Basic Math and Pre-Algebra
PART 1. The World of Numbers
CHAPTER 3. Order of Operations and Integers
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 ÷ 2 = 238 ÷ 2 = 119, but you might notice that 7(34 ÷ 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 x 5 + 4 x 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.
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 = 1,800
The distributive property is a great help in mental math. You might not think you can multiply 45 x 98 in your head, but if you think of it as 45(100 - 2), it's 4,500 - 90, which is 4,410.
When you see a minus sign in front of parentheses, as in 14 - (8 + 3), it says to do what’s in the parentheses and subtract the result from the number before the minus sign. The distributive property gives you the option to distribute the minus and rewrite the problem as 14 - 8 - 3. Taking away the sum of 8 and 3 is the same as taking away 8 and taking away 3.
Tell whether it’s easier to perform the operation in parentheses first or to distribute first. Then solve the problem.
6. 2(35 + 14)
8. 7(100 - 2)
7. 3(20 + 8)
9. 15(40 - 14)
10. 250(1,000 - 400)
By now you know how to do all your arithmetic, and you’ve memorized your addition facts and multiplication tables. You know that the commutative, associative, and distributive properties are handy helpers, and you’ve got PEMDAS totally under control. What else is there to say about arithmetic? Well, a lot actually. Some of it can wait for a later chapter, but this is a good time to tackle the half-truth you probably learned about subtraction.
As soon as you begin to subtract, you run into a problem. You can subtract 5 - 3 and get 2, but what happens if you subtract 3 - 5? The easy answer is, “you can’t take a bigger number away from a smaller one.” In some sense that’s true. If you only have 3 cookies, you can’t give me 5 cookies, which is sad, because I love cookies.
But what if you promised me 5 cookies? You owe me 5 cookies. If you give me 3 cookies, you still owe me 2 more cookies. You don’t have cookies, but you owe 2 cookies. In a sense, you have less than no cookies, because even if you get more cookies, you have to give me 2. How do we write, in symbolic form, that you owe me 2 cookies? That opposite-of-having idea is written by putting a negative sign in front of the 2. You have -2 cookies. The number -2, or negative 2, is the opposite of 2.
This notion of needing a way to express opposites leads you to a bigger set of numbers. Every number has an opposite, and the whole numbers didn’t take that into account. So you need a larger set of numbers that will include all the whole numbers and their opposites. Those numbers are called the integers.
Try to get in the habit of calling a number less than zero “negative” rather than “minus.” The word “minus” signals subtraction, and not everything you do with numbers less than zero is a subtraction. Talking about “negative 4” reminds you that it's the opposite of “positive 4.”