Topics
1. Operations with Fractions
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United States of AmericaVirginia
Grade 6

1.06 Solve problems with fraction operations

Lesson
Practice

Solve problems with fraction operations

We use fractions to solve many everyday problems. For example, in recipes, ingredients are often measured in fractions of a cup. If we wanted to know the total volume of the ingredients, we could use fraction addition.

We can use keywords to help us work out which operation we need to use to solve the problem. Here are the four operations and some common keywords that relate to them:

AdditionSubtractionMultiplicationDivision
morelessproductequally shared
addsubtractbyin each
all togetherhow many lefttimesper
totaldifferencegroups ofdivided by

Estimation can be a useful strategy for solving real-world problems, especially if the context of the problem doesn't require us to be exact.

Examples

Example 1

At a party, Bill makes a drink by combining 5 \, \dfrac{1}{3} \operatorname{ L} of water with 1 \, \dfrac{1}{2} \operatorname{ L} juice concentrate.

What is the total amount of the drink?

Worked Solution
Create a strategy

Identify the keyword in the story. The word "total" tells us we need to add the amounts for each part of the drink.

Apply the idea
\displaystyle \text{Total}\displaystyle =\displaystyle 5 \, \dfrac{1}{3} + 1 \, \dfrac{1}{2}Add the values
\displaystyle =\displaystyle 5 + \dfrac{1}{3} + 1 + \dfrac{1}{2}Split the mixed numbers into whole and fraction parts
\displaystyle =\displaystyle 6 + \dfrac{1}{3} + \dfrac{1}{2}Add the whole parts
\displaystyle =\displaystyle 6 + \dfrac{1 \cdot 2}{3 \cdot 2} + \dfrac{1 \cdot 3}{2 \cdot 3}Multiply for a common denominator
\displaystyle =\displaystyle 6 + \dfrac{2}{6} + \dfrac{3}{6}Evaluate the multiplication
\displaystyle =\displaystyle 6 + \dfrac{2 + 3}{6}Add the numerators over the common denominator
\displaystyle =\displaystyle 6 + \dfrac{5}{6}Evaluate the addition
\displaystyle =\displaystyle 6 \, \dfrac{5}{6} \operatorname{ L}Rewrite as a mixed number
Reflect and check

We could use estimation to see if our exact answer seems reasonable. For example, 5\dfrac{1}{3} is close to 5\dfrac{1}{2}, and if we add 1\dfrac{1}{2} we get 7. So our answer should be a little less than 7, which it is.

Example 2

Jack is making bags for his friends. He has 3 \, \dfrac{1}{2} \operatorname{ m} of fabric.

If each bag requires \dfrac{2}{5} \operatorname{ m} of fabric, how many bags can he make?

Express your answer as an improper fraction.

Worked Solution
Create a strategy

Identify the keyword in the story. The word "each" tells us we need to divide the length of fabric into equal sized pieces for each bag.

Apply the idea
\displaystyle \text{Number}\displaystyle =\displaystyle \dfrac{7}{2} \div \dfrac{2}{5}Divide the values
\displaystyle =\displaystyle \dfrac{7}{2} \cdot \dfrac{5}{2}Rewrite as multiplication using the reciprocal
\displaystyle =\displaystyle \dfrac{7 \cdot 5}{2 \cdot 2}Multiply the numerators and denominators separately
\displaystyle =\displaystyle \dfrac{35}{4}Simplify

Example 3

Jamal has \dfrac{5}{6}\operatorname{ m} of ribbon. After using some for a project, he has \dfrac{2}{5}\operatorname{ m} left. How much ribbon did he use?

Worked Solution
Create a strategy

To find out how much ribbon Jamal used, we need to subtract the amount of ribbon he has left from the total amount he started with. This means we will subtract two fractions: \dfrac{5}{6} - \dfrac{2}{5}.

Apply the idea

First, we find a common denominator for the two fractions. We can do this by multiplying the denominators together which gives us 6 \cdot 5 = 30.

\displaystyle \frac{5}{6} - \frac{2}{5}\displaystyle =\displaystyle \frac{5 \cdot 5}{6 \cdot 5} - \frac{2 \cdot 6}{5 \cdot6}Multiply for common denominator
\displaystyle \frac{5}{6} - \frac{2}{5}\displaystyle =\displaystyle \frac{25}{30} - \frac{12}{30}Evaluate the multiplication
\displaystyle =\displaystyle \frac{13}{30}Subtract the numerators

Jamal used \dfrac{13}{30}\operatorname{ m} of ribbon for his project.

Idea summary

Use keywords to help you identify which operation to use:

AdditionSubtractionMultiplicationDivision
morelessproductequally shared
addsubtractbyin each
all togetherhow many lefttimesper
totaldifferencegroups ofdivided by

Outcomes

6.CE.1

The student will estimate, demonstrate, solve, and justify solutions to problems using operations with fractions and mixed numbers, including those in context.

6.CE.1d

Estimate, determine, and justify the solution to single-step and multistep problems in context that involve addition and subtraction with fractions (proper or improper) and mixed numbers, with and without regrouping, that include like and unlike denominators of 12 or less. Answers are expressed in simplest form.

6.CE.1e

Estimate, determine, and justify the solution to single-step and multistep problems in context that involve multiplication and division with fractions (proper or improper) and mixed numbers that include denominators of 12 or less. Answers are expressed in simplest form.

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