Fractions, decimals and mixed numbers
These forms can name the same exact value. Keep the denominator non-zero. Simplify fractions when the question asks for lowest terms
\[\frac{a}{b}=a\div b,\qquad b\ne0\]
- Fraction to terminating decimal Divide the numerator by the denominator. An equivalent power-of-ten denominator also gives the decimal directly \(\displaystyle \begin{aligned}\frac{17}{25}&=\frac{68}{100}\\&=0.68\end{aligned}\)
- Decimal to fraction Use the final decimal place, then simplify by a common factor \(\displaystyle \begin{aligned}0.92&=\frac{92}{100}\\&=\frac{23}{25}\end{aligned}\)
- Mixed to improper For a non-negative mixed number with a positive denominator, multiply the whole part by the denominator and add the numerator. Keep the denominator \(\displaystyle \begin{aligned}6\frac{3}{7}&=\frac{6\times7+3}{7}\\&=\frac{45}{7}\end{aligned}\)
- Improper to mixed For a non-negative improper fraction with a positive denominator, divide the numerator by the denominator to find the whole part and the remainder. Write a non-zero remainder over the original denominator; a zero remainder gives a whole number \(\displaystyle \begin{aligned}52&=7\times7+3\\\frac{52}{7}&=7\frac{3}{7}\end{aligned}\)
- Exact recurring notation Put a bar over the repeating digits only. Do not replace exact equality with a stopped approximation \(\displaystyle \frac{11}{12}=0.91\overline{6}\)
Common mistakes
Reversing the division
✕\(\displaystyle \frac{17}{25}=25\div17\)✓\(\displaystyle \frac{17}{25}=17\div25\)
A fraction divides its numerator by its denominatorUsing the wrong place value
✕\(\displaystyle 0.092=\frac{92}{100}\)✓\(\displaystyle 0.092=\frac{92}{1000}\)
The final digit is in the thousandths place. The leading zero after the decimal point still counts as a placeStopping a recurring decimal
✕\(\displaystyle \frac{11}{12}=0.916\)✓\(\displaystyle \frac{11}{12}=0.91\overline{6}\)
A non-zero repeated remainder means more digits follow. The bar records the exact repeating partDifficulty levels
introRevise equivalent forms, simple mixed numbers and decimals with one or two places
standardConvert terminating decimals with up to four places and include whole units
challengeIdentify exact recurring notation and complete conversions in different forms
examChoose exact conversions and diagnose form errors using standard and challenge number shapes
Questions about this topic
When does a fraction have a terminating decimal?+
First simplify the fraction. Its decimal terminates exactly when the denominator has no prime factors other than two and five
What does a recurring bar mean?+
The digits under the bar repeat forever. Any digits before the bar occur before that repeating block
Must I write a mixed number or an improper fraction?+
Use the form requested. Both can represent the same value, but a question about a particular form checks that form as well as the value
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)