# Decimal to fraction

The decimal numeral system is a positional numeral system, which means that the position of a digit in a number determines its value. The decimal system is a base-ten system so each decimal place represents a factor of 10. Understanding this will provide a good basis for converting decimals to fractions.

### Converting terminating decimals to fractions

Use the following steps to convert terminating decimals.

- Write the decimal value as the numerator of a fraction with the denominator being 1
- Count the number of digits to the right of the decimal point, and call this number n
- Multiply the numerator and denominator by 10
^{n}. This removes the decimal point - Simplify the fraction

Example

Convert 0.375 to a fraction:

1.

2. There are 3 digits after the decimal point in 0.375 so n=3 and 10^{3} = 1000

3.

4.

In cases where the number before the decimal point is not zero, we can use the same steps, ignoring the whole number, and add the whole number value as a mixed fraction or improper fraction at the end.

Example

Convert 4.65 to both an improper fraction and mixed fraction:

First we'll ignore the whole number portion, 4 and read this fraction as 0.625. We can do this because 4.65 = 4 + 0.65; the whole number portion doesn't affect the conversion.

1.

2. There are 2 digits after the decimal point in 0.65 so n=2 and 10^{2} = 100.

3.

4.

Now we can add the whole number portion, 4, and write 4.65 in mixed fraction or improper fraction form:

Mixed fraction: | |

Improper fraction: |

### Converting repeating decimals to fractions

Repeating decimals, such as 0.3 require a slightly different process to convert than terminating decimals. There are two cases of repeating decimals to consider: all of the digits in the decimal repeat, or there is a mix of repeating and non-repeating digits. Below are two different ways to convert these two cases of repeating decimals to fractions. If you are comfortable with algebra, use the first method, as it is simpler to use for more complex repeating decimals. The steps in the second method are based on the algebraic method but don't use any algebra, so you have to follow some steps exactly that may seem random.

Algebraic method:

- Write an equation (1) setting a variable, x, equal to the decimal being converted
- When there is a mix of non-repeating and repeating decimals in our number, to be able to subtract the equations such that the repeating portion cancels, we need to multiply (1) by 10
^{m}where m is the number of non-repeating digits in the number. When this is the case, this new equation is referred to as equation (1) for the purposes of step 4. Refer to examples 3 and 4 below for clarification - Count the number of decimal places in the decimal being converted; call this number n
- Write a second equation (2), multiplying both sides of (1) by 10
^{n} - Subtract (1) from (2)
- Solve for x and reduce the fraction if necessary

Examples

1. Convert 0.6 to a fraction:

- Subtract (1) from (2):

2. Convert 0.4534 to a fraction:

- Subtract (1) from (2):

3. Convert 0.3362 to a fraction:

- Since 0.3362 has two non-repeating digits, we multiply our equation

by 10^{2}= 100 to get:

- Subtract (1) from (2)

4. Convert 0.652 to a fraction:

- Since 0.652 has two non-repeating digits, we multiply our equation

by 10^{2}= 100 to get:

- Subtract (1) from (2)

Non-algebraic method:

This method is relatively simple for certain cases, but quickly gets more tedious for others. Refer to the examples provided below for clarification.

- When all the digits repeat:
- Numerator:
- Write the repeating digit(s) without the decimal point. For 0.3 we would write 3
- Denominator:
- Write one 9 for each digit in the set of repeating digits in the decimal. For example, for 0.3, we would write one 9 (9). For 0.262 we would write three 9s (999)
- When there is a mix of repeating and non-repeating digits:
- Numerator:
- Write the non-repeating portion of the decimal followed by the difference between the repeating portion of the decimal and the non-repeating portion. For 0.3362 we would write 3329 (33 and (62 - 33 = 29)).
- If the repeating portion is smaller than the non-repeating portion, write out more digits of the repeating portion until it is large enough to subtract the non-repeating portion without resulting in a negative number: For 0.42, use 22 - 4. The numerator would then be 418.
- Denominator:
- Write one 9 for each digit in the set of repeating digits in the decimal, then write one 0 after the 9(s) for each digit in the non-repeating portion of the decimal. For 0.3362, we would write 9900. If the repeating portion is smaller than the non-repeating portion, as in the example described in the numerator section, 0.42, the number of 9s should reflect the number of digits of the repeating portion required to perform the subtraction in the numerator. We needed to use two digits, 22, so we write two 9s, and the denominator is 990.

Examples

When all the digits repeat:

1. Convert 0.6 to a fraction:

The numerator written without the decimal point, is 6. The denominator is 9, since the only repeating digit is 6. Note that the resulting fraction can be simplified. This is not always the case.

2. Convert 0.4534 to a fraction:

The numerator written without the decimal point is 4534. The denominator is 9999 since there are 4 digits in the repeating portion of the decimal.

When there is a mix of repeating and non-repeating digits:

1. Convert 0.3362 to a fraction:

The first part of the numerator is 33 since that is the non-repeating portion of the decimal. The second part of the numerator is the difference between the repeating portion and the non-repeating portion. Since in this case, the repeating portion is larger, we can just compute 62 - 33 = 29. So, the numerator is 3329.

The denominator is 9900 since there are two digits that don't repeat in the decimal, and two digits that do.

2. Convert 0.652 to a fraction:

The first part of the numerator is 65, which is larger than a single digit of the repeating portion, 2. Write out the repeating portion until the value is large enough to subtract without resulting in a negative number. 222 - 65 = 157, so our numerator is 65157.

When determining the numerator we needed to use 3 digits of the repeating portion to subtract the non-repeating portion of the decimal, so we write 999. Then we append two 0s since there are two non-repeating digits, making our denominator 99900.