Topics
2. Rational Numbers
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United States of America
Grade 7

2.03 Operations with fractions

Lesson
Practice

Introduction

We've already learned how to  add, subtract  ,  multiply and divide fractions  . Similarly, we've looked at each of these  operations with integers  .

The processes we know will be the same when we have questions with negative fractions - we'll just combine the two skills and their rules to complete the operation.

Operations with fractions

There are some key differences that we encounter when adding, subtracting, multiplying, and dividing fractions as opposed to other types of numbers. Here are some strategies that we can use with fractions:

  • When adding or subtracting fractions, ensure that there is a common denominator. Once the denominators are equal, we add the numbers across the top of the fraction bar and the denominator remains the same.

  • When multiplying fractions, multiply the numerators, multiply the denominators, and simplify.

  • When dividing fractions, take the reciprocal of the divisor and multiply. Then, follow the rules for multiplication.

  • When dealing with mixed numbers, change it into an improper fraction and proceed as seen above. You may be expected to change the solution back to a mixed number.

Using the number line or zero pairs may be helpful when working positive and negative fractions.

A number line with marks from negative 5 tenths to positive 4 tenths. A point is located on the 4 tenths mark and an arrow directs to the point on negative 3 tenths to show how the expression four tenths plus negative 7 tenths is illustrated.

For example, adding -\dfrac{7}{10} to \dfrac{4}{10} is the same as subtracting \dfrac{7}{10}.

On the number line, this is moving \dfrac{7}{10} units to the left of the \dfrac{4}{10}.

This gives us: \dfrac{4}{10}+ \left(-\dfrac{7}{10}\right) = -\dfrac{3}{10}

Examples

Example 1

Calculate 3+4-\left(-\dfrac{4}{5}\right).

Worked Solution
Create a strategy

Follow the order of operations.

Apply the idea
\displaystyle 3+4-\left(-\dfrac{4}{5}\right)\displaystyle =\displaystyle 7-\left(-\dfrac{4}{5}\right)Evaluate the addition
\displaystyle =\displaystyle 7+\dfrac{4}{5}Combine adjacent signs
\displaystyle =\displaystyle 7\dfrac45Write as a mixed number

Example 2

Evaluate -10\times \left(-2\dfrac{1}{4}\right), giving your answer as a mixed number.

Worked Solution
Create a strategy

Convert the mixed number to an improper fraction, then evaluate.

Apply the idea
\displaystyle -10\times \left(-2\dfrac{1}{4}\right)\displaystyle =\displaystyle -10\times \left(-\dfrac94\right)Convert the mixed number to improper fraction
\displaystyle =\displaystyle \dfrac{-10\times(-9)}{1\times4}Multiply the numerators and denominators
\displaystyle =\displaystyle \dfrac{90}4Evaluate the multiplications
\displaystyle =\displaystyle 22\dfrac24 Convert the improper fraction to a mixed number
\displaystyle =\displaystyle 22\dfrac12 Simplify

Example 3

Evaluate -8\dfrac{4}{7}+3\dfrac{3}{7}, writing your answer as a mixed number.

Worked Solution
Create a strategy

Convert the mixed numbers to improper fractions.

Apply the idea
\displaystyle -8\dfrac{4}{7}+3\dfrac{3}{7}\displaystyle =\displaystyle -\dfrac{60}{7}+\dfrac{24}{7}Convert mixed numbers to improper fractions
\displaystyle =\displaystyle \dfrac{-60+24}7Write as one fraction
\displaystyle =\displaystyle \dfrac{-36}7Evaluate the numerator
\displaystyle =\displaystyle -5\dfrac17Convert the improper fraction to a mixed number

Example 4

Evaluate 4\dfrac{2}{3}\div \left(-1\dfrac{2}{5}\right), giving your answer as a mixed number.

Worked Solution
Create a strategy

Convert the mixed numbers to improper fractions.

Apply the idea
\displaystyle 4\dfrac{2}{3}\div \left(-1\dfrac{2}{5}\right)\displaystyle =\displaystyle \dfrac{14}3\div\left(-\dfrac75\right)Convert the mixed numbers to improper fractions
\displaystyle =\displaystyle \dfrac{14}3\times \left(-\dfrac57\right)Multiply by the reciprocal
\displaystyle =\displaystyle \dfrac{14\times-5}{3\times7}Multiply the numerators and denominators
\displaystyle =\displaystyle -\dfrac{70}{21}Evaluate the multiplications
\displaystyle =\displaystyle -3\dfrac7{21}Convert the improper fraction to a mixed number
\displaystyle =\displaystyle -3\dfrac13Simplify
Idea summary

Operations with fractions:

  • When adding or subtracting fractions, be sure there is a common denominator. Then add or subtract the numerators and keep the denominator.

  • Multiply the numerators, multiply the denominators, and simplify.

  • When dividing, take the reciprocal of the divisor and multiply. Then, follow the rules for multiplication.

  • When there is a mixed number, change it to an improper fraction, and proceed as if this were a normal fraction operation.

Outcomes

7.NS.A.1

Apply and extend previous understandings of addition and subtraction to add and subtract rational numbers; represent addition and subtraction on a horizontal or vertical number line diagram.

7.NS.A.1.D

Apply properties of operations as strategies to add and subtract rational numbers.

7.NS.A.2

Apply and extend previous understandings of multiplication and division and of fractions to multiply and divide rational numbers.

7.NS.A.2.C

Apply properties of operations as strategies to multiply and divide rational numbers.

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