three twelfths to a quarter. So 𝐴 gets one-quarter of 6000 pounds. In a similar way, since 𝐵 gets five of the parts, we know 𝐵 gets five twelfths of the money. Finally, 𝐶 gets four parts. So 𝐶 gets four twelfths of the money. Four twelfths simplifies to a third. So 𝐶 gets one-third of 6000 pounds.

We’ll work out then how much 𝐴 gets by finding a quarter of 6000. That’s 6000 divided by four, which is 1500. We’ll come back to 𝐵 in a moment because it requires an extra step. But to find a third of 6000, we divide 6000 by three; that’s 2000. We’ll find for 𝐵 one twelfth of 6000 first. That’s 6000 divided by 12, which is 500. So one twelfth is 500 pounds. Five twelfths will be five lots of 500 pounds. That’s five times 500, which is 2500 pounds. The three friends receive 1500 pounds, 2500 pounds, and 2000 pounds, respectively.

In this video, we’ve seen that when we share in a ratio, we’re really interested in finding the value of one part of that ratio. We saw that there are a couple of methods that we can use. One of these is the add-divide-multiply method. We add the numbers in our ratio together, divide the original amount by this number, and that tells us what one part is worth. When we know what one part is worth, we multiply each bit of our original ratio by this number to share the amount.

But we also saw that sometimes it can be efficient to use fractions to calculate this. We saw that we can check our answers by either adding the final numbers we get and checking that this gives us the original amounts or by putting them into a ratio and simplifying them back. When we do this, we should get back to the original ratio.

Finally, we saw that very occasionally we won’t actually be given the entire amount that we’re sharing into. Instead, we’ll be given what one person receives. And very occasionally, we’ll be given the difference between what they receive. And so we should always make sure that we are being asked to share the amount given before following these steps.