- Group all the divisions
- Do all the divisions from left to right
- Convert all the subtractions to negative additions
- Do the additions from left to right
When you use these rules correctly, the above problem simplifies as shown in Table 5-1. As you read down-
ward, each expression is equal to the one above it.
Here’s a challenge!
Start with the integer 5, multiply by −4, then divide that result by −2, then multiply that result by 8, then
divide that result by −20, and finally divide that result by −4. What do you end up with?
Solution
We can break this down step-by-step, paying careful attention to signs and using parentheses when we
need them:
5 × (−4)=− 20
−20/(−2)= 10
10 × 8 = 80
80/(−20)=− 4
−4/(−4)= 1
The Commutative Law for Multiplication
In basic arithmetic, you learned that you can multiply numbers in any order and always get
the same product. But you can’t expect to do the same thing if there is division anywhere in
the process.
The Commutative Law for Multiplication 73
Table 5-1. This is a step-by-step simplification of a complicated expression
containing addition, subtraction, multiplication, and division without any
parentheses to indicate the order in which the operations should be done.
As you read down the left-hand column, each statement is equal to all the
statements above it.
Statements Reasons
2 + 48 / 4 × 6 − 2 × 5 + 12 / 2 × 2 − 5 Begin here
2 + 48 / (4 × 6) − (2 × 5) + 12 / (2 × 2) − 5 Group all the multiplications
2 + 48 / 24 − 10 + 12 / 4 − 5 Do all the multiplications
2 + (48/24) − 10 + (12/4) − 5 Group all the divisions
2 + 2 − 10 + 3 − 5 Do all the divisions
2 + 2 + (−10)+ 3 + (−5) Convert all the subtractions to negative additions
− 8 Do the additions from left to right
