A = (9 – 3) × 2 + 1 “Subtract 3 from 9, double your answer, and then add 1.”
B = 9 – 3 × 2 + 1 “Multiply 3 by 2, subtract from 9, and then add 1.”
C = 9 – (3 × 2 + 1) “Multiply 3 by 2, add 1, and subtract your answer from 9.”
D = 9 – 3 × (2 + 1) “Add 2 to 1, triple your answer, then subtract it from 9.”
Although they all consist of the same numbers and the same operation symbols, these numerical expressions are not calculated in the same way and each one gives a different result (A = 13, B = 4, C = 2, D = 0).
What are the rules for calculating such expressions? The clues are in the way the teacher read them out.
I. Order of operations
To be sure to get the correct answer, conventions have been agreed upon. So, to calculate a numerical expression without parentheses or brackets, we complete the calculations from left to right, starting with the multiplication and the division, which take priority over addition and subtraction.Note: If there is only addition and subtraction (or only multiplication and division), we complete the calculations from left to right.
Example: B = 9 – 3 × 2 + 1.
First, do the multiplication, which takes priority over the addition and the subtraction: B = 9 – 6 + 1.
Then do the subtraction and the addition, from left to right: B = 3 + 1 = 4.
II. Calculating with parentheses
If there are parentheses, start by doing the calculations inside the parentheses. Complete the calculations, following the priorities defined in section I.Example 1: C = 9 – (3 × 2 + 1).
Start by calculating 3 × 2 + 1, which is in parentheses. To do this, first work out the multiplication, which takes priority: C = 9 – (6 + 1).
Then complete the calculation inside the parentheses: C = 9 – 7 = 2.
Example 2:
This expression can be written in the following form:
G = 3 + (6 + 4) ÷ (7 – 2).
Start by working out 6 + 4 and 7 – 2: G = 3 + 10 ÷ 5.
Then do the division, which takes priority over the addition: G = 3 + 2 = 5.
Read more:
Applying the Distributive Law
Calculating a Numerical Expression (2)
Determining the Common Factors of Two Integers
Recognizing a Proportional Relationship
Rounding to the Nearest Unit
Writing a Numerical Expression That Corresponds to a Sequence of Operations