Percentages, Fractions and Decimals
A fraction, a decimal and a percentage are three ways to write the same number. In NSW Year 8 maths (NESA outcome MA4‑FRC‑C‑01) this subtopic moves between all three forms, including percentages above 100% and below 1%.
You will convert values such as \(2\frac{3}{5} = 260\%\) and \(8\frac{3}{4}\% = \frac{7}{80}\), keep an exact fractional percentage like \(16\frac{2}{3}\%\) or round a recurring result with \(\approx\), and put mixed lists in order by changing everything to decimals. A sign offering \(\frac{1}{3}\) off, netball shooting rates and school results show why one common form matters. It is part of Year 8 Fractions and Percentages.
Theory
A fraction, a decimal and a percentage are three ways to write the same number. Per cent means out of \(100\), so \(\dfrac{3}{4} = 0.75 = 75\%\). To make a percentage multiply by \(100\%\); to undo it divide by \(100\). This Year 8 maths page shows how.
Per cent means out of \(100\), so \(100\%\) is one whole.
Decimal and percentage. Multiply by \(100\) to make a percentage, and divide by \(100\) to go back. The digits move two places: \(0.025 = 2.5\%\) and \(4\% = 0.04\).
Decimal to fraction. Write the digits over the place value of the last digit, then simplify with the HCF: \(0.35 = \dfrac{35}{100} = \dfrac{7}{20}\). Fraction to decimal: divide the top by the bottom: \(\dfrac{3}{5} = 3 \div 5 = 0.6\).
Fraction to percentage. Multiply the fraction by \(100\%\): \(\dfrac{7}{25} \times 100\% = 28\%\). Percentage to fraction: write it over \(100\) and simplify: \(45\% = \dfrac{45}{100} = \dfrac{9}{20}\).
A fractional percentage. Change the mixed numeral to an improper fraction first, then divide by \(100\): \(4\dfrac{1}{2}\% = \dfrac{9}{2} \times \dfrac{1}{100} = \dfrac{9}{200}\).
Above and below. More than one whole is more than \(100\%\); less than one hundredth is less than \(1\%\). A percentage below \(1\%\) still divides by \(100\): \(0.3\% = 0.003\).
Recurring results. Some fractions give a recurring decimal. Keep the exact answer as a fractional percentage, or round it and write \(\approx\): \(\dfrac{5}{9} = 55\dfrac{5}{9}\%\), a little more than \(55.5\%\).
Ordering. Change every number to a decimal, pad them to the same number of places, then compare digit by digit.
Common equivalents:
| Fraction | Decimal | Percentage |
|---|---|---|
| \(\dfrac{1}{2}\) | \(0.5\) | \(50\%\) |
| \(\dfrac{1}{4}\) | \(0.25\) | \(25\%\) |
| \(\dfrac{1}{5}\) | \(0.2\) | \(20\%\) |
| \(\dfrac{1}{8}\) | \(0.125\) | \(12.5\%\) |
| \(\dfrac{1}{3}\) | \(0.333\ldots\) | \(33\dfrac{1}{3}\%\) |
| \(1\dfrac{1}{2}\) | \(1.5\) | \(150\%\) |
To a percentage
- Decimal: multiply by \(100\) (the decimal point moves two places to the right).
- Fraction: multiply by \(100\%\) and simplify.
- Recurring: give a fractional percentage, or round and write \(\approx\).
From a percentage
- To a decimal: divide by \(100\).
- To a fraction: write over \(100\) (improper fraction first if it is a mixed numeral), then simplify.
(ii) Change \(4\dfrac{1}{2}\%\) to a fraction.
| \(\dfrac{3}{4} \times 100\%\) | \(=\) | \(\dfrac{300}{4}\%\) |
| \(\) | \(=\) | \(75\%\) |
| \(4\dfrac{1}{2}\%\) | \(=\) | \(4\dfrac{1}{2} \div 100\) |
| \(\) | \(=\) | \(\dfrac{9}{2} \times \dfrac{1}{100}\) |
| \(\) | \(=\) | \(\dfrac{9}{200}\) |
(i) \(75\%\)
(ii) \(\dfrac{9}{200}\)
(ii) Change \(4\%\) to a decimal.
| \(0.025\) | \(=\) | \(0.025 \times 100\%\) |
| \(\) | \(=\) | \(2.5\%\) |
| \(4\%\) | \(=\) | \(4 \div 100\) |
| \(\) | \(=\) | \(0.04\) |
(i) \(2.5\%\)
(ii) \(0.04\)
(ii) Change \(0.35\) to a fraction.
(iii) Write \(0.24\) as a fraction and \(\dfrac{7}{40}\) as a percentage.
| \(3 \div 5\) | \(=\) | \(0.6\) |
| \(0.35\) | \(=\) | \(\dfrac{35}{100} = \dfrac{7}{20}\) |
| \(0.24\) | \(=\) | \(\dfrac{24}{100} = \dfrac{6}{25}\) |
| \(\dfrac{7}{40} \times 100\%\) | \(=\) | \(\dfrac{700}{40}\% = 17.5\%\) |
(i) \(0.6\)
(ii) \(\dfrac{7}{20}\)
(iii) \(\dfrac{6}{25}\) and \(17.5\%\). Check: \(7 \div 40 = 0.175\).
(i) \(225\%\)
(ii) \(0.6\%\)
| \(225 \div 100\) | \(=\) | \(2.25\) |
| \(\) | \(=\) | \(2\dfrac{25}{100} = 2\dfrac{1}{4}\) |
| \(0.6 \div 100\) | \(=\) | \(0.006\) |
| \(\) | \(=\) | \(\dfrac{6}{1000} = \dfrac{3}{500}\) |
(i) \(2.25 = 2\dfrac{1}{4}\), more than one whole.
(ii) \(0.006 = \dfrac{3}{500}\)
(ii) Write \(\dfrac{5}{12}\) as a percentage, exactly and to \(1\) decimal place.
| \(6\dfrac{1}{4}\%\) | \(=\) | \(\dfrac{25}{4} \div 100\) |
| \(\) | \(=\) | \(\dfrac{25}{400} = \dfrac{1}{16}\) |
| \(\dfrac{5}{12} \times 100\%\) | \(=\) | \(\dfrac{500}{12}\%\) |
| \(\) | \(=\) | \(41\dfrac{2}{3}\%\) |
(i) \(\dfrac{1}{16}\)
(ii) \(41\dfrac{2}{3}\%\). The decimal \(41.66\ldots\) recurs, so \(\approx 41.7\%\).
Change every score to a decimal with three places.
| \(\text{Ben}\) | \(=\) | \(4 \div 5 = 0.800\) |
| \(\text{Chloe}\) | \(=\) | \(79 \div 100 = 0.790\) |
| \(\text{Ava}\) | \(=\) | \(0.780\) |
| \(\text{Dan}\) | \(=\) | \(31 \div 40 = 0.775\) |
\(0.800 > 0.790 > 0.780 > 0.775\), so the order is Ben, Chloe, Ava, Dan.
Common pitfalls
Frequently asked questions
How do I change a fraction to a percentage?
Multiply the fraction by \(100\%\): \(\dfrac{3}{8} \times 100\% = 37.5\%\).
How do I change a percentage to a fraction?
Write the percentage over 100 and simplify. For a mixed number percentage, make it an improper fraction first.
How do I change a decimal to a percentage?
Multiply by \(100\): \(0.4 = 40\%\).
Can a percentage be more than 100%?
Yes. More than one whole is more than \(100\%\): \(1.5 = 150\%\).
How do I order fractions, decimals and percentages?
Change them all to decimals with the same number of decimal places, then compare digit by digit.