2.05 Order of operations with rational numbers

Lesson

Practice

Introduction

We've already learned about the order of operations, which tells us the steps to evaluating expressions with multiple operations.

Evaluate operations inside grouping symbols such as parentheses (\ldots) or absolute value |\ldots|.
Evaluate exponents such as squares, or cubes.
Evaluate multiplication and division going from left to right.
Evaluate addition and subtraction going from left to right.

We use the same order of operations for rational numbers as we do for integers.

Ideas

Order of operations with rational numbers

Order of operations with rational numbers

Let's look through some examples of questions involving rational numbers and the order of operations.

Examples

Example 1

Evaluate the expression 8.5 + 7.2 + (-1.3).

Worked Solution

Create a strategy

Follow the order of operations.

Apply the idea

\displaystyle 8.5 + 7.2 + (-1.3)	\displaystyle =	\displaystyle 15.7+ (-1.3)	Evaluate the first addition
	\displaystyle =	\displaystyle 15.7-1.3	Combine adjacent signs
	\displaystyle =	\displaystyle 14.4	Evaluate

Example 2

Calculate 86+\dfrac{3}{10}\times \left(-2\right).

Worked Solution

Create a strategy

Follow the order of operations.

Apply the idea

\displaystyle 86+\dfrac{3}{10}\times \left(-2\right)	\displaystyle =	\displaystyle 86-\dfrac{6}{10}	Evaluate the multiplication
	\displaystyle =	\displaystyle \dfrac{860}{10}-\dfrac{6}{10}	Write the whole number as a fraction
	\displaystyle =	\displaystyle \dfrac{854}{10}	Evaluate
	\displaystyle =	\displaystyle 85\dfrac{4}{10}	Convert the improper fraction to mixed number
	\displaystyle =	\displaystyle 85\dfrac25	Simplify the fraction

Idea summary

When evaluating multiple operations with rational numbers:

Evaluate operations inside grouping symbols such as parentheses (\ldots) or absolute value |\ldots|.
Evaluate exponents such as squares, or cubes.
Evaluate multiplication and division going from left to right.
Evaluate addition and subtraction going from left to right.

Outcomes

7.NS.A.3

Solve real-world and mathematical problems involving the four operations with rational numbers. (Note: computations with rational numbers extend the rules for manipulating fractions to complex fractions.)

7.EE.B.3

Solve multi-step real-life and mathematical problems posed with positive and negative rational numbers in any form (whole numbers, fractions, and decimals), using tools strategically. Apply properties of operations to calculate with numbers in any form; convert between forms as appropriate; and assess the reasonableness of answers using mental computation and estimation strategies.