PROPERTIES OF ARITHMETIC

The following arithmetic properties can be used to make computation easier. In the next section we will explore some shortcuts, called compensating techniques, that use these properties and simplify computation.

Even with calculators, some computation is still easier to do if it is reorganized. Consider looking for shortcuts whenever you are practicing problems. When you try new techniques they may seem like "longcuts" at first, but once you get used to them they can save crucial moments during a test. More importantly, the confidence you get from working quickly and efficiently will benefit your overall performance.

COMMUTATIVE PROPERTY FOR ADDITION

If a and b are real numbers, then a + b = b + a.

Example: 2 + 5 = 5 + 2

COMMUTATIVE PROPERTY FOR MULTIPLICATION

If a and b are real numbers, then a x b = b x a.

Example: 4.002 x 25 = 25 x 4.002

ASSOCIATIVE PROPERTY FOR ADDITION

If a, b, and c are real numbers, then (a + b) + c = a + (b + c)

Example:

(23.5 + 75) + 25 = 23.5 + (75 + 25)

98.5 + 25 = 23.5 + 100

125.5 = 123.5

Notice that 23.5 + (75 + 25) is easier to compute!

ASSOCIATIVE PROPERTY FOR MULTIPLICATION

If a, b, and c are real numbers, then (a x b) x c = a x (b x c)

Example:

=

=

=

Again note that the quantity on the right is easier to compute.

DISTRIBUTIVE PROPERTY OF MULTIPLICATION OVER ADDITION (OR SUBTRACTION)

If a, b, and c are real numbers, then a(b c) = ab ac

Example:

4(77 + 23) = 4(77) + 4(23)

4(100) = 308 + 92

400 = 400

Note how much easier the quantity on the left is to compute.

In this example, we know that multiplying 4 x 77 and 4 x 23 wouldn't take long with a calculator. However, 4 x 100 is so much simpler that not only is there no need to check your work, but when you do calculations this easily you have more confidence in your overall performance as you work.