Fraction operations
Choose the rule for the operation shown. Convert mixed numbers before multiplying or dividing. Using improper fractions also avoids borrowing errors in subtraction
- Addition and subtraction Match the denominators, combine the numerators in the given order, then simplify. A common denominator represents a common part size \(\displaystyle \begin{aligned}&2\frac{1}{8}-1\frac{5}{6}\\&=\frac{17}{8}-\frac{11}{6}\\&=\frac{51}{24}-\frac{44}{24}\\&=\frac{7}{24}\end{aligned}\)
- Multiplication Cancel the factor thirteen from the two 13s, then the factor four from 4 and 8. Only cancel factors in a product, never terms across addition or subtraction \(\displaystyle \begin{aligned}&1\frac{5}{8}\times\frac{4}{13}\\&=\frac{13}{8}\times\frac{4}{13}\\&=\frac{1}{8}\times\frac{4}{1}\\&=\frac{1}{2}\times\frac{1}{1}\\&=\frac{1}{2}\end{aligned}\)
- Division Multiply the first number by the reciprocal of the non-zero divisor. Here, cancel the factor three from 9 and 3, then the factor two from 14 and 10. A whole number has denominator one \(\displaystyle \begin{aligned}&\frac{9}{10}\div\frac{3}{14}\\&=\frac{9}{10}\times\frac{14}{3}\\&=\frac{3}{10}\times\frac{14}{1}\\&=\frac{3}{5}\times\frac{7}{1}\\&=\frac{21}{5}\end{aligned}\)
- Negative values Convert the magnitude of a negative mixed number, then keep its sign. In multiplication and division, like signs give a positive result and unlike signs give a negative result \(\displaystyle \begin{aligned}&2\frac{2}{7}\times\left(-\frac{7}{12}\right)\\&=\frac{16}{7}\times\left(-\frac{7}{12}\right)\\&=-\frac{4}{3}\end{aligned}\)
Common mistakes
Adding denominators
✕\(\displaystyle \frac{5}{8}+\frac{1}{12}=\frac{3}{10}\)✓\(\displaystyle \begin{aligned}&\frac{5}{8}+\frac{1}{12}\\&=\frac{15}{24}+\frac{2}{24}\\&=\frac{17}{24}\end{aligned}\)
The denominators name different part sizes. Rewrite both fractions in matching parts before adding their numeratorsReversing only the fractional subtraction
✕\(\displaystyle 2\frac{1}{8}-1\frac{5}{6}=1\frac{17}{24}\)✓\(\displaystyle \begin{aligned}&2\frac{1}{8}-1\frac{5}{6}\\&=\frac{17}{8}-\frac{11}{6}\\&=\frac{51}{24}-\frac{44}{24}\\&=\frac{7}{24}\end{aligned}\)
Subtracting the smaller fractional part from the larger changes the question. Improper fractions preserve the whole amounts and the original subtraction orderKeeping a denominator during multiplication
✕\(\displaystyle \frac{4}{9}\times\frac{5}{9}=\frac{20}{9}\)✓\(\displaystyle \frac{4}{9}\times\frac{5}{9}=\frac{20}{81}\)
The denominator stays fixed only when adding or subtracting matching parts. Multiplication multiplies both denominatorsInverting both fractions
✕\(\displaystyle \frac{9}{10}\div\frac{3}{14}=\frac{10}{9}\times\frac{14}{3}\)✓\(\displaystyle \begin{aligned}&\frac{9}{10}\div\frac{3}{14}\\&=\frac{9}{10}\times\frac{14}{3}\\&=\frac{21}{5}\end{aligned}\)
Only the divisor is replaced by its reciprocal. Keep the first number unchangedDifficulty levels
introUse proper fractions with matching or related denominators across all four operations
standardUse proper fractions with unlike denominators, mixed numbers and improper fractions. Multiply or divide with a whole number in either position
challengeSubtract mixed numbers with smaller fractional parts, or calculate with negative fractions
examChoose a simplified result for addition, subtraction, multiplication or division, including mixed-number operands
Questions about this topic
Why do addition and subtraction need a common denominator?+
The numerators must count equal-sized parts of the same whole. Equivalent fractions let you use that common part size
Why does division use the reciprocal?+
Multiplying the dividend and divisor by the same non-zero number leaves a quotient unchanged. The reciprocal makes the divisor one, leaving the dividend multiplied by that reciprocal
When can I cancel factors?+
You may divide a numerator and a denominator in a product by the same non-zero factor. For division, change to multiplication by the reciprocal first. Terms joined by addition or subtraction cannot be cancelled this way
Must I always convert mixed numbers before subtracting?+
No. You can subtract whole and fractional parts separately, exchanging a whole when needed. Improper fractions give one reliable method that also works for multiplication and division
Can I use a calculator?+
No. This topic is practised without a calculator
Which year level and chapter is this?+
Year 9 Mathematics, chapter 1: Number and financial mathematics
In this chapter
Evaluate an integer expressionFactors, multiples, HCF, LCM and primesRound to decimal places or significant figuresEstimate an expression using rounded inputsConvert between fractions, mixed numbers and decimalsClassify real numbers and compare exact valuesAdd, subtract, multiply and divide fractions and mixed numbersWrite a ratio from countsWrite ratios in simplest formComplete equivalent ratiosUse a constant rateDivide a quantity in a given ratioDecide which option is the better valueConvert between fractions, decimals and percentagesFind a percentage of a quantityExpress one quantity as a percentage of another1F Percentages and money: Find the original amount from a percentageIncrease or decrease a quantity by a percentageFind percentage changePercentage error against a reference value1G Percentage change and percentage error: Find the original amount from a percentageProfit, mark-up and discountIncome and taxSimple interestCompound interest (optional extension)