Lesson video

Hello and welcome to this lesson on ratio and proportion.

In this lesson, we're going to be solving problems involving ratio.

You may have looked at ratio already, you may have used some representations, and you may have thought about how solving ratio problems relates to multiplication and division.

So let's have a look at what's in today's lesson.

So in the first part of today's lesson, we're going to be really focusing on how we can use division to help us to solve ratio problems. We'll probably end up using multiplication as well, but we're gonna have a real focus on thinking about how division helps us in solving of ratio problems. So our context today is about Sofia and she's making pancakes for a party with her friends.

So let's look at her recipe.

So to make 12 pancakes, Sofia uses two eggs, three spoons of sugar, and six spoons of flour.

So that's her basic recipe.

So let's have a look at some problems then and help Sofia to make sure she can make the right number of pancakes for her party.

So we're asked here how would Sofia make 30 pancakes? And we've got a table of values here.

You may have collected ratio values in tables in other lessons.

So you can see in the top row of our table, we've got that basic recipe that was on the previous slide, that for every two eggs, Sofia needs three spoons of sugar and six spoons of flour, and she'll make 12 pancakes.

Oh, we talked about variables I think in the past with this.

So how many variables have we got there? What could we change? Well, we could change the number of eggs, the number of spoons of sugar, the number of spoons of flour, and the number of pancakes.

And the question we are being asked is how would Sofia make 30 pancakes? So in this case, it's the number of pancakes that has changed.

So which row of the table is going to help us? So we're gonna have to do some work with this, aren't we? Sometimes we can start with that basic ingredients list, that basic ratio, and work up.

But 30 and 12, well, 30 isn't a multiple of 12, so I'm gonna have to go somewhere else to help me.

So I think I might be able to use the fourth row down, which has 60 pancakes.

So let's have a look at that row and see if we can think how that can help us.

So I've isolated that row and I've filled in my 30 pancakes.

Now can we see a relationship between 60 and 30? I think we can, and I think we can see that 60 divided by two is equal to 30.

We've divided the number of pancakes by two, or we've halved the number of pancakes.

So we can think about dividing by two, or finding a half of multiplying by a half.

So we can say that Sofia has made one half times the number of pancakes.

So we'll need one half times the amount of each ingredient.

So we can divide those ingredients by two to work out how much of the other ingredients she needs.

30 divided by two is 15 or half, of 30 is 15.

Half the number of eggs, or divide the number of eggs by two, 10 divided by two is five, or 10 multiplied by a half is five.

Why have I left the spoons of sugar 'till last? Have you spotted? Yeah, that's not an even number.

So halving an odd number, we are going to end up with a fraction of a spoon of sugar.

So half of 15, or 15 divided by two is 7.

5, 7 1/2.

So to make 30 pancakes, Sofia will need half the ingredients she used for 60.

So she'll need five eggs, 7 1/2 spoons of sugar, 15 spoons of flour, and she'll make 30 pancakes.

So you can see how we've used division there to go from a row that we have, some information that we have to create some new information, a different number, a different amount of the ingredients in order to make a different number of pancakes.

Now, we could have used a different row.

I wonder if some of you spotted this.

We could perhaps have used that row to make six pancakes, because we know that six multiplied by five is equal to 30.

So we could have used this row and multiplied by five, but we are really thinking about division at the moment.

So I picked a row where we could use division.

So let's have a look at another context, this time using a bar model rather than a table.

So we are having a quiz at school and each team in a school quiz is made up of two adults and three children and that language of ratio, so for every two adults there are three children.

So this time we're told that there are 45 people doing the quiz and we're asked how many of them are children? So here's our basic bar model that for every two adults, there are three children.

So there are five people in one quiz team.

So could we draw a bar model to help us to work out if there are 45 people doing the quiz, how many of them are children? Let's have a look.

So now we know that that whole bar, all the people taking part in the quiz is 45 people.

We know that those 45 people are two parts adults and three parts children, and those parts are all equal.

So we need to find out what those three question mark boxes are worth.

So what calculation are we going to do to work out the value of one of those boxes representing the adults and the children, one of those parts of our bar? Well, there's five parts to that bar.

So we need to divide 45 into five equal groups, and then to find out how many are children, we need to find the value of three of those groups.

So to divide 45 into five equal groups, we're going to do 45 divided by five is equal to nine.

So each of those sections of the bar is worth nine, and we need to find out the value of three of them.

So we need to do nine multiplied by three to equal 27.

So we can see that each of those sections of the bar would be worth nine because nine times five is 45.

But we need to know the value of three of those sections.

Nine multiplied by three is equal to 27.

Let's have a look at that in a table representation as well.

So we know that in each individual team is two adults and three children.

For every two adults there are three children and the total number of people is five.

We know that we want our total number of people to equal 45.

So what calculations are we thinking about to turn five into 45, or 45 into five? Well, we can think about it going either way.

We can do the five to the 45, or the 45 to the five, and we're either going to divide by nine.

45 divided by nine equals five, or five multiplied by nine is equal to 45.

So we can either think of this as five multiplied by nine is equal to 45, or 45 divided by nine is equal to five.

And the table perhaps allows us to think of multiplication or division and how both of them would work.

Perhaps more easy than the bar.

And the bar really leads us to think about division because we can see that 45 and we can see that we've got to divide it into five equal parts.

In the table, we are looking at that relationship between five and 45, and we can go straight to thinking about it as a multiplication relationship.

And then we know we've got to have nine times as many children.

So three multiplied by nine is equal to 27.

So we sort of done the same calculations, but we might have thought more about multiplication with the table and we might have thought initially more about division with the bar model.

So what I've just done there is have a think about what's the same and what's different between these representations? Okay, time for us to have a bit of a reflection and to check where we're up to, really thinking about how we are using division and how we're using those different representations to help us to make sense of the problem.

So we are still sticking with these school quiz teams made up of two adults and three children.

For every two adults there are three children.

And this time the problem says, if there are 12 adults, what is the total number of people? And I'd like you to have a go at drawing a bar model and using a table to solve the problem.

So you might want to pause the video here and draw your representations.

Okay, how did you get on? I've started with a bar model here.

So this time I know that there are 12 adults.

So what does that mean for my bar model? Well, I know that for adults, two parts of the bar are adults.

So to work out the value for each of those sections of the bar relating to the adults, I've got to take that 12 and divide it by two.

So I'm going to say that 12 divided by two is equal to six.

So I know that the value for each of my parts that represent the adults is six.

I don't need to know how many children are taking part this time, because the question asks me what is the total number of people taking part? So I know that that bar is made up of two parts adults and three parts children.

So there are five parts to that bar.

So I need to multiply six by five.

So six multiplied by five equals 30.

So I can see that the total number of people taking part when there are 12 adults is 30.

So I've drawn my top line of my table, which shows me my basic ratio.

For every two adults, there are three children, and the total number of people in one team is five.

So there's my information for one team.

The extra bit of information I'm given this time is that there are 12 adults.

So again, how can I think about the relationship between that number of adults? Do you think straightaway of division, or do you think of multiplication first? Well, we could definitely say that 12 divided by two is equal to six.

So we've divided by six.

But then to go the other way from the two to the 12, we are thinking about multiplying by six.

So if I've got six times as many adults taking part in the quiz, there will be six times as many people in total taking part in the quiz.

So five multiplied by six is equal to 30.

So what is the same and what is different with these representations? Well, in our bar model, I'm more likely to think about that division, aren't I? I can see the division there, I can see that I've got 12 adults and I've got to divide it by two to find out how many adults are represented in each of those sections of the bar for my two parts adults for the team.

And then I can see that six repeating along and I can see my multiplication.

So I'm dividing by two because I can see those two parts.

If I look at the table, I can see my two adults and my 12 adults, and I might straightaway see that multiplied by six, which is the multiplication that we did once we got to the six from our division.

And the division I might think about in the table is actually 12 divided by six is equal to two.

So I've used that idea that my factors of 12 are two and six, but I've thought about them slightly differently depending on whether it's the bar model or whether it's the table.

Eventually though, both are going to let me get to the point where I think, well, I've got to multiply by six and I've got to do a six multiplied by five or a five multiplied by six.

So you can see that my divisors and my quotient are different.

So that I've thought about 12 divided by two is equal to six in my bar model, and 12 divided by six is equal to two for my table.

And in both cases, I've multiplied six and five.

But in my bar model, I've looked at that one part being six and multiplied by five to find the total.

In my table, I knew that my total number for one team was five and I've then multiplied by six to get to 30.

So we can really see that fact that in the multiplication it's commutative, and that in division, my divisor and my quotient can swap places as long as my dividend stays the same.

The key thing is to see what's the same and what's different with those representations.

And maybe to start to think about which ones help you best with the way you want to think about finding missing values and calculating answers involving ratio.

Time for you to put this thinking into practise now.

So we've got a slightly different context, here we're thinking about a bag of fruit.

And in this bag of fruit, for every three apples there is one orange.

In a, we've got the value of the whole and we're trying to work out a value of one part.

In b, we are given information about one part and we are looking at working out the value of the other part.

And it might be interesting to think about how that relates to your bar model and how that relates to your table.

And in C, it says how many apples are there if there are 240 pieces of fruit all together? So again, we've got information about the whole and we're asked to work out the value of a part.

So really think about how those parts and wholes fit together in your representations.

And I'd like you to draw a bar model and a table to solve each problem.

So you want to pause the video here, have a go, and then we'll have a look at the solutions together.

How did you get on? I've drawn a bar model and a table to get me started with solving a.

So part a asked how many apples are there if there are 20 pieces of fruit altogether? So let's start with my bar model.

So my top bar shows that 20 pieces of fruit in total.

So it shows me the whole, and I can see there that that 20 pieces of fruit is made up of three parts of apples and one part of oranges, which relates to that initial information.

For every three apples, there is one orange.

So I need to work out what values go into those sections of the bar.

So I can see that my 20 pieces of fruit is made up of four parts.

So I can divide the 20 pieces of fruit by four, and I know that each of those sections is then worth five, 'cause 20 divided by four is equal to five, so I can put those fives in.

So I've definitely used division to help me to make sense of what's going on.

I can see now that there are three parts of the bar that represent the apples.

So five multiplied by three gives me 15 apples.

So if I have 20 pieces of fruit altogether, 15 of them will be apples.

Okay, how about the table? What can I see here? So I filled out the table with my basic information that I'm given.

If there are three apples, there's one orange and there are four pieces of fruit.

And I've put my value of 20 in because I know that there are 20 pieces of fruit all together.

So how can I think about the information here? Well, I can think with division and I can think that 20 divided by five is equal to four.

So I'm dividing by five, 20 divided by five is equal to four.

So I could then think what divided by five is equal to three for my apples? But I could also think about it with multiplication, can't I? So if 20 divided by five is equal to four, then four multiplied by five is equal to 20.

So I've got five times as many pieces of fruit.

So I will need five times as many apples, and three multiplied by five is equal to 15.

So what's the same and different with these representations? I did say I was gonna ask this question.

I really want you to think about it.

So you might have had a think already.

Again, it's around where is it easier to see a division, or where might it be easier to see a multiplication to help us to work things out? And just looking, it's interesting, isn't it? That we've divided by four when we thought about the bar model, but when we used the table, we thought about dividing by five.

And we know that 20 divided by four equals five and 20 divided by five equals four.

They're very closely linked because of that commutative in our multiplication.

Okay, so b asked us how many oranges are there if there are 27 apples.

So this time we're working from one part to another part.

So I've drawn my bar model again, I'm not really interested in all the fruit this time, but I've put that whole in there 'cause we can still see that that whole is made up of three parts apples and one part oranges.

So this time I know that those three parts of apples are worth 27.

So to find out what one part is worth, I need to divide by three.

So 27 divided by three is equal to nine.

So I know that if those three sections are worth nine, then I must have nine oranges as well, because those parts of the bar are equal.

So we can divide the number of apples by three.

Each part has the same value, so there will be nine oranges.

Let's have a look at the table.

This time, I've only got two columns.

I've only got the apples and the oranges because it doesn't matter about the whole number of fruits altogether this time.

So what can I see there as a relationship? Well, I can see 27 divided by nine is equal to three, but I can also see very quickly that I've got to multiply by nine there.

If I've got nine times as many apples, I must have nine times as many oranges, so I will have nine oranges.

So there's that question again, what's the same and what's different? Can you see where we've done the calculations that we've done in the table, in the bar model? Can you see the way we've calculated in the bar model in the table? What's the same, what's different? Which one works for you? That's the key question.

Let's have a look at part c.

Okay, so part c said how many bags are there if there are 240 pieces of fruit altogether.

Let's have a look.

So my bar model there, 240 pieces of fruit.

How many bags are there? Well, how many pieces of fruit are there in each bag? Okay, I'm gonna divide by four because that bar gives me the pieces of fruit.

So I can divide by four because there are four pieces of fruit in each bag.

So I need to know how many fours are there in 240, because there are four pieces of fruit in each bag.

So I'm not actually interested in apples or oranges in this question.

I'm interested in the fact that there are four pieces of fruit in each bag.

So how many bags will there be if there are 240 pieces of fruit? So 240 divided by four is equal to 60.

We can use some of our times table knowledge there, can't we? Because 240 is 24 tens, and I know that 24 divided by four is equal to six.

So 24 tens divided by four will be equal to six tens, and six tens is equal to 60.

So there must be 60 bags of fruit.

Thinking about our table, we can see that 240 divided by 60 is equal to four, but equally, we can see that four multiplied by 60 is equal to 240.

So if I've multiplied my number of pieces of fruit by 60, I will have multiplied my number of bags of fruit by 60.

So if I have 240 pieces of fruit, I will have 60 bags of fruit.

Let's have a think about that division equation and what the different numbers within it represent.

So what does the 240 represent? Well, the 240 represents the pieces of fruit that we have altogether.

What about the four? Well, the four represents the four pieces of fruit in one bag.

So each bag contained four pieces of fruit, three apples and one orange.

So what about the 60? The 60 represents the number of bags we fill if we have 240 pieces of fruit, 240 pieces of fruit, four in each bag.

At that point, we are not really, the apples and oranges don't necessarily come into it, except that we know that altogether, there are four pieces of fruit in a bag, because our basic ratio says that for every three apples, there is one orange in a bag.

And that question, again, what's the same and what's different with the representations? I'll leave you to have a think about that.

Right, so we spent some time thinking about division in particular and how it relates to solving ratio problems. Now we're going to think about multiplication and division, and using them together to solve some more problems. So let's have a look at some more ratios.

Okay, so we've been thinking and asking this question about tables and bar models.

So you might want to just pause at the moment before we get into solving some more problems and really think, do you agree that the table is best for solving a ratio problem, or a bar model is best for solving a ratio problem? You might have a view for yourself, but it's also worth remembering that sometimes a table might be good and sometimes a bar model might be good.

So just pause the video for a moment and just reflect on how you've been using bar models and tables.

When have they been working for you and which do you prefer at the moment? So you've just been thinking about bar models and ratio tables.

Jacob here is thinking of sandwiches.

So let's help Jacob make some sandwiches.

So in one sandwich there are two slices of bread, three slices of cheese, and four slices of tomato.

So what would the bar model look like if you used 16 slices of tomato? So if our tomato is 16, oh, I'm not sure how many sandwiches we're making now, what's our bar model going to look like? What do we need to do to complete that bar model? Well, we know that there are four parts tomato in our sandwich.

So if we've got 16 slices of tomato, then we're going to do 16 divided by four, which equals four.

So each of those parts of the bar will be worth four.

Then the other parts will also be equal to four, and that will also mean that we will be making four sandwiches, and we can do a check on that because we know that there are two slices of bread in each sandwich.

So if we've got eight slices of bread, we will be making four sandwiches, four times as many slices of bread, four times as much cheese, and four times as many slices of tomato.

So let's have a look at the table now.

So what would the table look like if we wanted to add a row that used 16 slices of tomato? Well, it's possibly two we could start from here.

So we could think about multiplying the the the first row where there's four slices of tomato by four, or we could double the second row for eight slices of tomato.

So we are gonna think about that division, 16 divided by what is equal to four? So 16 divided by four is equal to four, but equally, four multiplied by four is equal to 16.

So if we use 16 slices of tomato, we can multiply the top row by four.

So one multiplied by four is four for our number of sandwiches.

Two slices of bread multiplied by four is eight, and three slices of cheese multiplied by four is 12.

And we know that that then matches our four slices of tomato multiplied by four, which is equal to 16.

You could say have doubled that second row and had twice as many slices of bread and twice as many slices of cheese and twice as many slices of tomato.

So time for some practise on using multiplication and division.

And this time draw a table or a bar model to identify the calculations you need to use.

This time we're making smoothies and you're given the ingredients for two smoothies, one banana, 20 strawberries, and 200 millilitres of milk.

What will you need to make four smoothies, 20 smoothies, and one smoothie? That's question one.

And question two says, if you use one litre of milk, how many strawberries will you need? Okay, you might want to pause the video now, have a go at this task, and then we'll look through the answers together.

I've decided to draw a table to help me to solve these first three parts of the question.

So I've put in our basic recipe for two smoothies.

I use one banana, 20 strawberries, and 200 millilitres of milk.

And I've now added in the fact that I'm gonna make four smoothies, 20 smoothies, and one smoothie.

So how am I going to use multiplication and/or division in order to solve those problems? So let's look at a, making four smoothies.

So I can see that if I multiply my initial recipe, my basic recipe by two, I will make four smoothies.

Two multiply by two is equal to four.

So if I am making twice as many smoothies, I will need twice as many bananas, I will need twice as many strawberries, and I will need twice as much milk.

So for four smoothies, I will need two bananas, 20 multiplied by two, which equals 40 strawberries, and 200 multiplied by two, 400 millilitres of milk.

Okay, what about 20 smoothies? So I'm making my change now from two to 20.

So what am I multiplying by? How many times as many smoothies am I making? Well, I could think 20 divided by 10 is two, but I can also think two multiplied by 10 is equal to 20.

So this time I need 10 times as much of each ingredient in order to make my 20 smoothies.

And you may have used that language before.

If I'm making 10 times as many smoothies, I will need 10 times as much of each ingredient.

So for bananas, one times 10 is 10, for the strawberries, 20 times 10 is equal to 200, and for the milk 200 multiplied by 10 is 2,000 millilitres.

And I also know that 2,000 millilitres is the same as two litres.

Okay, what about my one smoothie? So this time I'm actually making fewer smoothies and I am thinking about division.

So this time I am dividing my number of smoothies by two, or finding half of my ingredients.

So this time I need half as much of each ingredient in order to make one smoothie.

So I need half a banana, I need 10 strawberries, and I need 100 millilitres of milk.

So this time I was using division to calculate my values, not as a sort of step on the way to a multiplication because this time the number of smoothies I was making was smaller than my original recipe.

So I had to halve my original recipe in order to work out how much I needed to make my smoothies.

So question two asked, if you have one litre of milk, how many strawberries will you need? So we can think about one litre of milk as 1,000 millilitres.

Our original recipe was in millilitres, so it helps if the units are the same so that we can think about the relationship between the values.

So we could think about dividing here.

So what do we divide a thousand by to get to 200? Well, we divide by five, because a thousand divided by 200 is equal to five.

We could think there's a 10 divided by two bit in there, isn't there? So I've got 10 hundreds divided by two hundreds, which is equal to five.

So a thousand divided by 200 is equal to five.

But I can also think of it as a multiplication, that I've now got five times as much milk, because 200 multiplied by five is equal to 1,000.

So I've got a multiplied by five relationship going on here.

So if I've got five times as much milk, I'm going to need five times as many strawberries.

So I'm going to have 20 multiplied by five, which is a hundred strawberries.

Let's just have a look at how the bar model works for these.

So if I am making different numbers of smoothies, how do I change my bar model in order to represent this new value of smoothies? Well, I could start thinking about crossing numbers out and realising that I've doubled it to make four smoothies, and then doubling it.

But the bar itself stays the same and the bar stays the same because it's representing that basic ratio.

For every one banana, there are 20 strawberries and 200 millilitres of milk.

So the proportions will stay the same.

Otherwise, I might have a very banana flavoured milkshake, or a very thick milkshake if I didn't have the right proportion of milk.

So our bar stays the same, but we think about how we are changing those values.

So let's have a look.

We had two smoothies to start with.

What will we need to make four smoothies? So we can look at it to think, well, four smoothies is two times the number of smoothies.

So you'll need two times the quantity of ingredients.

So you'll need two bananas, 40 strawberries, and 400 millilitres of milk.

For b, for 20 smoothies, 20 smoothies is 10 times the number of smoothies.

So you'll need 10 times the quantity of ingredients.

So we can alter our bar and make everything 10 times bigger.

And for part c, one smoothie is a half times the number of smoothies, so you'll need a half times the quantity of ingredients.

So question two asks about this one litre of milk and how many strawberries will we need? So how can we adjust our bar model to help us here? So our original bar model is for 200 millilitres of milk.

We're now thinking about one litre, or 1,000 millilitres of milk.

So we've got 1,000 millilitres of milk, which is five times the amount of milk.

So if we've got five times the amount of milk, we'll need five times the amount of strawberries.

So one litre of milk is 1,000 millilitres, five times the amount in the recipe.

So you'll need five times the number of strawberries, 20 multiplied by five is a hundred.

So is the bar model as easy to use as the table for these problems? What do you think? And how will you make use of bar models and tables to solve other problems involving ratio? Thank you for all your hard work in this lesson.

I hope that you are feeling confident about how division works when we're thinking about ratio problems and that you are much more confident now in using multiplication and division to solve ratio problems. We've looked at the similarities and differences between tables and bar models.

So I hope that you are feeling confident again in being able to use those to help you to make sense of a problem and to identify the calculations and equations that you need in order to solve them.