Decimals and Fractions
Theory
A terminating decimal is a fraction with a denominator of \(10\), \(100\), \(1000\), \(\ldots\) written in place-value form. The place of the last digit gives the denominator. This Year 7 maths page shows how to change decimals to fractions and simplify them, change fractions to decimals, and link fractions, decimals and percentages.
The number of decimal places matches the number of zeros in the denominator:
| Place of last digit | Decimal places | Denominator |
|---|---|---|
| tenths | 1 | \(10\) (1 zero) |
| hundredths | 2 | \(100\) (2 zeros) |
| thousandths | 3 | \(1000\) (3 zeros) |
| ten thousandths | 4 | \(10\,000\) (4 zeros) |
Decimal to fraction: put the digits over the denominator given by the last place, then simplify by dividing by the HCF (highest common factor): \(0.64 = \dfrac{64}{100} = \dfrac{16}{25}\).
Fraction to decimal: if the denominator is \(10\), \(100\), \(1000\), \(\ldots\), fill the places from the right: \(\dfrac{37}{1000} = 0.037\). Otherwise make an equivalent fraction with one of those denominators, or divide the top by the bottom.
Per cent means out of \(100\): \(0.64 = \dfrac{64}{100} = 64\%\).
Fractions worth knowing by heart:
| Fraction | Decimal | Percentage |
|---|---|---|
| \(\dfrac{1}{2} = \dfrac{50}{100}\) | \(0.5\) | \(50\%\) |
| \(\dfrac{1}{4} = \dfrac{25}{100}\) | \(0.25\) | \(25\%\) |
| \(\dfrac{1}{5} = \dfrac{20}{100}\) | \(0.2\) | \(20\%\) |
| \(\dfrac{1}{10} = \dfrac{10}{100}\) | \(0.1\) | \(10\%\) |
| \(\dfrac{1}{20} = \dfrac{5}{100}\) | \(0.05\) | \(5\%\) |
| \(\dfrac{1}{25} = \dfrac{4}{100}\) | \(0.04\) | \(4\%\) |
| \(\dfrac{1}{8}\) | \(0.125\) | \(12.5\%\) |
How to write a decimal as a fraction
- Find the place of the last digit: tenths, hundredths or thousandths.
- Write the digits after the point over \(10\), \(100\) or \(1000\).
- Simplify by dividing the top and bottom by their HCF.
How to write a fraction as a decimal
- Denominator 10, 100, 1000: count the zeros; that is the number of decimal places.
- Other denominators: multiply the top and bottom to make \(10\), \(100\) or \(1000\).
- Or divide the top by the bottom with short division.
(i) \(\dfrac{87}{100}\)
(ii) \(\dfrac{55}{10\,000}\)
(iii) six and forty-five thousandths
(i) \(100\) has \(2\) zeros: \(2\) decimal places.
(ii) \(10\,000\) has \(4\) zeros: \(4\) decimal places.
(iii) \(6 + \dfrac{45}{1000}\)
(i) \(0.87\)
(ii) \(0.0055\)
(iii) \(6.045\)
(i) \(0.9\)
(ii) \(0.041\)
(iii) \(0.003\)
Find the place of the last digit: tenths, thousandths and thousandths.
(i) \(\dfrac{9}{10}\)
(ii) \(\dfrac{41}{1000}\)
(iii) \(\dfrac{3}{1000}\)
| \(0.85\) | \(=\) | \(\dfrac{85}{100}\) |
| \(\) | \(=\) | \(\dfrac{85 \div 5}{100 \div 5}\) |
| \(\) | \(=\) | \(\dfrac{17}{20}\) |
\(0.85 = \dfrac{17}{20} = 85\%\)
| \(3.4\) | \(=\) | \(3 + \dfrac{4}{10}\) |
| \(\) | \(=\) | \(3\dfrac{2}{5}\) |
| \(\) | \(=\) | \(\dfrac{17}{5}\) |
\(3\dfrac{2}{5} = \dfrac{17}{5}\) (simplify \(\dfrac{4}{10}\) by the HCF \(2\); \(3 = \dfrac{15}{5}\)).
| \(\dfrac{11}{20}\) | \(=\) | \(\dfrac{11 \times 5}{20 \times 5}\) |
| \(\) | \(=\) | \(\dfrac{55}{100}\) |
| \(\) | \(=\) | \(0.55\) |
\(0.55 = 55\%\)
Write \(9\) as \(9.000\). Divide the digits by \(8\), carrying each remainder: \(9 \div 8\) is \(1\) remainder \(1\), \(10 \div 8\) is \(1\) remainder \(2\), \(20 \div 8\) is \(2\) remainder \(4\), and \(40 \div 8\) is exactly \(5\).
The digits are \(1125\), with the point after the first: \(\dfrac{9}{8} = 1.125\). The remainder is \(0\), so the decimal stops.
| \(0.14\) | \(=\) | \(\dfrac{14}{100}\) |
| \(100 \div 50\) | \(=\) | \(2\) |
| \(14 \div 2\) | \(=\) | \(7\) |
\(\square = 7\). Check: \(\dfrac{7}{50} = \dfrac{14}{100} = 0.14\).
Common pitfalls
Frequently asked questions
How do you change a decimal to a fraction?
Use the place of the last digit as the denominator, then simplify. \(0.64 = \dfrac{64}{100} = \dfrac{16}{25}\).
How many decimal places does a fraction over 1000 have?
Three, one for each zero. \(\dfrac{37}{1000} = 0.037\).
How do you change 3/4 to a decimal?
Make the denominator \(100\): \(\dfrac{3}{4} = \dfrac{75}{100} = 0.75\).
What does HCF mean?
Highest common factor: the biggest number that divides both the top and the bottom.
How do decimals link to percentages?
Per cent means out of \(100\), so \(0.64 = 64\%\).