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.
Well the first property is closure, and that means when you multiply two integers together, the result you get is always another integer.
And integer multiplication is also commutative, and that means if you’ve got two integers, it doesn’t matter what order you multiply them together in. You’re gonna get the same answer. So if I’ve got two integers 𝑎 and 𝑏, 𝑎 times 𝑏 is gonna give me the same answer as 𝑏 times 𝑎.
And the third property is associativity: If I’ve got three integers, I could choose to pick the first two and multiply those together and multiply that result by the third or I could pick the second two, multiply those together, and multiply that result by the first; and I’ll still get the same answer.
Then we’ve got the distributive property: if I’ve got 𝑎 times the sum of 𝑏 and 𝑐, I can also write that as 𝑎 times 𝑏 plus 𝑎 times 𝑐 and I’ll get the same result.
And then I’ve got the identity property, and that basically means if I take an integer and multiply it by one I just get the same integer; and if I take an integer and multiply it by zero, I get the answer zero.
So we can choose between these properties to find a way of doing this calculation more easily. Well let’s look at the numbers first, eight, fifteen, and a hundred and twenty-five.
Well I know that four times twenty-five is a hundred, so eight times twenty-five is gonna be two hundred. Eight times a hundred is eight hundred, so eight hundred plus two hundred is a thousand. So eight times a hundred twenty-five is a thousand. So I’ve gonna have fifteen times a thousand, and it looks like that’s gonna be the way that’s gonna be easiest for us to tackle this calculation.
Now my first step is gonna be to use the commutative property to say that fifteen times a hundred and twenty-five is the same as a hundred and twenty-five times fifteen. That’s gonna bring the eight and the hundred and twenty-five together in the calculation.
Next, I’m gonna use the associative property to decide to multiply the first two numbers together before multiplying by the third number.
Next, in preparation for the distributive property, I’m gonna split a hundred and twenty-five up into a hundred plus twenty-five. And the distributive property is gonna let me say that that’s the same as doing 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 I also know that eight hundred plus two hundred is a thousand.
And that leaves us with a thousand times fifteen, which is fifteen thousand. So my answer is fifteen thousand. But because the question told us to use the properties of multiplication of integers, it’s very important that I’ve pointed out that this step was commutative, this step was associative, and this step was distributive. That’s all part of my answer.