Video: Multiplying Three Integers Using the Commutative and the Associative Properties

Evaluate 50 × 123 × 2 using the commutative and associative properties.

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Video Transcript

Evaluate 50 multiplied by 123 multiplied by two using the commutative and associative properties.

This problem asked us to evaluate or find the value of a multiplication that involves three numbers. Now, at first glance, this might look like a complicated calculation to work out. But we’re given two clues as to how to find the answer. We’re told to use the commutative and associative properties. What does each of these words mean?

The commutative property of multiplication states that we can change the order when we multiply and the product will always stay the same. And it’s this property of multiplication that we use when we learn that the three-by-five array represents exactly the same amount as a five-by-three array.

We can write this property as 𝑎 multiplied by 𝑏 equals 𝑏 multiplied by 𝑎. Importantly, it’s this property that we use if ever we want to make a calculation easier to work out mentally.

Let’s remind ourselves what the associative property of multiplication means. Well, this property says that we can group numbers in our multiplication in different ways, but the product will always stay the same. We can write this rule as 𝑎 multiplied by 𝑏 multiplied by 𝑐, where we put the brackets in different places to show the different ways we can group the numbers.

In the first example, we multiply 𝑏 by 𝑐 first because that’s in brackets and then we multiply 𝑎 by the total. In the second example, we find the answer to 𝑎 multiplied by 𝑏 first and then we multiply the total by 𝑐.

Different ways of grouping the same multiplication, but the answer will be the same. It’s this property that we use when we learn that four lots of two multiplied by three is exactly the same as three lots of four multiplied by two. Again, we can use this property to make mental calculations a lot easier.

Well, how can we use these two properties to help us find the answer to our problem? Firstly, can we see any numbers that we could multiply in a different order to help us? Working out 50 multiplied by 123 to start with might be tricky. So we could apply the commutative property and rearrange the calculation.

50 multiplied by two is a lot easier to start with than 50 multiplied by 123. And we know that by using the commutative property in this way, our product is still going to be the same at the end.

Let’s apply the associative property now. How can we group these numbers to make them easier to work out mentally? Well, multiplying 50 by two first gives us a total of 100. This makes our calculation into 100 multiplied by 123, which is a lot easier to work out than it was to begin with.

By the commutative and associative properties, we found that 50 multiplied by 123 multiplied by two equals 12300.

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