To convert a decimal to a fraction, write the digits after the decimal point as the numerator, use the place value of the last digit (10, 100, 1,000 and so on) as the denominator, and simplify. For 0.75, the last digit is in the hundredths place, so 0.75 = 75/100 = 3/4.
What are the steps to convert a terminating decimal?
OpenStax Prealgebra gives a four-step routine:
- Look at the number left of the decimal point. If it is zero, the result is a proper fraction. If not, keep that whole number to form a mixed number.
- Find the place value of the final digit: tenths, hundredths, thousandths and so on.
- Write the digits after the point as the numerator and that place value as the denominator.
- Simplify.
The number of decimal places tells the denominator directly: one place gives 10, two places give 100, three places give 1,000.
Worked example: 0.286
The final digit, 6, is in the thousandths place, so 0.286 = 286/1,000. Both numbers are even, so divide by 2 to get 143/500. Since 143 = 11 × 13 and 500 = 2² × 5³ share no factor, the fraction is in lowest terms.
Worked example: 4.09 and 3.7
The whole parts, 4 and 3, stay in front. In 4.09 the 9 is in the hundredths place, so 4.09 = 4 9/100. In 3.7 the 7 is in the tenths place, so 3.7 = 3 7/10. For an improper fraction, multiply the whole number by the denominator and add the numerator: 4 9/100 = (4 × 100 + 9)/100 = 409/100.
How do you simplify the fraction?
Divide the numerator and denominator by their greatest common factor (GCF). For 0.125 = 125/1,000, the GCF is 125, giving 1/8. For 0.05 = 5/100, the GCF is 5, giving 1/20. When the GCF is hard to spot, divide by small common factors repeatedly (2, 5, 3) until none remains, or use the GCF calculator.
Decimal to fraction chart
These values come up often. Each fraction is in lowest terms.
| Decimal | Fraction | Decimal | Fraction |
|---|---|---|---|
| 0.05 | 1/20 | 0.6 | 3/5 |
| 0.0625 | 1/16 | 0.75 | 3/4 |
| 0.125 | 1/8 | 0.875 | 7/8 |
| 0.25 | 1/4 | 1.75 | 1 3/4 (or 7/4) |
| 0.3125 | 5/16 | 2.5 | 2 1/2 (or 5/2) |
| 0.4 | 2/5 | 0.5 | 1/2 |
How do you convert a repeating decimal to a fraction?
A repeating decimal has a block of digits that never stops, such as 0.777… A fraction cannot be read off the place value, so use algebra. The procedure in OpenStax Contemporary Mathematics counts the repeating digits, shifts the decimal with a power of 10, and subtracts so the repeating part cancels.
For 0.777…:
- Let x = 0.777…
- One digit repeats, so multiply by 10: 10x = 7.777…
- Subtract the first line: 10x − x = 7.777… − 0.777…, so 9x = 7.
- Divide: x = 7/9.
For a repeating block of two digits, multiply by 100 instead. For 0.4545…: 100x = 45.4545…, so 99x = 45 and x = 45/99 = 5/11. The denominator of a purely repeating decimal is a string of nines, one for each digit in the block, before simplifying. See repeating decimal in the glossary.
What about 0.33333?
It depends on whether the 3s stop. The finite decimal 0.33333 is 33,333/100,000. The repeating decimal 0.333… (written with a bar over the 3) is exactly 1/3. Calculators that show 0.3333333 usually truncate a repeating value, so check whether the decimal is exact before converting.
Which fractions give terminating decimals?
A fraction in lowest terms gives a terminating decimal only when the denominator's prime factors are 2 and 5 only, because a terminating decimal is a fraction with a power of 10 in the denominator and 10ⁿ = 2ⁿ × 5ⁿ. So 3/40 terminates (40 = 2³ × 5) and 0.075 is its decimal form, while 1/3 and 5/11 repeat. The University of Toronto Question Corner explains this test.
Common mistakes
- Using the wrong denominator: 0.05 is 5/100, not 5/10.
- Stopping before the fraction is simplified (25/100 is 1/4).
- Dropping the whole number from a decimal such as 2.5.
- Treating a rounded calculator display of a repeating decimal as an exact value.
Once a decimal is a fraction, the ratio calculator can scale it or compare it with another quantity.