This online Decimal To Fraction Converter converts a positive decimal number (including a repeating decimal) to a fraction. The repeated part of the fraction must be enclosed in parentheses. In case the resulting fraction is an improper fraction (when the numerator is greater than the denominator) we present the result also in the form of a mixed number.
How to Convert Decimals to Fractions
Writing decimals as fractions is a frequently performed mathematical operation. For non repeating decimals the algorithm of conversion is quite simple.
1) Make a fraction with the decimal number as the numerator and write 1 as the denominator. Let’s take 0.125 as an example and rewrite this number as a fraction: 0.125 = 0.125 / 1.
2) Multiply the numerator and denominator of this fraction by 10 until the decimal point in the numerator moves to the far right position. In our example we will have: 0.125 = 0.125 / 1 = 1.25 / 10 = 12.5 / 100 = 125 /1000.
3) Reduce the fraction dividing both the numerator and denominator by the greatest common divisor (GCD). In our example we can simplify the fraction dividing the numerator and denominator by GCD = 125: 0.125 = 125 / 1000 = 1 / 8.
4) In case the numerator is greater than the denominator you can further simplify the remaining fraction converting it to a mixed number fraction. So, if we have the fraction 3.125 we could have 3.125 = 3125 / 1000 = 25 / 8 = 3 1/8. Or, even simpler, in view that 3.125 = 3 + 0.125 we can convert the fractional part only and have as the result: 3.125 = 3 + 0.125 = 3 + 1 / 8 = 3 1/8.
How to Convert Repeating Decimals to Fractions
A repeating decimal (or recurring decimal) is decimal representation of a number whose digits are periodic (repeating its values at regular intervals) and the infinitely repeated portion (the repetend) is not zero. It can be shown that a number is rational if and only if its decimal representation is repeating or terminating.
Since a repeating decimal is rational, it can be represented as a fraction. For example, let’s take 0.636363… which has 63 as the repetend. Perform the following simple calculations:
2) 100x = 63.636363… (multiply each side of the above equation by 100)
3) 99x = 63 (subtract the first equation from the second one)
4) x = 63 / 99 = 7 / 11 (reduce the required fraction)
Notice, that in general case of a repeating decimal of the form x = 0.a1a2…an … with an n-digit period, we can get the result x = a1a2…an / (10n-1).
Consider a more complex case of a repeating decimal: 5.4121212… Using the ideas and results presented above, we can easily get:
5.4121212… = 5.4 + 0.0121212… = 54/10 + (12/99)/10 = (54*99 + 12)/(10*99) = 5358/990 = 893/165 = 5 68/165
All these calculations can be instantly performed using our Decimal To Fraction Converter. It is important to remember that when entering a repeated decimal, the repeated part of the fraction must be enclosed in parentheses. So, the number from the above example should look like 5.4(12).
In case the resulting fraction is an improper fraction it’s also presented in the form of a mixed number. We use tiny Unicode characters to present the fractional part of the mixed number.
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