Topics
2. Rational Numbers
topic badge
United States of America
Grade 7

2.06 Problem solving with rational numbers

Lesson
Practice

Introduction

We now see our four primary operations, addition, subtraction, multiplication, and division, with different rational numbers. In this lesson, we will use  order of operations  and apply  properties of operations  to solve problems involving rational numbers.

Problem solving with rational numbers

When working with real-life application problems, it's important to be able to identify keywords. These keywords can help us determine which operation(s) can be used to solve the problem. Here are some examples of keywords which indicate the use of a certain operation.

additionsubtractionmultiplicationdivision
morelessproductequally shared
addsubtractbyin each
all togetherhow many lefttimesper
totaldifferencegroups ofdivided by
increasedecreasedoublehalf
combinedfewermultiplysplit

There are many different strategies we can use to solve application problems. In the questions below, we will just show one approach, but it's important to recognize that there are many different ways to approach the problems.

Examples

Example 1

Consider the following phrase: "The quotient of -7 by the sum of 6 and -4."

a

Without simplifying the result, translate this sentence into a mathematical expression.

Worked Solution
Create a strategy

Quotient refers to division and sum refers to addition.

Apply the idea
\displaystyle \text{Expression}\displaystyle =\displaystyle \dfrac{-7}{6+(-4)}Divide -7 by the sum of 6 and -4
b

Evaluate the expression.

Worked Solution
Create a strategy

Evaluate the answer from part (a).

Apply the idea
\displaystyle \dfrac{-7}{6+(-4)}\displaystyle =\displaystyle -\dfrac{-7}{2}Evaluate the addition

Example 2

How many 0.26 \text{ L} glasses can a 20.8 \text{ L} water bottle fill?

Worked Solution
Create a strategy

Divide the amount of liquid in the water bottle by the amount in each glass.

Apply the idea
\displaystyle \text{Number of glasses}\displaystyle =\displaystyle \dfrac{20.8}{0.26}Divide the bottle liters by the glass liter
\displaystyle =\displaystyle \dfrac{20.8 \times 100}{0.26 \times 100}Multiply both by 100
\displaystyle =\displaystyle \dfrac{2080}{26}Evaluate
\displaystyle =\displaystyle 80Evaluate

Example 3

Jenny takes out a loan of \$2200. She pays back \$42.60 each month and doesn't have to pay interest.

If she has made 5 repayments so far, how much does Jenny still owe?

Worked Solution
Create a strategy

Subtract the total repaid from the total amount borrowed.

Apply the idea
\displaystyle \text{Balance}\displaystyle =\displaystyle 2200-(42.60\times 5)Subtract the total repaid from borrowed amount
\displaystyle =\displaystyle 2200-213Evaluate the multiplication
\displaystyle =\displaystyle \$1987.00Evaluate

Example 4

A bottle is \dfrac27 full of orange juice. If 230 milliliters of orange juice is added to it, the bottle is \dfrac56 full. How many milliliters does the bottle hold when full?

Worked Solution
Create a strategy

Divide the added amount by the difference of first full amount from the second full amount.

Apply the idea
\displaystyle \text{Full bottle}\displaystyle =\displaystyle 230\div\left({\dfrac56-\dfrac27}\right)Divide the added amount by the amount difference
\displaystyle =\displaystyle 230\div\left(\dfrac{35}{42}-\dfrac{12}{42}\right)Find the common denominator
\displaystyle =\displaystyle 230\div\dfrac{23}{42}Evaluate the parenthesis
\displaystyle =\displaystyle \dfrac{230\times 42}{1\times23}Multiply by the reciprocal
\displaystyle =\displaystyle \dfrac{9660}{23}Evaluate the multiplication
\displaystyle =\displaystyle 420\text{ mL}Evaluate
Idea summary

Recall the following operations and keywords when solving worded problems.

additionsubtractionmultiplicationdivision
morelessproductequally shared
addsubtractbyin each
all togetherhow many lefttimesper
totaldifferencegroups ofdivided by
increasedecreasedoublehalf
combinedfewermultiplysplit

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.

What is Mathspace

About Mathspace