Using the properties of multiplication of integers, calculate eight times fifteen
times one hundred and twenty-five. Okay, first let’s think about what the properties of multiplication of integers are.
The first property is closure: when you multiply two integers together, the result
is always another integer.
Next, we’ve got commutativity. And that means it doesn’t matter what order you multiply your integers together
in; you’ll always get the same answer. So if I’ve got two integers 𝑎 and
𝑏, whether I do 𝑎 times 𝑏 or 𝑏 times 𝑎 I’m still gonna get the same answer.
Thirdly, we’ve got associativity. And that means if I’ve got three integers that I’m
multiplying together, it doesn’t matter if I multiply the first two together first and take
that result and multiply it by the third integer or whether I take the second two integers and
multiply those together and take that result and multiply it by the first integer. I’m still
gonna get the same answer.
Fourthly, integer multiplication is distributive. And that means if I’ve got one
integer times the sum of two other integers like 𝑎 times 𝑏 plus 𝑐 then
that can also be written as 𝑎 times 𝑏 plus 𝑎 times 𝑐.
And fifthly, we have the property of identity. And that’s basically, if I multiply any integer by one I just get the same integer
as an answer; and if I multiply any integer by zero, I just get zero as an answer.
So we’ve gotta use these properties in order to make the calculation of this a
little bit easier. At first, I’m gonna take a look at these numbers: eight, five, and a hundred and
Well I know that four times twenty-five is one hundred, so eight times twenty-five is
gonna be two hundred. And I know that eight times one hundred is eight hundred, so eight times one
hundred and twenty-five is eight hundred plus two hundred, which is a thousand. So I’ve then got
fifteen times a thousand, so it looks like that’s gonna be a good approach.
So my first step is gonna be to rearrange that calculation. It doesn’t matter
what order I multiply them together in; I’m gonna get the same answer. So I’m gonna swap
the fifteen and the hundred and twenty-five so that I can then tackle that eight times a hundred and
twenty-five. So that step is the commutative property in action.
Next, I’m gonna decide to multiply eight times twenty-five before multiplying by fifteen, and
that’s the associative property.
And then in preparation for the distributive step, I’m gonna split up a
hundred and twenty-five into a hundred plus twenty-five. Now the distributive property tells me that’s the same as eight times a hundred plus
eight times twenty-five.
And now I know that eight times a hundred is eight hundred and eight times twenty-five
is two hundred. And of course, eight hundred plus two hundred is a thousand.
And lastly, a thousand times fifteen is fifteen thousand. So the answer is fifteen thousand; but because the question told us to use the
properties of multiplication of integers, it was very important that I wrote the fact that we
had this step being commutative, this step being associative, and this step being distributive.