Middle School Mathematics
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Middle school mathematics

Working with Fractions

Learning objectives

Adding and subtracting fractions

Fractions can be added only when they have the same denominator. If they do not, rewrite them over a common denominator, for example the lcm of the two denominators.

\[\frac{a}{b}\pm\frac{c}{d}=\frac{ad\pm bc}{bd}\]
Worked example

Compute \(\frac34+\frac56\).

The lcm of 4 and 6 is 12, so \(\frac34+\frac56=\frac{9}{12}+\frac{10}{12}=\frac{19}{12}\).

Worked example

Compute \(\frac78-\frac13\).

The lcm of 8 and 3 is 24: \(\frac{21}{24}-\frac{8}{24}=\frac{13}{24}\).

Mixed numbers

Definition

A mixed number such as \(2\frac35\) means \(2+\frac35\). A fraction whose numerator is at least its denominator is an improper fraction.

Example

\(2\frac35=\frac{2\cdot 5+3}{5}=\frac{13}{5}\)

Conversely \(\frac{17}{4}=4\frac14\), because \(17=4\cdot 4+1\).

Multiplying and dividing fractions

Theorem · Multiplying fractions

\[\frac{a}{b}\cdot\frac{c}{d}=\frac{ac}{bd}\]

Cancel common factors first to keep the numbers small.

Worked example

Compute \(\frac23\cdot\frac{9}{10}\).

\[\frac23\cdot\frac{9}{10}=\frac{2\cdot 9}{3\cdot 10}=\frac{18}{30}=\frac35\]

Example

\(\frac34\) of 20 is \(\frac34\cdot 20=15\).

Definition

The reciprocal of \(\frac{c}{d}\) is \(\frac{d}{c}\) (for \(c\neq 0\)). Dividing by a fraction means multiplying by its reciprocal: \(\frac{a}{b}\div\frac{c}{d}=\frac{a}{b}\cdot\frac{d}{c}\).

Worked example

Compute \(\frac35\div\frac{9}{10}\).

\[\frac35\div\frac{9}{10}=\frac35\cdot\frac{10}{9}=\frac{30}{45}=\frac23\]

Remark · Why this works

Dividing by \(\frac12\) asks how many halves fit in a number: \(3\div\frac12=6\), which equals \(3\cdot 2\).

Fractions on the number line

Use the sliders to place \(\frac58\) and then \(\frac23\) (set 2 and 3). Compare their decimals: which one is farther to the right?

Worked example · A recipe

A recipe needs \(2\frac12\) cups of flour. How much flour is needed for \(\frac34\) of the recipe?

\[2\frac12\cdot\frac34=\frac52\cdot\frac34=\frac{15}{8}=1\frac78\]

So \(1\frac78\) cups of flour are needed.

Exercises

Compute \(\frac25+\frac34\).

Show solution

\(\frac{8}{20}+\frac{15}{20}=\frac{23}{20}=1\frac{3}{20}\)

Compute \(5\frac13-2\frac34\).

Show solution

\(\frac{16}{3}-\frac{11}{4}=\frac{64}{12}-\frac{33}{12}=\frac{31}{12}=2\frac{7}{12}\)

Compute \(\frac49\cdot\frac38\).

Show solution

\(\frac{4\cdot 3}{9\cdot 8}=\frac{12}{72}=\frac16\)

Compute \(\frac{7}{12}\div\frac56\).

Show solution

\(\frac{7}{12}\cdot\frac65=\frac{42}{60}=\frac{7}{10}\)

A tank is \(\frac34\) full. Two thirds of the water is used. What fraction of the whole tank was used?

Show solution

\(\frac23\) of \(\frac34\) is \(\frac23\cdot\frac34=\frac{6}{12}=\frac12\) of the tank.

Which is larger, \(\frac79\) or \(\frac56\)?

Show solution

Cross-multiply: \(7\cdot 6=42\) and \(5\cdot 9=45\). Since 42 < 45, \(\frac79<\frac56\).

Summary