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5.10 Add and subtract fractions


Are you ready?

If we can find  equivalent fractions  it will help us in this lesson. Let's try a problem to review.


Example 1

Fill in the blank to find an equivalent fraction to \dfrac{1}{3}:

\dfrac{1}{3}= \dfrac{⬚}{6}

Worked Solution
Create a strategy

Use fraction area models.

Apply the idea
A rectangle divided into 3 equal parts with 1 shaded part.

On the left of the equals sign we have \dfrac{1}{3} which looks like this.

A rectangle divided into 6 equal parts with 2 parts shaded.

1 out of the 3 squares are shaded. We want to write this as a fraction of 6. Dividing the model into 6 parts would look like this.

We can see that 2 out of 6 parts are shaded to get the same area. So:

\displaystyle \frac{1}{3}\displaystyle =\displaystyle \frac{2}{6}
Reflect and check

We also could have multiplied the numerator and denominator by 2 since 3\times 2=6.

\displaystyle \dfrac{1}{3}\displaystyle =\displaystyle \dfrac{1 \times 2}{3 \times 2}Multiply the numerator and denominator by 2
\displaystyle =\displaystyle \dfrac{2}{6}
Idea summary

Equivalent fractions look different but have the same value.

You need to multiply or divide both the numerator and the denominator of a fraction by the same number to work out the equivalent fraction.

Add and subtract fractions with different denominators

This video shows how to add and subtract fractions by finding common denominators.

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Example 2

Find the value of \,\dfrac{3}{5}+\dfrac{3}{4}.

Worked Solution
Create a strategy

We need to find the smallest common multiple of the denominators.

Apply the idea

The smallest common multiple of 5 and 4 is 20 since 5\times 4 =20. So the least common denominator of the two fractions is 20.

We need to multiply both the numerator and denominator of \dfrac{3}{5} by 4 to get a denominator of 20.

\displaystyle \dfrac{3}{5}\displaystyle =\displaystyle \dfrac{3\times 4}{5\times 4}Multiply numerator and denominator by 4
\displaystyle =\displaystyle \dfrac{12}{20}

We need to multiply both the numerator and denominator of \dfrac{3}{4} by 5 to get a denominator of 20.

\displaystyle \dfrac{3}{4}\displaystyle =\displaystyle \dfrac{3\times 5}{4\times 5}Multiply numerator and denominator by 5
\displaystyle =\displaystyle \dfrac{15}{20}

Now we can use these equivalent fractions in the addition.

\displaystyle \dfrac{3}{5}+\dfrac{3}{4}\displaystyle =\displaystyle \dfrac{12}{20}+\dfrac{15}{20}Use the new fractions
\displaystyle =\displaystyle \dfrac{12+15}{20}Add the numerators
\displaystyle =\displaystyle \dfrac{27}{20}
Idea summary

Before we add or subtract fractions, we must first make sure that the fractions have the same denominator.



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