Arithmetic drills

Fraction calculation practice

Fraction arithmetic rests on three skills: common denominators, cancelling and reciprocals.

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Reducing and common denominators

To reduce a fraction, divide the numerator and denominator by a common factor, ideally the greatest one: $\frac{18}{24} = \frac{3}{4}$ .

To add or subtract fractions with different denominators, rewrite them with a common denominator, usually the least common multiple.

\frac{5}{6} + \frac{3}{4} = \frac{10}{12} + \frac{9}{12} = \frac{19}{12}

Tip

Always check whether your answer can be reduced.

Multiplying and dividing

Rules

\frac{a}{b} \times \frac{c}{d} = \frac{a \times c}{b \times d},\qquad \frac{a}{b} \div \frac{c}{d} = \frac{a}{b} \times \frac{d}{c}

Dividing by a fraction is the same as multiplying by its reciprocal. Cancel common factors before multiplying to keep the numbers small.

Example: $\frac{6}{5} \div \frac{9}{7} = \frac{6}{5} \times \frac{7}{9} = \frac{14}{15}$

Mixed numbers and decimals

Convert mixed numbers to improper fractions first: $2\frac{1}{3} = \frac{7}{3}$ .

When decimals and fractions appear together, convert the decimals to fractions ( $0.75 = \frac{3}{4}$ , $1.2 = \frac{6}{5}$ ).

The order of operations is the same as with whole numbers:

\frac{3}{4} + \frac{2}{3} \times \frac{3}{8} = \frac{3}{4} + \frac{1}{4} = 1

Q1BasicFractions

Reduce the fraction to lowest terms.

\frac{49}{84}

Answer

\frac{7}{12}

Solution

Divide both by their greatest common divisor, 7.
$\frac{49}{84} = \frac{49 \div 7}{84 \div 7} = \frac{7}{12}$

Q2StandardFractions

Calculate.

\frac{5}{12} \div \frac{1}{2} \div \frac{3}{8}

Answer

\frac{20}{9}

Solution

Rewrite as multiplication only, then cancel.
$\begin{aligned}\frac{5}{12} \div \frac{1}{2} \div \frac{3}{8} &= \frac{5}{12} \times 2 \times \frac{8}{3} \\ &= \frac{20}{9}\end{aligned}$

Q3AdvancedFractions

Calculate using a shortcut.

\frac{1}{12} + \frac{1}{20} + \frac{1}{30} + \frac{1}{42} + \frac{1}{56}

Answer

\frac{5}{24}

Solution

Each denominator is a product of consecutive integers, so split each term into a difference.
$\left(\frac{1}{3} - \frac{1}{4}\right) + \left(\frac{1}{4} - \frac{1}{5}\right) + \cdots + \left(\frac{1}{7} - \frac{1}{8}\right)$
Neighbouring terms cancel, leaving the first and the last.
$\frac{1}{3} - \frac{1}{8} = \frac{5}{24}$

Q1BasicFractions

Write the mixed number as an improper fraction.

2\frac{5}{7}

Q2StandardFractions

Calculate.

2\frac{1}{3} \div 1\frac{1}{3}

Q3AdvancedFractions

Calculate.

2\frac{2}{5} \times 2\frac{7}{8} + \frac{1}{2}

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Fraction calculation practice

Reducing and common denominators

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Arithmetic

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