Terminating Decimals and Fractions
Terminating Decimals and Fractions
A terminating decimal is one that ends after a finite number of decimal places — for example, \( 0.375 \) or \( 0.25 \). Every terminating decimal can be written exactly as a fraction, and some fractions can be written as terminating decimals. Fractions, decimals and percentages are three different representations of the same value; being able to convert freely between them is an essential skill throughout mathematics.
Decimal to Fraction
Write the decimal digits as the numerator and use the appropriate power of 10 as the denominator. Then simplify.
\[ 0.35 = \frac{35}{100} = \frac{7}{20} qquad 0.125 = \frac{125}{1000} = \frac{1}{8} \]The denominator matches the number of decimal places: 1 decimal place → ÷ 10, 2 places → ÷ 100, 3 places → ÷ 1000.
Fraction to Decimal
Divide the numerator by the denominator (short or long division). Equivalently, find an equivalent fraction with denominator 10, 100 or 1000.
\[ \frac{3}{8} = 3 \div 8 = 0.375 qquad \frac{7}{20} = \frac{35}{100} = 0.35 \]Which Fractions Terminate?
A fraction \( \frac{p}{q} \) in its lowest terms gives a terminating decimal if and only if the denominator \( q \) has no prime factors other than 2 and 5 — i.e. \( q = 2^a \times 5^b \) for non-negative integers \( a \) and \( b \). If \( q \) contains any other prime factor (3, 7, 11, ...), the decimal will recur.
| Fraction | Denominator factors | Terminating? | Decimal |
|---|---|---|---|
| \( \frac{3}{8} \) | \( 8 = 2^3 \) | Yes | 0.375 |
| \( \frac{7}{20} \) | \( 20 = 2^2 \times 5 \) | Yes | 0.35 |
| \( \frac{1}{6} \) | \( 6 = 2 \times 3 \) | No (3 is a factor) | \( 0.1\overline{6} \) |
| \( \frac{2}{9} \) | \( 9 = 3^2 \) | No | \( 0.\overline{2} \) |
FDP Reference Table
These conversions are worth knowing from memory — they appear regularly in non-calculator papers.
| Fraction | Decimal | Percentage |
|---|---|---|
| \( \frac{1}{2} \) | 0.5 | 50% |
| \( \frac{1}{4} \) | 0.25 | 25% |
| \( \frac{3}{4} \) | 0.75 | 75% |
| \( \frac{1}{5} \) | 0.2 | 20% |
| \( \frac{2}{5} \) | 0.4 | 40% |
| \( \frac{3}{5} \) | 0.6 | 60% |
| \( \frac{1}{8} \) | 0.125 | 12.5% |
| \( \frac{3}{8} \) | 0.375 | 37.5% |
| \( \frac{1}{10} \) | 0.1 | 10% |
| \( \frac{1}{100} \) | 0.01 | 1% |
Worked Examples
Convert 0.36 and 0.875 to fractions in simplest form.
\[ 0.36 = \frac{36}{100} = \frac{9}{25} \] \[ 0.875 = \frac{875}{1000} = \frac{7}{8} \]For 0.875: divide by HCF(875, 1000) = 125. Alternatively, notice \( 0.875 = 0.5 + 0.25 + 0.125 = \frac{1}{2} + \frac{1}{4} + \frac{1}{8} = \frac{7}{8} \).
Convert \( \frac{11}{16} \) and \( \frac{9}{25} \) to decimals.
\( \frac{11}{16} \): divide 11 by 16, or note that \( 16 = 2^4 \) so the decimal terminates.
\[ \frac{11}{16} = \frac{11 \times 625}{16 \times 625} = \frac{6875}{10000} = 0.6875 \]\( \frac{9}{25} \): \( 25 = 5^2 \) so terminates. Multiply top and bottom by 4:
\[ \frac{9}{25} = \frac{36}{100} = 0.36 \]Arrange in ascending order: \( \frac{3}{5}, 0.61, 58\%, \frac{5}{8} \)
Convert all to decimals for comparison:
\[ \frac{3}{5} = 0.6,quad 0.61,quad 58\% = 0.58,quad \frac{5}{8} = 0.625 \]Ordering: \( 0.58 < 0.6 < 0.61 < 0.625 \)
\[ 58\% < \frac{3}{5} < 0.61 < \frac{5}{8} \]Key Takeaways
- Decimal to fraction: place digits over the appropriate power of 10 and simplify — one decimal place over 10, two over 100, three over 1000.
- Fraction to decimal: divide numerator by denominator, or find an equivalent fraction with denominator 10, 100 or 1000.
- A fraction in its lowest terms terminates as a decimal only if its denominator has no prime factors other than 2 and 5.
- Percentage ↔ decimal: divide by 100 (or move decimal 2 places left) to go from % to decimal; multiply by 100 to go the other way.
- To order a mixed set of fractions, decimals and percentages: convert all to decimals first, then compare.