Equations with Fractions

This page provides examples and practice questions on linear equations that have fractions. To understand this page, we need to use these four facts:

  1. A fraction is an alternative way to express division.

        \[\frac{9}{4}=9 \div 4 = 2.25\]

  2. A number divided by itself is equal to one, for example

        \[\frac{4}{4}=1\]

  3. Multiplying by 1 is a ‘do nothing’ move. For example

        \[\frac{4}{4}\times 9 = 9\]

  4. The inverse operation of division is multiplication.

Straight to practice questions: One fraction More fractions

Explanations, Examples and Practice Questions

We begin with one fraction and then get more complicated:

Type Solving strategy
One fraction Multiply all terms on both sides by the denominator
More than one fraction Multiply all terms by the least common multiple of all denominators
Fraction equal fraction Cross multiply
Denominator is an algebraic expression Multiply all terms by that expression

Multiplying by the denominator ‘clears’ the denominator.

Example 1: A one step equation. Solve

    \[\frac{x}{5}=4\]

Multiply both sides by 5:

    \begin{align*}5\times \frac{x}{5}&=5\times 4\\[10pt]\frac{5x}{5}&=20\\[10pt]\frac{\cancel{5}x}{\cancel{5}}&=20\\[10pt] x&=20\end{align*}

It makes sense that x=20. The original question asks: \frac{x}{5}=4, and we know that \frac{20}{5}=4.

Thinking about solving linear equations with fractions

Example 2: When there is one fraction

    \[\frac{58-x}{5}=2x+5\]

In this case, the denominator is 5. Remember to multiply both sides by 5.

    \begin{align*} 5\times \Big(\frac{58-x}{5}\Big)&=5\times \Big(2x+5\Big)\\[10pt] \frac{5(58-x)}{5}&=10x+25\\[10pt]\frac{\cancel{5}(58-x)}{\cancel{5}}&=10x+25\\[10pt] 58-x&=10x+25\\[10pt]58&=11x+25\\[10pt]33&=11x\\[10pt]x&=3\end{align*}

Alternatively, you might notice that when there is one fraction the denominator multiplies all other terms in the equation:

This saves a whole bunch of lines!

Example 3: Another term(s) on the same side as the fraction

    \[\frac{x+7}{2}+13=2x\]

The denominator is 2, so we will multiply by 2. We have a choice, to start with the move ‘subtract 13’ or with the move ‘multiply by 2’. Whichever way, that 13 should become 26.

    \begin{align*}\frac{x+7}{2}+13&=2x\\[10pt]2\times \Big(\frac{x+7}{2}+13\Big)&=2\times \Big(2x\Big)\\[10pt]\frac{2(x+7)}{2}+2(13)&=4x\\[10pt]\frac{\cancel{2}(x+7)}{\cancel{2}}+26&=4x\\[10pt]x+7+26&=4x\\[10pt]x+33&=4x\\[10pt]33&=3x\\[10pt]x&=11 \end{align}

Remember to substitute your answer into the original equation to see if your answer makes the equation work or if you have made a mistake in your algebra.

Here are some questions to practice with:

When there is more than one fraction

Multiply the whole equation by the least common multiple of all denominators

We can clear a fraction by multiplying by the denominator, or by any multiple of the denominator. Our goal is to resolve the fractions to integers.

When we need to clear several fractions at once, we multiply by a common multiple of all denominators.

Example 4: More than one denominator

    \[\frac{x+9}{4}-\frac{2x+7}{6}=\frac{x+7}{3}\]

The least common multiple of 4, 6 and 3 is 12.

    \begin{align*}12\Big(\frac{x+9}{4}-\frac{2x+7}{6}\Big)&=12\Big(\frac{x+7}{3}\Big)\\[10pt]12\Big(\frac{x+9}{4}\Big)-12\Big(\frac{2x+7}{6}\Big)&=12\Big(\frac{x+7}{3}\Big)\\[10pt]^3\cancel{12}\Big(\frac{x+9}{\cancel{4}}\Big)-^2\cancel{12}\Big(\frac{2x+7}{\cancel{6}}\Big)&=^4\cancel{12}\Big(\frac{x+7}{\cancel{3}}\Big)\\[10pt] 3(x+9)-2(2x+7)&=4(x+7) \\[10pt] 3x+27-4x-14&=4x+28\\[10 pt] -x+13&=4x+28\\[10pt]13&=5x+28\\[10pt]-15&=5x\\[10pt] x&=-3 \end{align*}

x=-3 substituted to the original line gives:

    \begin{align*}\frac{6}{4}-\frac{1}{6}=\frac{4}{3}\\[10pt]\frac{18}{12}-\frac{2}{12}=\frac{16}{12}\checkmark \end{align}

Here are some questions to pratice with:

This set of ten questions all have the same answer. Click to see the answer. Choose one question. Write the lines of algebra to derive the answer.

applet link

Printable textbook equations with fractions from corbettmaths

Online interactive exercise on solving equations with fractions from transum math

Video on the special case of (fraction) = (fraction) by Brian McLogan: Cross Multiply

Check out these online/interactive/printable learning resources.

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