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2.03 Adding and subtracting fractions

Lesson

Introduction

Consider the addition 2+3. This is the same as 2 wholes plus 3 wholes.

5 circles with 2 coloured green and 3 coloured blue.

We can see from the diagram above that 2+3=5. That is, 2 wholes plus 3 wholes equals 5 wholes.

3 circles with 1 coloured blue and 2 with green and blue stripes.

If we wanted to find 3-2 we would take 2 wholes away from 3 wholes and get 1 whole.

The important thing when adding and subtracting is that we are adding and subtracting quantities of the same amount.

Add and subtract fractions with the same denominator

Suppose we want to find \dfrac{2}{7}+\dfrac{3}{7}.

2 circles each divided into 7 parts. One circle has 2 parts shaded green, the other has 3 parts shaded blue.

Here we have a circle with two sevenths shaded and a circle with three sevenths shaded.

Notice that the parts of each circle are the same size.

A circle divided into 7 parts. 2 parts are shaded green, and 3 parts are shaded blue.

We can place one circle on top of the other. Now five sevenths of the circle are shaded in.

So we can conclude that \dfrac{2}{7}+\dfrac{3}{7}=\dfrac{5}{7}, or 2 sevenths plus 3 sevenths equals 5 sevenths.

When the denominators are the same, we are adding quantities of the same amount. So we can add the numerators and keep the same denominator.

Suppose we want to find \dfrac{3}{7}-\dfrac{2}{7}. Using the same circles as above, we can take two sevenths away from three sevenths.

A circle divided into 7 parts. 2 parts are have green and blue stripes, and 1 part is shaded blue.

The part that remains is one seventh of the circle. So we can conclude that \dfrac{3}{7}-\dfrac{2}{7}=\dfrac{1}{7}, or 3 sevenths minus 2 sevenths equals 1 seventh.

When the denominators are the same, we are subtracting quantities of the same amount. So we can subtract the numerators and keep the same denominator.

Examples

Example 1

Evaluate \dfrac{2}{6}+\dfrac{2}{6} and simplify your answer.

Worked Solution
Create a strategy

Add the numerators over the same denominator and simplify.

Apply the idea
\displaystyle \dfrac{2}{6}+\dfrac{2}{6}\displaystyle =\displaystyle \dfrac{4}{6}Add the numerators
\displaystyle =\displaystyle \dfrac{4 \div 2}{6\div 2}Divide the numerator and denominator by 2
\displaystyle =\displaystyle \dfrac{2}{3}Simplify
Idea summary

When two fractions have the same denominator we can add or subtract them by adding or subtracting the numerators over the same denominator.

Add and subtract fractions with different denominators

If the denominators are different then we are not adding quantities of the same amount. Consider \dfrac{2}{9}+\dfrac{3}{4}. These two fractions look like this.

This image shows 2 circles with different shaded portions. Ask your teacher for more information.

Before we can add these two fractions we rewrite them with the same denominator. Since we can change the denominator by multiplying the numerator and denominator by the same number, we want to first find a common multiple of the two denominators.

The denominators here are 9 and 4, so a common multiple of the denominators is 4\times9=36.

In this case, the denominators, 4 and 9 have no common factors. This means that 36 is the lowest common multiple of the denominators. This is sometimes called the lowest common denominator.

If the denominators were 4 and 6 instead, then we could find a common multiple the same way. That is, 4\times6=24. However, 24 is not the lowest common multiple of 4 and 6, because the lowest common multiple is 12.

In such a case, we could use either number, because they are both multiples.

This gives us two methods for adding fractions with different denominators. We can always find a common multiple by multiplying the two denominators. However, this will also mean that we will need to simplify the fraction resulting from the addition. Which method is better is a matter of preference.

Now we can rewrite the fractions. Multiplying the numerator and denominator of \dfrac{2}{9} gives \dfrac{2\times4}{9\times4}=\dfrac{8}{36}. Multiplying the numerator and denominator of \dfrac{3}{4} gives \dfrac{3\times9}{4\times9}=\dfrac{27}{36}. Now these fractions look like this.

This image shows 2 circles with different shaded portions. Ask your teacher for more information.

And now that the denominators are the same, we can add the fractions together.

This image shows a circle with 35 shaded parts out of 36. Ask your teacher for more information.

We can see that \dfrac{2}{9}+\dfrac{3}{4}=\dfrac{8}{36}+\dfrac{27}{36}=\dfrac{35}{36}

So when the denominators are different, we rewrite the fractions with the same denominator, and then we can follow the procedure for fractions with the same denominator.

Exploration

The following applet demonstrates adding fractions with area models.

Loading interactive...

We can use area models to change fractions to equivalent fractions so the denominators are the same. Then we can add the equivalent fractions.

Suppose we want to find \dfrac{3}{4}-\dfrac{2}{9}. Since the denominators are different, we rewrite the fractions with the same denominator before we subtract them. From the previous example we know that \dfrac{3}{4}=\dfrac{27}{36} and \dfrac{2}{9}=\dfrac{8}{36}.

This image shows a circle with 19 shaded parts out of 36. Ask your teacher for more information.

When we take \dfrac{8}{36} away from \dfrac{27}{36} we are left with \dfrac{19}{36}.\dfrac{3}{4}-\dfrac{2}{9}=\dfrac{27}{36}-\dfrac{8}{36}=\dfrac{19}{36}

So when the denominators are different, we rewrite the fractions with the same denominator, and then we can follow the procedure for fractions with the same denominator.

Examples

Example 2

Evaluate \dfrac{3}{4}-\dfrac{1}{8}.

Worked Solution
Create a strategy

Find the lowest common multiple of the two denominators.

Apply the idea

The lowest common multiple of 4 and 8 is 8.

Since the denominator of \dfrac{1}{8} is already 8 we only need to find the equivalent fraction of \dfrac{1}{4}.

\displaystyle \dfrac{3}{4}\displaystyle =\displaystyle \dfrac{3\times2}{4\times2}Multiply the numerator and denominator by 2
\displaystyle =\displaystyle \dfrac{6}{8}Evaluate
\displaystyle \dfrac{3}{4}-\dfrac{1}{8}\displaystyle =\displaystyle \dfrac{6}{8}-\dfrac{1}{8}Substitute the new fraction
\displaystyle =\displaystyle \dfrac{5}{8}Subtract the numerators
Idea summary

When two fractions have different denominators we first rewrite the fractions with the same denominator. Then we can add or subtract the numerators over the same denominator.

Add and subtract mixed numbers

Mixed numbers have a whole number part and a fraction part. The best way to add or subtract mixed numbers is to convert the mixed numbers into improper fractions. Then we can rewrite the improper fractions with the same denominator and add or subtract the fractions.

For example, to find the value of 2\,\dfrac{3}{4}-1\,\dfrac{5}{6}we can start by rewriting both mixed numbers as improper fractions:

\displaystyle 2\,\dfrac{3}{4}\displaystyle =\displaystyle \dfrac{8}{4}+\dfrac{3}{4}Write the whole part as a fraction
\displaystyle =\displaystyle \dfrac{11}{4}Add the fractions
\displaystyle 1\,\dfrac{5}{6}\displaystyle =\displaystyle \dfrac{6}{6}+\dfrac{5}{6}Write the whole part as a fraction
\displaystyle =\displaystyle \dfrac{11}{6}Add the fractions

Rewriting these improper fractions with the same denominator gives:

\displaystyle \dfrac{11}{4}\displaystyle =\displaystyle \dfrac{11\times3}{4\times3}Multiply the numerator and denominator by 3
\displaystyle =\displaystyle \dfrac{33}{12} Evaluate
\displaystyle \dfrac{11}{6}\displaystyle =\displaystyle \dfrac{11\times2}{6\times2}Multiply the numerator and denominator by 2
\displaystyle =\displaystyle \dfrac{22}{12} Evaluate
\displaystyle 2\,\dfrac{3}{4}-1\,\dfrac{5}{6}\displaystyle =\displaystyle \dfrac{33}{12}-\dfrac{22}{12}Subtract the improper fractions
\displaystyle =\displaystyle \dfrac{11}{12}Evaluate

Examples

Example 3

Evaluate 2\,\dfrac{3}{11} + 4\,\dfrac{7}{11}.

Worked Solution
Create a strategy

Add the whole number parts and the fraction parts separately.

Apply the idea
\displaystyle 2\,\dfrac{3}{11} + 4\,\dfrac{7}{11}\displaystyle =\displaystyle 2+4 + \dfrac{3}{11}+\dfrac{7}{11}Group the whole numbers and the fractions
\displaystyle =\displaystyle 6 + \dfrac{10}{11}Add the whole numbers and the fractions
\displaystyle =\displaystyle 6\,\dfrac{10}{11}Write as a mixed number
Idea summary

To add or subtract mixed numbers, we first write them as improper fractions and then we can use the same process to add or subtract them.

Outcomes

VCMNA243

Solve problems involving addition and subtraction of fractions, including those with unrelated denominators.

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