Lesson Video: Properties of Multiplication in a Set of Rational Numbers Mathematics • 7th Grade

In this video, we will learn how to identify the properties of the multiplication operation in the set of rational numbers.

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Video Transcript

In this video, we’ll learn how to identify the key properties of the multiplication operation in the set of rational numbers. And we recall, of course, that a rational number is a number that can be written in the form 𝑎 over 𝑏 where 𝑎 and 𝑏 are integers; they’re whole numbers. Knowing these properties is really important as it can help us to solve problems which might initially seem quite tricky.

There are six properties involving multiplication that can help make problems easier to solve. The first is the commutative property. And this says that when two numbers are multiplied together, the result is the same regardless of the order of the multiplication. So, for example, two times three is equal to three times two. The next is the associative property. And this time we know that when three or more numbers are multiplied, the product is the same regardless of how the numbers are grouped. So, for example, two times three and then multiplied by four is the same as two multiplied by three times four.

Next, we have the multiplicative identity property. And this simply says that when we multiply any number by one, we just get that number. So, five times one is equal to five. The zero product property tells us that the product of a rational number and zero is zero. So, for example, five times zero is simply zero. Then we have the distributive property. This says that the sum of two numbers multiplied by a third number is equal to the sum of each of those individual numbers multiplied by that third number. In other words, this is when we deal with parentheses. So, three plus four times two is equal to three times two plus four times two.

Our sixth and final property is the multiplicative inverse property. And this says the product of a rational number and its reciprocal is one. So five times one-fifth is equal to one. Essentially, when we say multiplicative inverse, it’s just another way of saying the reciprocal. Let’s begin by just identifying some of these properties from examples.

Which property of multiplication is used in the equation negative four-ninths times negative nine over four equals one?

Let’s identify what’s actually happening in this multiplication problem. We take a rational number, negative four-ninths, and then we multiply that by negative nine over four. And so we should ask ourselves, what’s the relationship between negative four-ninths and negative nine over four? Well, we know that to find the reciprocal of a number, we divide one by that number. So, the reciprocal of negative four-ninths is one divided by negative four-ninths, and that’s equal to negative nine over four.

Another way to find the reciprocal of a number in fractional form is to simply switch the numerator and the denominator. And so, here we’re multiplying a number by its reciprocal, and we get one. Another word that we have or a phrase that we have for the word reciprocal is the multiplicative inverse. It’s so called because when we find the inverse, we essentially undo an operation. And in this case, we multiply it by something that gives us the result of one. And so, the property of multiplication that’s used here is the multiplicative inverse property. This says that if we multiply a rational number by its reciprocal, we’ll get one.

Let’s have a look at another example.

What property of multiplication is used in the sum eight times zero equals zero?

In this example, we’re finding the product of the number eight and zero. Eight is a rational number. And we see that the product of this rational number and zero is zero. This has a special name. Because we’re finding the product of a rational number and zero, we call this the zero product property. And so, this is the property of multiplication used in this problem. And it says that the product of any rational number and the number zero is zero.

What property of multiplication is used in the sum four times one equals four?

Here we have the product of the number four and one. And we see that the result is actually equal to the first number. It’s equal to four. So what’s the name of this? Well, this is the multiplicative identity property. This property tells us that the product of any rational number and one is that rational number. Here, four is the rational number, and we’re left with four when we multiply it by one.

Now that we’ve practiced identifying some of these properties, we’re going to see how their application can help us to simplify problems.

Fill in the blank: three times seven plus three times nine is equal to blank times seven plus nine.

Let’s look carefully at what’s going on here. We have the sum of two products. Remember, the product is when we multiply two numbers together. We’re adding three times seven and three times nine. Notice how each of these products has a common factor of three. This should make us think of the distributive property of multiplication. This says that the sum of two numbers times a third number is equal to the sum of each of those individual numbers multiplied by that third number. For instance, two plus three times four is equal to two times four plus three times four. You might consider this to be a little bit like distributing parentheses. We take the four, and we multiply it by each value inside our parentheses.

And so this looks a little bit like what we’ve got in our question, but we’re going to be reversing that process. In the right-hand side of our sum, we have something times seven plus nine. Well here’s the seven, and here’s the nine, and they’re both being multiplied by three. And then we’re finding the sum. This means that this sum must be equal to seven plus nine all multiplied by three. And so the three must go in the blank. Let’s check this by reversing the process. We’re going to multiply the three by the seven and then the three by the nine. And we see that three times seven plus nine is the same as three times seven plus three times nine.

We can check this further by actually calculating the answer to both parts of this sum. Seven plus nine is 16. So the left-hand side is three times 16, which is 48. Then three times seven is 21, and three times nine is 27. So the right-hand side is 21 plus 27, which is also 48. And we see that both sides of this equation are indeed equal. And so, the number that goes in our blank is three.

We’ll now consider another property of multiplication.

Which equation shows the commutative property of multiplication? Is it (A) two-thirds times one equals two-thirds? Is it (B) 0.5 times three is equal to 0.5 plus 0.5 plus 0.5? Is it (C) 3.5 times two is equal to two times 3.5? Is it (D) a half times four minus a quarter times four is equal to a half minus a quarter times four? Or, is it (E) a half times a quarter times two-thirds is equal to a half times a quarter times two-thirds?

Let’s recall what we mean by the commutative property of multiplication. The commutative property says that when two numbers are multiplied together, the result will be the same no matter the order of the numbers. For example, two times three is the same as three times two. And actually, this is a really useful property. For example, if you know you feel unconfident with a certain times tables, knowing that three times seven is the same as seven times three means you can think about the times table that you do know by heart. So, which of these examples shows this property? Well, it’s not (A). In fact, (A) is the multiplicative identity property. This says if you multiply a number by one, you get the original number.

Is it (B) 0.5 times three is 0.5 plus 0.5 plus 0.5? Well no, this just shows us that we can think about multiplication as repeated addition. So, what about (C) 3.5 times two is equal to two times 3.5? Well, yes we’ve shown that the result is the same even if we change the order in which we perform the multiplication. So, it’s (C). Let’s look at (D) and (E) next. (D) says that a half times four minus a quarter times four is equal to the difference between a half and a quarter times four. This is called the distributive property of multiplication.

And finally, (E) does look like it’s going to be correct, but in fact, this is all to do with grouping. This is called the associative property. And it says that when three or more numbers are multiplied together, the product is the same regardless of how those numbers are grouped. Notice that we use the parentheses to show the grouping here. And so the answer to this question is (C). The commutative property of multiplication is demonstrated by the equation 3.5 times two equals two times 3.5.

In our final example, we’ll look at how we can simplify a rather nasty-looking calculation by using a combination of these properties.

Calculate eight multiplied by 94 times seven multiplied by one-eighth.

Now, what we’re not going to do is work out the calculations inside our parentheses. Yes, the order of operations would usually tell us to do this, but we can use something called the associative property here. The associative property says that when we have three or more numbers being multiplied, the result is the same regardless of how those numbers are grouped. And so we’re going to group together eight and one-eighth and also 94 and seven. And so, this is the same as eight times one-eighth times 94 times seven.

At this stage, you might be wondering why this actually makes our life easier. Well, let’s look at eight times one-eighth. The multiplicative inverse property says that the product of a number and its reciprocal is equal to one. Well, the reciprocal of eight is one-eighth, so eight times one-eighth must be equal to one. And so this problem becomes one times 94 times seven. But the multiplicative identity property then says that if we multiply something by one, the result is just that original number. And so, one times 94 times seven must be the same as 94 times seven. But what do we do now?

Well, we’re going to use the distributive property. This says that the sum of two numbers times a third number is equal to the sum of each of those numbers times that third number. And so we’re going to write 94 as 90 plus four. And then we’re going to multiply this by seven by first multiplying the 90 by seven and then multiplying the four by seven. So, our problem becomes 90 times seven plus four times seven. Well, nine times seven is 63, so 90 times seven is 630. Then, four times seven is 28, so we get 630 plus 28 which is equal to 658. Eight times 94 times seven times one-eighth is 658.

We’re now going to recap the key parts of this video. In this video, we learned that the properties of the multiplication operation in the set of rational numbers can help us to solve quite difficult problems. We have the commutative property, and this says that when two numbers are multiplied together, the result is the same no matter the order we do this. For example, four times two is equal to two times four. We have the associative property, and this says that when we multiply three or more numbers together, it doesn’t matter which order we group these. So, for example, two times three times four would be the same as two times three times four.

We then have the multiplicative inverse property. This says that if we multiply a number by its reciprocal, we get one. Then, there’s the multiplicative identity property. And this says that if we multiply a number by one, we get that original number. We have the zero product property. And that says if we multiply a number by zero, we get zero. And finally, there’s the distributive property. This says that the sum of two numbers times a third number is equal to the sum of each of those numbers times the third number. So, for instance, two plus three times four is equal to two times four plus three times four.

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