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Evaluate 50 multiplied by 123 multiplied by two using the commutative and associative properties.
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This problem asked us to evaluate or find the value of a multiplication that involves three numbers.
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Now, at first glance, this might look like a complicated calculation to work out.
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But weβre given two clues as to how to find the answer.
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Weβre told to use the commutative and associative properties.
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What does each of these words mean?
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The commutative property of multiplication states that we can change the order when we multiply and the product will always stay the same.
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And itβs 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.
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We can write this property as π multiplied by π equals π multiplied by π.
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Importantly, itβs this property that we use if ever we want to make a calculation easier to work out mentally.
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Letβs remind ourselves what the associative property of multiplication means.
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Well, this property says that we can group numbers in our multiplication in different ways, but the product will always stay the same.
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We can write this rule as π multiplied by π multiplied by π, where we put the brackets in different places to show the different ways we can group the numbers.
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In the first example, we multiply π by π first because thatβs in brackets and then we multiply π by the total.
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In the second example, we find the answer to π multiplied by π first and then we multiply the total by π.
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Different ways of grouping the same multiplication, but the answer will be the same.
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Itβs 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.
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Again, we can use this property to make mental calculations a lot easier.
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Well, how can we use these two properties to help us find the answer to our problem?
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Firstly, can we see any numbers that we could multiply in a different order to help us?
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Working out 50 multiplied by 123 to start with might be tricky.
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So we could apply the commutative property and rearrange the calculation.
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50 multiplied by two is a lot easier to start with than 50 multiplied by 123.
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And we know that by using the commutative property in this way, our product is still going to be the same at the end.
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Letβs apply the associative property now.
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How can we group these numbers to make them easier to work out mentally?
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Well, multiplying 50 by two first gives us a total of 100.
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This makes our calculation into 100 multiplied by 123, which is a lot easier to work out than it was to begin with.
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By the commutative and associative properties, we found that 50 multiplied by 123 multiplied by two equals 12300.