Use real number properties to evaluate 75 multiplied by nine multiplied by eight mentally.
In this problem, we’re given a calculation, and we’re asked to work it out mentally by using real number properties. Before we think about real number properties, let’s just have a look at the calculation that we’re being asked to work out. The dots in this calculation represent numbers that are being multiplied. So we can read the calculation as 75 multiplied by nine multiplied by eight. The brackets show us which part of the calculation to work out first. So if we were working out this the way it’s written, we’d find the answer to nine multiplied by eight and then we’d multiply 75 by whatever that answer was. We can do a quick mental calculation and say that nine eights are 72, but then we’re faced with 75 multiplied by 72, which is a lot harder to work out mentally.
And this is why our problem asked us to use real number properties. By applying these properties, we can make the calculation a lot easier to solve. The distributive property says that we can multiply numbers in any order, and they always make the same total. So we could find the answer to nine multiplied by 75 multiplied by eight or eight multiplied by nine multiplied by 75. However we arrange the numbers in a multiplication, they’ll always make the same product. So how can we arrange these numbers to make the calculation easier? When any number is multiplied by eight, it’s the same as doubling it, then doubling it, then doubling it again. 75 doubled is a 150, doubled again is 300, and doubled again is 600. So by grouping the numbers in this way, we’ve changed the calculation into 600 multiplied by nine. The associative property of multiplication says that we can group numbers in a multiplication in different ways, work out different parts of the multiplication first, and still get the same product.
So let’s go through and find the answer mentally. As we’ve seen already, 75 multiplied by eight is 600. So our calculation becomes 600 multiplied by nine. We know that six nines are 54, and so 600 nines are 54 hundreds or 5400. And so by changing the order of the numbers and also changing the way that we group them, we were able to calculate the answer much more quickly: 75 multiplied by nine multiplied by eight equals 5400.