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
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
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.