Order of Operations

Math expressions and equations often involve several operations. For example, to convert from Celsius to Fahrenheit, you first multiply the number of Celsius degrees by the fraction image and then add 32 degrees. To convert from Fahrenheit to Celsius, you subtract 32 degrees from the number of Fahrenheit degrees and then multiply the result by image . A rule called the order of operations is used to write math expressions clearly so that anyone seeing the formula or equation would know whether multiplication was the first step or the second step.

Order of Operations

  1. Evaluate all expressions within parentheses.

  2. Evaluate all terms with exponents.

  3. Multiply and divide from left to right.

  4. Add and subtract from left to right.

Example 1

Temperature Conversions

Convert these temperatures.

  1. 37 °C to degrees Fahrenheit

  2. 48 °F to degrees Celsius

Solution

Substitute the known value into each equation and then solve using the order of operations.

a. Substitute 37° into the equation. image
Multiply. F = 66° + 32°
Add. F = 98 °F
b. Substitute 48° into the equation. image
Subtract. image
Multiply. C = 9 °C

A-6

Example 2

Parentheses, Exponents, and Fractions

Evaluate these expressions.

  1. image

  2. image

Solution

Evaluate the expression in the parentheses first.

a. Original expression image
Subtract the numbers within the parentheses. image
Multiply image by 81. = 45

b. The fraction line acts like parentheses. In fact, when the expression is entered into a calculator, parentheses are required around the 2 · 5.

Original expression. image
Evaluate the expressions above and below the fraction line. image
Divide 36 by 10 and then add 12. = 15.6

Practice Exercises

  1. Evaluate these expressions.

    1. 3 · 24 ÷ 8

    2. 3 + 24 · 8

    3. 3 – 24 + 8

    4. (3 + 24) · 8

    5. (3 + 21) ÷ 8

    6. 3 – (24 + 8)

  2. Calculate the value of each expression.

    1. –2 + 5– (–8)

    2. (– 52) – (–3)2

    3. –0.3 · 20 + 15

  3. Insert parentheses as needed to make each equation true.

    1. 1 5 ÷ 3 + 7 – 4 = –48

    2. 1 5 ÷ 3 + 7 – 4 = 8

    3. –42 + –32 = –7

Show Answers

Answers

  1. 1a. 9

  2. 1b. 195

  3. 1c. –13

  4. 1d. 216

  5. 1e. 3

  6. 1f. –29

  7. 2a. 11

  8. 2b. –34

  9. 2c. 9

  10. 3a. (15 ÷ 3 + 7)(–4) = –48

  11. 3b. 15 ÷ 3 + 7 – 4 = 8

  12. 3c. –42 + (–3)2 = –7