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Convert between fractions and decimals, and between mixed and improper forms

Start a set of 8
Step 3 of 9

Change the form, keep the value

Use division to convert a fraction to a decimal. A terminating decimal ends. A recurring decimal has a block of digits that repeats forever

  1. 1Terminating decimals Scale both fraction parts to make a power-of-ten denominator, or divide until the remainder is zero \(\displaystyle \begin{aligned}\frac{9}{16}&=\frac{5625}{10000}\\&=0.5625\end{aligned}\)
  2. 2Decimals to fractions Use the place of the final decimal digit for the denominator. Then simplify both parts by their highest common factor \(\displaystyle \begin{aligned}0.58&=\frac{58}{100}\\&=\frac{29}{50}\end{aligned}\)
  3. 3Mixed and improper forms For non-negative values with positive denominators, count the fractional parts in all the wholes, then add the extra parts. Reverse this by finding the number of complete wholes and the remainder \(\displaystyle \begin{aligned}4\frac{2}{9}&=\frac{4\times9+2}{9}\\&=\frac{38}{9}\end{aligned}\)
  4. 4Recurring decimals A repeated remainder starts a repeated block. Put a bar over that block only. A stopped decimal is an approximation \(\displaystyle \begin{aligned}\frac{11}{18}&=0.6\overline{1}\\\frac{11}{18}&\ne0.611111\end{aligned}\)
Year 11, VCE Units 1 & 2, Chapter 1, 4 difficulty levels, No calculatorBuild a worksheet

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

\[\frac{a}{b}=a\div b,\qquad b\ne0\]
  1. Fraction to terminating decimal Divide the numerator by the denominator. An equivalent power-of-ten denominator also gives the decimal directly \(\displaystyle \begin{aligned}\frac{17}{25}&=\frac{68}{100}\\&=0.68\end{aligned}\)
  2. Decimal to fraction Use the final decimal place, then simplify by a common factor \(\displaystyle \begin{aligned}0.92&=\frac{92}{100}\\&=\frac{23}{25}\end{aligned}\)
  3. 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 \(\displaystyle \begin{aligned}6\frac{3}{7}&=\frac{6\times7+3}{7}\\&=\frac{45}{7}\end{aligned}\)
  4. 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 \(\displaystyle \begin{aligned}52&=7\times7+3\\\frac{52}{7}&=7\frac{3}{7}\end{aligned}\)
  5. Exact recurring notation Put a bar over the repeating digits only. Do not replace exact equality with a stopped approximation \(\displaystyle \frac{11}{12}=0.91\overline{6}\)

Worked example

Write the decimal as a fraction in lowest terms

\[1.68\]
  1. \(\displaystyle 1.68=\frac{168}{100}\)The final decimal digit is in the hundredths place
  2. \(\displaystyle \frac{168}{100}=\frac{168\div4}{100\div4}\)The highest common factor of the numerator and denominator is four
  3. \(\displaystyle \frac{168\div4}{100\div4}=\frac{42}{25}\)Divide both parts. No common factor greater than one remains

Common mistakes

Reversing the division
✕\(\displaystyle \frac{17}{25}=25\div17\)✓\(\displaystyle \frac{17}{25}=17\div25\)
A fraction divides its numerator by its denominator
Using 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 place
Stopping 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 part

Difficulty 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 11, VCE Units 1 & 2, Specialist Mathematics, chapter 1: Number, sets and proof

In this chapter

Classify real numbers and recognise undefined formsExplain number-system structureConvert between fractions and decimals, and between mixed and improper formsConstruct a fraction from a recurring decimalWork with sets and interval notationCount subsets and power-set elementsExpress a number in prime-factor index formInterpret propositions and quantifiersReason with direct and indirect proof methodsBuild and check mathematical induction arguments