Lesson Video: Word Problems: Comparison Problems | Nagwa Lesson Video: Word Problems: Comparison Problems | Nagwa

# Lesson Video: Word Problems: Comparison Problems Mathematics

In this video, we will learn how to solve additive and multiplicative comparison problems by drawing a bar model to help us choose the correct operation.

17:36

### Video Transcript

Word Problems: Comparison Problems

In this video, we’re going to learn how to solve comparison problems that we need to use addition or multiplication to solve. And we’re going to use bar models to help us choose the correct operation each time. Now, if you were asked to list some superheroes, I’m sure you could think of a few names, but have you ever thought, these are the superheroes that are around now. What about those superheroes that have gotten a bit older and have retired? What does they do with their time? Well, they mostly sit around drinking cups of tea or knitting and looking back at the good old days, talking about what they used to get up to, comparing what they can do now with what they could do back then.

For example, if we look at what this superhero is saying, she’s saying, “When I was younger, I could clear six more buildings in a leap than I can do now.” Can you see the comparison that’s going on in this sentence? She’s comparing what she could do then and what she can do now. Let’s imagine that now she can clear only two buildings in a single leap. We could draw a bar model now to help show the comparison. We’ll draw a bar to represent what she can do now, and that’s to leap over two buildings, and a bar to represent what she could do back in the good old days. And we can see that the top bar is longer than the bottom bar, can’t we? We know this because she could clear six more buildings in a leap than she can do now.

And now that we’ve drawn this bar model, we can see how to find out what this superhero could do when she was younger. To find the value of the top bar, we need to add two and then six more. Back in her prime, this superhero could clear eight buildings in one leap. We solved this comparison using addition. And the bar model helped us understand that it was addition we needed. It showed us which operation to use. And did you notice there was also a clue in what the superhero was saying. The words six more give us an idea that we’re going to have to add, don’t they?

Let’s have a think about another type of problem. Let’s imagine that this particular superhero is looking back and saying, “Back in the good old days, I could lift five times as many cars as I can now.” And let’s also imagine that we know that now he can only lift three at a time. Again, we could use a bar model to help us work out what this particular superhero used to be able to do. We could draw a bar to represent what he can do now, and we’ll label it three to represent the number of cars he can lift. Now, we know by looking at what he’s said that this particular superhero used to be able to lift five times as many cars, so that’s five times as many as three. This is the same as five lots of three.

In our last problem, we used addition to compare the amounts. But can you see we might use a different operation this time? Although you could possibly find the answer by working out three plus three plus three plus three plus three, it will be a lot quicker to calculate five times three. And so we can write an equation now that’s going to help us to solve this problem: five times three equals what. And it’s not just the bar model that we can use to help us. If we look closely at what the superhero has said, he said he could lift five times as many. And that little phrase five times tells us that we’re probably going to need to use multiplication to find the answer.

Now we can complete the equation and use it to help us solve the problem. We know five times three is 15, and so we can say that back in the good old days, this particular superhero could lift 15 cars. You can see now why he’s complaining about only being able to manage three now. So once again, we’ve solved the problem that needed us to compare two values together. But this time, we needed to use multiplication to solve it. And the bar model was really useful again in helping us to understand which operation to use. We’re going to try some questions now that are all about comparing values together. And to solve each word problem, we’re either going to need to add or multiply. To help us choose which operation to use, we’re going to be drawing bar models whenever we can. Let’s see if we can apply what we’ve learned.

Select the problem that can be solved by the equation five times 18 equals 90. Noah is 18 years old. Sophia is five years older than Noah. How old is Sophia? Or Noah is 18 years old? Sophia is five times as old as Noah. How old is Sophia? Or finally, Noah is 90 years old. Sophia is five times as old as Noah. How old is Sophia?

In this question, we’re given three problems. And they’re all very similar, aren’t they? They’re to do with the ages of Noah and Sophia. These are what we call comparison problems because we’re comparing one person’s age to the other. Now, one of the things we do when we’re solving a word problem is to work out what we need to do to find the answer. And often, we can write an equation that’s going to help us to find it. And in the first sentence, we’re told that one of these problems can be solved by working out five times 18 equals 90. So, in a way, this is a sort of backwards question, isn’t it? We’ve got the answer. We just need to work out what the problem was.

One thing that we know can be really helpful when we’re trying to solve a word problem is sketching a bar model to try and show what the question is asking. So let’s read through each of our possible answers really carefully and try sketching bar models. In the first problem, we’re told that Noah is 18 years old. And then comes the comparison, Sophia is five years older than Noah. So we could draw a bar to represent Noah’s age and label it 18 and another bar to represent the five years that Sophia is older than Noah. And then we can see how to find Sophia’s age. This is an addition problem. And to find Sophia’s age in this problem, we will need to add together 18 and five.

Our second problem begins in the same way. We’re told that Noah is 18 years old. But something slightly different comes next. Instead of Sophia being five years older than Noah, we’re told that she’s five times as old as Noah. Now, this means something completely different. If we draw a bar labeled 18 to represent Noah’s age again, we can show Sophia’s age by drawing five times this amount, in other words, five bars all the same length labeled 18. Can you see this is no longer an addition problem, is it? We solve this problem by using multiplication. We’d need to find the answer to five lots of 18, five multiplied by 18. This is the equation we were looking for, and we know already that the answer is going to be 90. Looks like this is the correct problem, doesn’t it? Let’s just check the last one.

Our final problem is very similar to the one before it. Can you see there’s only one difference? Instead of Noah being 18 years old, he’s now 90 years old. So we can draw a very similar bar model to represent this problem. We can start off drawing a bar to represent Noah’s age. And as we say, instead of 18, it’s now 90. And again, we can show Sophia’s age by drawing five times this bar, each one labeled 90. Can you see that this problem doesn’t really make a lot of sense, does it? To find Sophia’s age in this problem, we’d have to multiply five by 90. The answer is going to be over 400. And this isn’t very realistic at all. It’s also not the same as the equation we were looking for.

We found the problem that can be solved by the equation five times 18 equals 90. And the way we did it was by sketching a bar model for each of the problems and thinking about how we might find the answer. The correct problem is the one that reads, “Noah is 18 years old. Sophia is five times as old as Noah. How old is Sophia?”

Select the problem that can be solved by the addition equation 26 plus 12 equals 38. In the morning, 26 boats set sail from a harbor. In the afternoon, 12 more boats set sail. How many boats are out sailing by the afternoon? Or in the morning, 26 boats set sail from a harbor. In the afternoon, 12 times as many boats set sail than in the morning. How many boats set sail in the afternoon?

Sometimes, when we’re solving a word problem, we can see a lot of words on the page. Well, in this problem, we’ve got enough words for about three problems. Our two possible answers are word problems. And then the original question itself is a word problem, too. So there are lots of words here. We need to go through the whole problem really carefully. To begin with, we’re told to select or to choose the problem that can be solved by an addition equation. We know that sometimes we solve problems by adding, subtracting, maybe multiplying, dividing, sometimes by doing more than one thing.

But the problem we’re looking for is the problem that can be solved by using a particular addition equation. And we’re told what it is. 26 plus 12 equals 38. And we’re given two word problems as possible answers. If we were to try to work out the answer to these problems, we’d only be able to solve one of them by working out 26 plus 12 equals 38. But which one? Now, there are no pictures or diagrams to help us understand these problems. So in cases like this, sometimes a bar model can be really helpful. Let’s look at the first problem to begin with.

We’re told that in the morning 26 boats set sail from a harbor. So we could start off by drawing a bar and labeling it 26. These are the boats that go out in the morning, but we’re then told something else happens in the afternoon: 12 more boats set sail. So we can draw a little bit extra to our bar and we’ll label it 12 for the 12 more boats. Then finally, we’re asked how many boats are out sailing by the afternoon. We could draw one long bar that sums up this amount. We know that the answer is going to be the 26 boats that set sail in the morning and then the 12 more, in other words, 26 plus 12.

This problem can be solved by using addition. And when we look at the addition equation in our first sentence, it looks like we found the right answer already. But it’s very important that we check all possible answers. So let’s look at our final word problem. This problem starts off in a similar way: In the morning, 26 boats set sail from a harbor. But can you see where the difference lies? In the afternoon, 12 times as many boats set sail than in the morning. We’re going to need a little bit more space for this bar model, aren’t we? If we draw a bar to represent the number of boats that set sail in the morning, which is 26 again, then to show the number of boats that set sail in the afternoon, we’re going to have to make this 12 times as long.

To answer this problem, we’d need to find 12 lots of 26 or 12 times 26. This is a multiplication problem. In both of the problems, we compared the number of boats that set sail from a harbor in the morning to in the afternoon; but they’re both solved in very different ways. We used bar models to work out which operation to use. The problem that can be solved by the addition equation 26 plus 12 equals 38 is the one that reads, “In the morning, 26 boats set sail from a harbor. In the afternoon, 12 more boats set sail. How many boats are out sailing by the afternoon?”

Jacob started training to join the basketball team. In the first week, he scored a total of 23 baskets. In the second week, he scored 12 more baskets than in the first week. Should I add or multiply to find the number of baskets he scored in the second week? Why? Because “a total of” tells us to use addition. Because “12 more” tells us to use multiplication. Because “a total of” tells us to use multiplication. Because “12 more” tells us to use addition. Or because 12 more tells us to use subtraction.

Questions like this are really interesting because they’re not asking us to find the answer; they’re asking us how we would find the answer. So we don’t have to work any answers out to our calculation here. We just need to work out what the calculation is. To begin with, let’s read that word problem once again. And as we do, we could sketch a bar model to help us understand what we need to do. So, firstly, we’re told that Jacob started training to join the basketball team. In the first week, he scored a total of 23 baskets. So we could draw a bar to represent week one. And we’ll label it 23 to represent all those baskets he scored in week one.

But it seems that Jacob’s training pays off, doesn’t it, because we’re told that in week two, he scored 12 more baskets than in the first week. How could we show the idea of 12 more than 23 on our bar model? Well, we could draw an extra bar labeled 12. Those are the 12 more that he scored. And then we could draw a new bar that covers all of this length. And this is the number of baskets that Jacob scores in the second week. Now, as we’ve said already, we’re not asked to find out how many baskets Jacob scores in the second week. We’re just asked how we would do this. Should we add or multiply? Well, we can see by looking at our bar model, can’t we, that the length of the second bar is worth 23 plus 12 more. We’re going to need to add to find the answer, aren’t we?

And then comes a really important question: Why? How do we know that we need to add and not multiply? What is it about the problem that we were given that shows us that we need to add to find the answer. We’re given five possible explanations here. Now perhaps the first time we read through these, it might have sounded a little complicated, but we can get rid of some of these answers straight away because we’ve already decided we need to add to find the answer. We don’t need to use multiplication, so we can cross through this sentence. And we know this sentence is wrong, too. And we definitely don’t need to use subtraction, do we? So we’re only left with two possible answers.

And the only difference between these answers are two little phrases. Did we see the words “a total of” in the question and thought to ourselves, oh, we need to use addition? Or did we see the phrase “12 more” and that’s what told us we needed to use addition? Well, firstly, we know that finding the total of something does mean that we need to use addition. But when we go back to the question and look at how those words are used, we’re just told in the first week he scored a total of 23 baskets. In other words, this was the amount he scored in week one. It’s got nothing to do with us finding the answer to the question, has it? It’s just saying that was the whole amount.

Now, if we find the phrase “12 more,” we can see that this is all about comparing the two weeks together, isn’t it? In week two, he scored 12 more than in the first week. This is where we find our answer, and this is how we know we need to use addition. In this question, we didn’t solve a word problem. We just thought about how we would solve it. To begin with, we used a bar model to work out that we needed to add to find the number of baskets that Jacob scored in the second week. And we know we need to add because “12 more” tells us to use addition.

So what have we learned in this video? We have learned how to solve comparison problems by using addition or multiplication. We’ve also used bar models to help us choose which operation to use.

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