Decimal to Fraction calculator
The Decimal to Fraction calculator is an easy-to-use online tool that converts decimal numbers to proper fractions or mixed numbers. The calculator takes terminating or recurring decimals as inputs and returns the answer in the form of a proper fraction or a mixed number.
Directions for the Use of the Fraction calculator
To use the calculator, enter the given number in decimal form. Then enter the number of repeating decimals (see the explanation below) and press "Calculate".
How to Enter the Number of Repeating Trailing Decimal Places
Repeating, or recurring, trailing decimal places are those digits after the decimal sign that are infinitely repeated in a number.
For example, suppose you need to enter a recurring decimal \$0.333\ldots=0.\bar{3}\$. In this case, you should first enter 0.3 into the "Enter a Decimal Number" field. Then enter 1 into the second input field since this number has only one trailing decimal place – 3. (The answer will be \$\frac{1}{3}\$.)
If you need to enter a recurring decimal such as \$0.454545\ldots=0.\bar{45}\$, first enter 0.45 into the "Enter a Decimal Number" field. Then enter 2 into the second input field since this number has two trailing decimal places – 45. (The answer will be \$\frac{5}{11}\$.)
If you need to enter a decimal, such as \$2.83333333\ldots=2.8\bar{3}\$, first enter 2.83 into the "Enter a Decimal Number" field. Then enter 1 into the second input field since this number has only one trailing decimal place – 3. (The answer will be \$2\frac{5}{6}\$.)
For a decimal such as \$0.285714285714\ldots=0.\bar{285714}\$, first, enter 0.285714 into the "Enter a Decimal Number" field. Then enter 6 into the second input field since this number has six trailing decimal places – 285714. (The answer will be \$\frac{2}{7}\$.)
The calculator accepts positive as well as negative decimal numbers as inputs.
After you enter the decimal and the number of trailing decimal places, the calculator will perform the conversion to a fraction or a mixed number and display the answer, as well as a detailed explanation of the solution.
Important Definitions
Decimal numbers
Decimal numbers can be divided into two big groups: terminating and non-terminating decimal numbers. Decimal numbers with a finite number of digits after the decimal point are terminating since they terminate or stop at some point. On the contrary, decimal numbers with an infinite number of digits after the decimal point are called non-terminating. These non-terminating numbers can be divided into two groups: recurring and non-recurring. If some digits after the decimal point are infinitely repeated, this number is called a recurring decimal. Examples of such decimals are:
$$16.3333333\ldots=16.\bar{3}$$
or
$$3.961961961\ldots=3.\bar{9}61$$
Non-terminating decimal numbers, where every digit after the decimal point is different, are called non-recurring decimal numbers. You can never write out such numbers completely. Therefore, it is impossible to use them as inputs for the decimal to fraction conversion. An example of a non-recurring decimal is:
$$6.7102984637\ldots$$
Fractions and mixed numbers
This decimal to fraction converter re-writes the given decimal number in fraction or mixed number form. In fraction form, the calculator always uses the proper fraction – the fraction representing a number less than 1 – meaning that the numerator will be less than the denominator. Examples of proper fractions are:
$$\frac{4}{9}\ or \ \frac{3}{7}$$
We call a fraction improper if it represents a number greater than or equal to 1, meaning that the numerator will be greater than or equal to the denominator. Examples of improper fractions are:
$$\frac{11}{7}\ or \ \frac{13}{2}$$
If a number consists of a whole number and a proper fraction, it is called a mixed number. Examples of mixed numbers are:
$$3\frac{3}{5}\ or \ 6\frac{17}{31}$$
The calculator will answer either as a proper fraction or a mixed number.
Converting decimals to fractions
You should follow the steps below to convert a decimal to a fraction or a mixed number.
Any decimal number x can be represented as a fraction with 1 as the denominator \$\frac{x}{1}\$. As the first step, re-write the given number as a fraction, with the number itself as the numerator, and 1 as the denominator.
Next, count the number of digits after the decimal point, and multiply the numerator and the denominator by 10 in a corresponding power. If your number has n digits after the decimal point, the fraction's numerator and denominator must be multiplied by \${10}^n\$.
Find the numerator's greatest common factor (GCF) and the denominator of the resulting fraction. Reduce the fraction by dividing the numerator and the denominator by the GCF.
If, after simplification, you have an improper fraction, convert it to a mixed number.
Calculation example (terminating decimals)
Let's convert the decimal number 0.125 to a fraction. Following the steps above, we get:
Represent the number as a fraction with 1 in the denominator:
$$0.125=\frac{0.125}{1}$$
This number has 3 digits after the decimal point: 125. Therefore, we need to multiply both the numerator and the denominator by \${10}^3\$:
$$\frac{0.125}{1}×\frac{1000}{1000}=\frac{125}{1000}$$
The greatest common factor of the numerator and the denominator is 125. Therefore, to simplify this fraction, we need to divide both the numerator and the denominator by 125:
$$\frac{125\div125}{1000\div125}=\frac{1}{8}$$
This is already a proper fraction. Therefore, no further simplification is required.
Answer: \$0.125=\frac{1}{8}\$
Converting decimals to fractions (recurring decimals)
You should follow the steps below to convert a recurring decimal to a fraction.
Write an equation where the variable (e.g., x) equals the decimal number, with the recurring digits included only once. For example, if you have a decimal number \$5.61111\ldots=5.6\bar{1}\$, the equation should look as follows:
$$x=5.6\bar{1}$$
Identify the number of digits in the repeating decimal group n, and multiply both sides of the equation with \${10}^n\$. In our case, there is only one repeating digit: 1. Therefore, both sides of the equation have to be multiplied by \${10}^1=10\$:
$$10x=56.1\bar{1}$$
Subtract the first equation from the second equation. In our example, we get:
$$10x=56.1\bar{1}$$
$$x=5.6\bar{1}$$
$$9x=50.5$$
Solving for x, we get:
$$x=\frac{50.5}{9}$$
To eliminate decimal places, multiply the numerator and the denominator of the number by 10 to the power of n, where n is the number of digits after the decimal point. In our case, there is only one digit after the decimal point – 5. Therefore, we need to multiply by 10:
$$\frac{50.5}{9}×\frac{10}{10}=\frac{505}{90}$$
Find the numerator's greatest common factor (GCF) and the denominator of the resulting fraction. Reduce the fraction by dividing the numerator and the denominator by the GCF. In our case, the GCF is 5 therefore:
$$\frac{505\div5}{90\div5}=\frac{101}{18}$$
Simplify the improper fraction:
$$\frac{101}{18}=5\frac{11}{18}$$
In conclusion, \$5.6\bar{1}=5\frac{11}{18}\$.
