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Addition & Subtraction with Fractions

Method 1: with a Formula

In this section we learn a method for adding and subtracting with fractions.

This is the first of two methods that we'll learn and relies upon a formula.

We start by learning the formula for addition and subtraction of fractions as well as watch a tutorial. We'll then work our way through some questions.

By the end of this section we should be comfortable adding, or subtracting any two fractions.

Tutorial

In the following tutorial we review the method for adding and subtracting fractions and work through some well-explained examples.

Formula for Addition & Subtraction

Addition

When adding two fractions, \(\frac{a}{b}\) and \(\frac{c}{d}\), we can use the following formula: \[\frac{a}{b}+\frac{c}{d} = \frac{a\times d + b\times c}{b\times d}\]

Subtraction

When subtracting two fractions, \(\frac{a}{b} - \frac{c}{d}\), we can use exactly the same formula. The only difference is that we replace the \(+\) by a \(-\) (since we're dealing with a subtraction).
So the formula for subtraction with fractions is: \[\frac{a}{b}-\frac{c}{d} = \frac{a\times d - b\times c}{b\times d}\]

Now that we know the method and the formula, we move-on to work our way through some exercises.
Make sure to try each of the following questions before looking at the answer.

Exercise 1 - Adding Fractions

Using the method we've just learned, calculate each of the following and write your answer in its simplest form:

  1. \(\frac{1}{4} + \frac{2}{3}\)

  2. \(\frac{1}{8}+\frac{3}{4}\)

  3. \(\frac{3}{8} + \frac{2}{5}\)

  4. \(\frac{2}{5} + \frac{1}{3}\)

  5. \(\frac{2}{7} + \frac{3}{5}\)

  6. \(\frac{3}{8} + \frac{1}{2}\)

Answers Without Working

  1. \(\frac{1}{4} + \frac{2}{3} = \frac{11}{12}\)

  2. \(\frac{1}{8}+\frac{3}{4} = \frac{7}{8}\)

  3. \(\frac{2}{7} + \frac{3}{5} = \frac{31}{35}\)

  4. \(\frac{3}{8} + \frac{5}{12} = \frac{76}{96}\)

    This fraction can also be simplified and written:

    \(\frac{3}{8} + \frac{1}{2} = \frac{14}{16} = \frac{7}{8}\)

Solution With Working

  1. To calculate \(\frac{1}{4}+\frac{2}{3}\) we use the formula: \[\begin{aligned} \frac{1}{4}+\frac{2}{3} & = \frac{1\times 3 + 4\times 2}{4\times 3} \\ & = \frac{3+8}{12} \\ & = \frac{11}{12} \end{aligned}\] It's always worth checking if we can simplify the fraction. To do that we look for the highest common factor, HCF, of \(11\) and \(12\). Since \(HCF(11,12) = 1\) the fraction can't be simplified any further.

    So the final answer is: \[\frac{1}{4} + \frac{2}{3} = \frac{11}{12}\]

  2. To calculate \(\frac{1}{8}+\frac{3}{4}\) we use the formula: \[\begin{aligned} \frac{1}{8}+\frac{3}{4} & = \frac{1\times 4 + 8 \times 3}{8\times 4} \\ & = \frac{4 + 24}{32} \\ & = \frac{28}{32} \end{aligned}\] We now check to see whether we can simplify this fraction.
    The highest common factor of \(28\) and \(32\) is \(4\), that's \(HCF(28,32)=4\).
    So we can simplify the fraction by cancelling-out the \(HCF\): \[\begin{aligned} \frac{28}{32} &= \frac{4\times 7}{4\times 8}\\ & = \frac{7}{8} \end{aligned} \] And that's the final answer: \[\frac{1}{8}+\frac{3}{4} = \frac{7}{8}\]

  3. To calculate \(\frac{3}{8} + \frac{2}{5}\) we use the formula: \[\begin{aligned} \frac{3}{8}+\frac{2}{5} & = \frac{3\times 5 + 8 \times 2}{8\times 5} \\ & = \frac{15 + 16}{40} \\ & = \frac{31}{40} \end{aligned}\] Again, we check whether we can simplify this fraction.
    Since the highest common factor of \(31\) and \(40\) is \(1\), \(HCF(31,40)=1\), the fraction can't be simplified any further.
    The final answer is therefore: \[\frac{3}{8}+\frac{2}{5} = \frac{31}{40}\]

  4. To calculate \(\frac{2}{5}+ \frac{1}{3}\) we use the formula: \[\begin{aligned} \frac{2}{5}+ \frac{1}{3} & = \frac{2\times 3 + 5 \times 1}{5\times 3} \\ & = \frac{6+5}{5\times 3} \\ & = \frac{11}{15} \end{aligned}\] We now check whether we can simplify this fraction. The highest common factor of \(11\) and \(15\) is \(1\), \(HCF(11,15)=1\), so the fraction can't be simplified any further.
    The final answer is therefore: \[\frac{2}{5}+\frac{1}{3} = \frac{11}{15}\]

  5. To calculate \(\frac{2}{7} + \frac{3}{5}\) we use the formula: \[\begin{aligned} \frac{2}{7} + \frac{3}{5} & = \frac{2\times 5 + 7 \times 2}{7\times 5} \\ & = \frac{10 + 14}{35} \\ & = \frac{24}{35} \end{aligned}\] We now check whether we can simplify this fraction. The highest common factor of \(24\) and \(35\) is \(1\), \(HCF(24,35)=1\), so the fraction can't be simplified any further.
    The final answer is therefore: \[\frac{2}{7}+\frac{3}{5} = \frac{24}{35}\]

  6. To calculate \(\frac{1}{3} + \frac{1}{4}\) we use the formula: \[\begin{aligned} \frac{1}{3} + \frac{1}{4} & = \frac{1\times 4 + 3 \times 1}{3 \times 4} \\ & = \frac{4 + 3}{12} \\ & = \frac{7}{12} \end{aligned}\] We now check whether we can simplify this fraction. The highest common factor of \(7\) and \(12\) is \(1\), \(HCF(7,12)=1\), so the fraction can't be simplified any further.
    The final answer is therefore: \[\frac{1}{3}+\frac{1}{4} = \frac{7}{12}\]

Exercise 2 - Subtracting Fractions

Using the method we've just learned, calculate each of the following and write your answer in its simplest form:

  1. \(\frac{3}{8} - \frac{1}{4}\)

  2. \(\frac{11}{10} - \frac{3}{20}\)

  3. \(\frac{7}{25} - \frac{5}{8}\)

  4. \(\frac{2}{3} - \frac{1}{2}\)

  5. \( \frac{7}{18} - \frac{2}{9}\)

Answers Without Working

Solution With Working