Fraction, Decimal & Percent Converter
Type a fraction, a mixed number, a decimal, a repeating decimal or a percentage, and get all the other forms at once — exactly, with the steps: long division for a fraction to a decimal, place value or the repeating-decimal trick for a decimal to a fraction.
Three Ways to Write a Part
A fraction, a decimal and a percentage are three ways of writing the same number. A fraction becomes a decimal when you divide the numerator by the denominator. The decimal ends when the simplified denominator only has the prime factors 2 and 5 (as 8 = 2³ in 3/8 = 0.375); otherwise it repeats for ever (1/3 = 0.333…, 1/7 = 0.142857142857…).
Every ending or repeating decimal can be written as a fraction: these are exactly the rational numbers. Numbers like π and √2 have decimals that never end and never repeat, so they are not fractions. A percentage is the decimal times 100: 0.375 = 37.5%.
Step-by-Step Solution
Common Fractions
| Fraction | Decimal | Percentage |
|---|
Worked Techniques
Three decimal places means thousandths. Then simplify by the greatest common divisor.
Multiply by powers of 10 so that the repeating parts line up, then subtract: the endless tail disappears.
Long division: when the remainder becomes 0 the decimal ends; when a remainder comes back, the digits start to repeat.
Sources & Verification
- H.-H. Wu, Understanding Numbers in Elementary School Mathematics, American Mathematical Society, 2011 — Part 2 (Decimals, and converting fractions to decimals; repeating decimals).
- G. H. Hardy & E. M. Wright, An Introduction to the Theory of Numbers, 6th ed., Oxford University Press, 2008 — Ch. 9 (The representation of numbers by decimals).
Frequently Asked Questions
How do you convert a fraction to a decimal?
Divide the numerator by the denominator. For 3/8: 3 ÷ 8 = 0.375, found by long division (30 ÷ 8 = 3 remainder 6, 60 ÷ 8 = 7 remainder 4, 40 ÷ 8 = 5 remainder 0). When the remainder becomes 0 the decimal ends; when a remainder comes back, the digits start to repeat, as in 2/3 = 0.666…
How do you convert a decimal to a fraction?
Write the digits over 10, 100, 1000 … depending on the number of decimal places, then simplify. Example: 0.375 has three decimal places, so it is 375/1000; dividing the top and bottom by their greatest common divisor 125 gives 3/8.
How do you convert a repeating decimal to a fraction?
Use the trick of shifting and subtracting. Example: x = 0.1666…
1. Multiply so that the repeating parts line up: 100x = 16.666… and 10x = 1.666…
2. Subtract: 100x − 10x = 90x = 15, because the endless tails cancel.
3. So x = 15/90 = 1/6. In the calculator, type the repeating digits in brackets: 0.1(6).
How do you convert a fraction or decimal to a percentage?
Multiply the decimal by 100 and add the % sign: 3/8 = 0.375 = 37.5%. The other way round, divide by 100: 37.5% = 0.375 = 3/8.
Which fractions have a decimal that ends?
Exactly those whose denominator, in lowest terms, has no prime factors other than 2 and 5. So 3/8 = 0.375 and 7/20 = 0.35 end, but 1/3, 1/6 and 1/7 repeat for ever. 1/7 = 0.142857142857…, with a repeating block of six digits.
Is 0.999… really equal to 1?
Yes. With the repeating-decimal trick: x = 0.999…, so 10x = 9.999…, and 10x − x = 9x = 9, which gives x = 1. Another way to see it: 1/3 = 0.333…, and three times that is 0.999… = 3/3 = 1. Type 0.(9) in the calculator to see it.