Percentages and Fractions
Per cent means out of 100, so every percentage can be written as a fraction. This NSW Year 7 maths subtopic (NESA outcome MA4‑FRC‑C‑01) converts between the two forms and treats both as rational numbers.
You will make a denominator of 100 (\(\frac{16}{25} = 64\%\)), multiply by 100 when that is not possible, and write percentages as fractions in simplest form, including fractional percentages such as \(7\frac{1}{2}\% = \frac{3}{40}\) and values above 100%. A test score of 42 out of 60 becomes 70%, and you can compare a quarter off with 30% off. This subtopic is part of Year 7 Percentages.
Theory
Per cent means “out of \(100\)”, so a percentage is a fraction out of \(100\): \(47\% = \dfrac{47}{100}\). This Year 7 maths page shows how to change fractions to percentages and back, how to handle fractional percentages such as \(12\dfrac{1}{2}\%\) and percentages above \(100\%\), and how to compare.
A percentage is a fraction whose denominator is \(100\): \(15\% = \dfrac{15}{100}\).
Fraction to percentage. Make the denominator \(100\) by multiplying the top and the bottom by the same number: \(\dfrac{11}{20} = \dfrac{55}{100} = 55\%\). When the denominator will not make \(100\), multiply the fraction by \(100\%\): \(\dfrac{2}{5} \times 100\% = 40\%\) and \(\dfrac{5}{8} \times 100 = 62.5\), so \(\dfrac{5}{8} = 62.5\%\).
Percentage to fraction. Write the percentage over \(100\), then simplify using the HCF: \(25\% = \dfrac{25}{100} = \dfrac{1}{4}\) and \(80\% = \dfrac{80}{100} = \dfrac{4}{5}\).
A fractional percentage such as \(1\dfrac{1}{2}\%\): write the mixed numeral as an improper fraction, then divide by \(100\).
\(100\%\) is one whole. A percentage above \(100\%\) is more than one whole: \(120\% = \dfrac{6}{5} = 1\dfrac{1}{5}\). Fractions and percentages are both rational numbers (numbers that can be written as \(\dfrac{a}{b}\) with integers \(a\), \(b\) and \(b \ne 0\)).
Fractions worth knowing by heart:
| Fraction | % | Fraction | % |
|---|---|---|---|
| \(\dfrac{1}{2}\) | \(50\%\) | \(\dfrac{1}{10}\) | \(10\%\) |
| \(\dfrac{1}{4}\) | \(25\%\) | \(\dfrac{1}{8}\) | \(12.5\%\) |
| \(\dfrac{3}{4}\) | \(75\%\) | \(\dfrac{1}{3}\) | \(33\dfrac{1}{3}\%\) |
| \(\dfrac{1}{5}\) | \(20\%\) | \(1\) | \(100\%\) |
Fraction to percentage
- Simplify the fraction if you can.
- Make the denominator \(100\), or multiply the fraction by \(100\).
- Write the answer with a \(\%\) sign (a mixed numeral if it is not whole).
Percentage to fraction
- Write the percentage over \(100\) (change a fractional percentage to an improper fraction and divide by \(100\)).
- Simplify by dividing the top and bottom by the HCF.
(i) \(\dfrac{13}{20}\)
(ii) \(\dfrac{21}{35}\)
| \(\dfrac{13}{20}\) | \(=\) | \(\dfrac{13 \times 5}{20 \times 5} = \dfrac{65}{100}\) |
| \(\dfrac{21}{35}\) | \(=\) | \(\dfrac{3}{5} = \dfrac{60}{100}\) |
(i) \(65\%\)
(ii) \(60\%\) (simplify first: the HCF of \(21\) and \(35\) is \(7\))
(i) \(45\%\)
(ii) \(140\%\)
| \(\dfrac{45}{100}\) | \(=\) | \(\dfrac{9}{20}\) |
| \(\dfrac{140}{100}\) | \(=\) | \(\dfrac{7}{5} = 1\dfrac{2}{5}\) |
(i) \(\dfrac{9}{20}\) (HCF \(5\))
(ii) \(1\dfrac{2}{5}\) (HCF \(20\); more than one whole)
(i) \(12\dfrac{1}{2}\%\)
(ii) \(6\dfrac{1}{4}\%\)
| \(\dfrac{25}{2} \times \dfrac{1}{100}\) | \(=\) | \(\dfrac{25}{200} = \dfrac{1}{8}\) |
| \(\dfrac{25}{4} \times \dfrac{1}{100}\) | \(=\) | \(\dfrac{25}{400} = \dfrac{1}{16}\) |
(i) \(\dfrac{1}{8}\) (the table gives \(\dfrac{1}{8} = 12.5\%\))
(ii) \(\dfrac{1}{16}\)
(i) \(\dfrac{5}{8}\)
(ii) \(\dfrac{7}{12}\)
| \(\dfrac{5}{8} \times 100\) | \(=\) | \(\dfrac{500}{8} = 62\dfrac{1}{2}\) |
| \(\dfrac{7}{12} \times 100\) | \(=\) | \(\dfrac{700}{12} = 58\dfrac{1}{3}\) |
(i) \(62\dfrac{1}{2}\%\) (or \(62.5\%\))
(ii) \(58\dfrac{1}{3}\%\)
| \(\dfrac{34}{40} \times 100\) | \(=\) | \(85\) |
Kai scored \(85\%\).
| \(\dfrac{5}{6} \times 100\) | \(=\) | \(\dfrac{250}{3} = 83\dfrac{1}{3}\) |
\(83\dfrac{1}{3}\% > 82\%\), so Ella has the better record.
| \(\dfrac{3}{10}\) | \(=\) | \(30\%\) |
| \(100 - 30 - 52\) | \(=\) | \(18\) |
| \(\dfrac{18}{100}\) | \(=\) | \(\dfrac{9}{50}\) |
\(18\%\), or \(\dfrac{9}{50}\), come by car.
Common pitfalls
Frequently asked questions
What does per cent mean?
Out of \(100\). \(47\% = \dfrac{47}{100}\).
How do you change a fraction to a percentage?
Multiply by \(100\%\), or make the denominator \(100\). \(\dfrac{2}{5} = 40\%\).
How do you change a percentage to a fraction?
Write it over \(100\) and simplify. \(25\% = \dfrac{1}{4}\).
How do you change 12 1/2% to a fraction?
\(\dfrac{25}{2} \div 100 = \dfrac{25}{200} = \dfrac{1}{8}\).
Can a percentage be more than 100%?
Yes. It is more than one whole: \(120\% = 1\dfrac{1}{5}\).