Adding and Subtracting Fractions

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Adding and Subtracting Fractions

In order to add and subtract fractions, you need to find a common denominator – some value that can become the denominator of both fractions. There are two main methods for choosing a common denominator:

Use the lowest common multiple (LCM) of the two denominators.
Use the product of the two denominators.

Take Note

Before adding and subtracting fractions, it’s important to understand that if you multiply both top and bottom of a fraction by the same value, the fraction’s value doesn’t change. For example,

$dfrac{2}{3}=dfrac{4}{6},,,,,text{ and },,,,dfrac{2}{3}=dfrac{10}{15}$

Example 1: Adding Fractions

Evaluate $dfrac{3}{5} + dfrac{1}{4}$

[2 marks]

To find a common denominator here, we will take the product of the two denominators: $5times 4=20$ .

To make sure we aren’t changing the value of the fraction, we also multiply the top by $4$ .

$dfrac{3}{5}=dfrac{3times textcolor{red}{4}}{5times textcolor{red}{4}}=dfrac{12}{20}$

For the second fraction’s denominator to be $20$ , we’ll have to multiply it by $5$ . So, we will also have to multiply the top by $5$ .

$dfrac{1}{4}=dfrac{1times textcolor{red}{5}}{4times textcolor{red}{5}}=dfrac{5}{20}$

Now, to add two fractions with the same denominator, simply add the numerators together. Doing so, we get,

$dfrac{3}{5} + dfrac{1}{4}=dfrac{12}{20} + dfrac{5}{20}=dfrac{17}{20}$

Example 2: Subtracting Fractions

Evaluate $dfrac{4}{7} – dfrac{2}{5}$

[2 marks]

Our choice of common denominator in this case will be $textcolor{red}{7}times textcolor{red}{5}=35$

To make the denominator of the first fraction $35$ , we’ll have to multiply its top and bottom by $5$ . To make the denominator of the second fraction be $35$ , we’ll have to multiply its top and bottom by $7$ . This looks like,

$begin{aligned}dfrac{4}{7}-dfrac{2}{5} &= dfrac{5times 4}{35}-dfrac{7times 2}{35} &=dfrac{20}{35} – dfrac{14}{35} =dfrac{6}{35}end{aligned}$

Remember, in the final step you subtract the numerators, and the denominator is unchanged.

Example 3: Adding Fractions and Whole Numbers

Evaluate $8+dfrac{5}{6}$

[2 marks]

To add a whole number to a fraction, we turn the whole number into a fraction by dividing by $1$ . Thus,

$dfrac{8}{1}+dfrac{5}{6}$

This time, since $1times 6=6$ , the common denominator will be $6$ , meaning we’ll only have to change the first fraction – we will multiply its top and bottom by $6$ . Doing so, we get

$begin{aligned}dfrac{8}{1}+dfrac{5}{6} &= dfrac{8times 6}{1times 6}+dfrac{5}{6} &=dfrac{48}{6} + dfrac{5}{6} = dfrac{53}{6} end{aligned}$

Example 4: Adding Mixed Fractions

Evaluate $2dfrac{5}{9}+dfrac{1}{3}$

[3 marks]

To add a mixed number to a fraction, first convert the mixed number to an improper fraction.

$2dfrac{5}{9}=dfrac{(2times 9)+5}{9}=dfrac{23}{9}$

Now the calculation looks like

$dfrac{23}{9}+dfrac{1}{3}$

Now, for the common denominator we could use $27$ (since it’s the product of $9$ and $3$ ). However, the LCM of $9$ and $3$ is just $9$ , and if we use $9$ as our common denominator we will only have to change the second fraction. Doing this, we get

$begin{aligned}dfrac{23}{9}+dfrac{1}{3} &= dfrac{23}{9}+dfrac{1times 3}{3times 3} &=dfrac{23}{9} + dfrac{3}{9} = dfrac{26}{9} end{aligned}$

This fraction is already in its simplest form, so we’re done. However, if we had chosen $27$ as our common denominator, we would’ve had to simplify the fraction at the end.

Adding and Subtracting Fractions Example Questions

Question 1: Evaluate $dfrac{1}{8}+dfrac{5}{12}$

[2 marks]

Give your answer in its simplest form.

To add fractions, they must first share a common denominator.

This can be achieved by first multiplying the top and bottom of the first fraction by $3$ , and then multiplying the top and bottom of the second fraction by $2$ . Thus,

$begin{aligned}dfrac{1}{8}+dfrac{5}{12} &= dfrac{1times 3}{24}+dfrac{5times 2}{24} &=dfrac{3}{24} + dfrac{10}{24} =dfrac{13}{24} end{aligned}$

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Question 2: Evaluate $dfrac{9}{10} + 5$

[2 marks]

Give your answer in its simplest form.

Writing $5$ as $dfrac{5}{1}$ , the calculation becomes,

$dfrac{9}{10}+dfrac{5}{1}$

To add fractions, they must first share a common denominator. This can be achieved by first multiplying the top and bottom of the second fraction by $10$ . Thus,

$begin{aligned}dfrac{9}{10}+dfrac{5}{1} &= dfrac{9}{10}+dfrac{5times 10}{1times10} &=dfrac{9}{10} + dfrac{50}{10} =dfrac{59}{10} end{aligned}$

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Question 3: Evaluate $dfrac{4}{5} + dfrac{5}{3}$

[2 marks]

Give your answer in its simplest form.

To add fractions, they must first share a common denominator.

This can be achieved by first multiplying the top and bottom of the first fraction by $3$ , and then multiplying the top and bottom of the second fraction by $5$ . Thus,

$begin{aligned}dfrac{4}{5}+dfrac{5}{3} &= dfrac{4times 3}{15}+dfrac{5times 5}{15} &=dfrac{12}{15} + dfrac{25}{15} = dfrac{37}{15} end{aligned}$

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Question 4: Evaluate $4dfrac{1}{2} + dfrac{4}{3}$

[3 marks]

Give your answer in its simplest form.

Firstly we have to convert the mixed fraction to an improper fraction,

$4dfrac{1}{2} + dfrac{4}{3} = dfrac{9}{2} + dfrac{4}{3}$

To add fractions, they must first share a common denominator. This can be achieved by first multiplying the top and bottom of the first fraction by $3$ , and then multiplying the top and bottom of the second fraction by $2$ . Thus,

$begin{aligned}dfrac{9}{2} + dfrac{4}{3} &= dfrac{9times 3}{2times3}+dfrac{4times 2}{3times2} &=dfrac{27}{6} + dfrac{8}{6} = dfrac{35}{6} end{aligned}$

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Question 5: Evaluate $5dfrac{2}{3} + 2dfrac{3}{4}$

[3 marks]

Give your answer in its simplest form.

Firstly we have to convert the mixed fractions to an improper fraction,

$5dfrac{2}{3} + 2dfrac{3}{4} = dfrac{17}{3} + dfrac{11}{4}$

To add fractions, they must first share a common denominator. This can be achieved by first multiplying the top and bottom of the first fraction by $4$ , and then multiplying the top and bottom of the second fraction by $3$ . Thus,

$begin{aligned}dfrac{17}{3} + dfrac{11}{4} &= dfrac{17times 4}{3times4}+dfrac{11times 3}{4times3} &=dfrac{68}{12} + dfrac{33}{12} = dfrac{101}{12 }end{aligned}$

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Specification Points Covered

Number – 2. apply the four operations, including formal written methods, to integers, decimals and simple fractions (proper and improper), and mixed numbers – all both positive and negative; understand and use place value (e.g. when working with very large or very small numbers, and when calculating with decimals)