36 Part 1: The World of Numbers
The Distributive Property
Occasionally, when you’re following the order of operations, you many notice that you have a
little wiggle room in the rules. For example, at the end of Example 3 above, I said 7[34] z 2 =
238 z 2 = 119, but you might notice that 7(34 z 2) = 7(17) is also 119. That’s because dividing
the product of 7 and 34 by 2 is equivalent to dividing one of them by 2 and then multiplying.
Dividing is so closely related to multiplication that you have a little flexibility. But it’s risky to try
to cheat the rules. Yes, sometimes in a problem involving only multiplication and division or only
addition and subtraction, you might be able to change things around a bit, but most of the time,
you’ll want to stay with PEMDAS.
There’s an important rule in arithmetic that gives you a choice of what to do first when you have
multiplication and addition or subtraction, however. That rule is called the distributive property.
It says that if you’re asked to do an addition or subtraction and then multiply the result by a
number, you can do just that, or you can choose to spread out the multiplication. Before you add,
you multiply each of the addends by the multiplier, like this: 4(5 + 3) = 4 v 5 + 4 v 3 = 20 + 12
= 32. You get the same answer if you do the addition first 4(5 + 3) = 4(8) = 32. The distributive
property gives you the chance to decide which you think will be easier.DEFINITION
The distributive property says that for any three numbers a, b, and c, c(a + b) = ca + cb.
In other words, the answer you get by first adding a and b and then multiplying the
sum by c will be the same as the answer you get by multiplying a by c and b by c and
then adding the results.Consider the problem 17(86 + 14).
The easiest way to do this one is to add first.
17(86 + 14) = 17(100) = 1,700
Now think about 4(125 + 325).
This might be easier if you distribute.
4(125 + 325) = 4(125) + 4(325) = 500 + 1,300 = 1,800WORLDLY WISDOM
The distributive property is a great help in mental math. You might not think you can
multiply 45 v 98 in your head, but if you think of it as 45(100 – 2), it’s 4,500 – 90,
which is 4,410.