Decimal to Fraction Calculator

Basic Calculator

Calculators

Math

Fractions

Decimal to Fraction Calculator

Decimal to a Fraction or Mixed Number

Enter a Decimal Number:

How many trailing decimals places
above are repeating? help

Answer:

\[ 0.\overline{6} = \frac{2}{3} \]Showing the work

Let X equal the decimal number
Equation 1:\[ X = 0.\overline{6} \]With 1 digits in the repeating decimal group,
create a second equation by multiplying
both sides by 10¹ = 10
Equation 2:\[ 10 X = 6.\overline{6} \]Subtract equation (1) from equation (2)\[ \eqalign{10 X &= &\hfill6.6...\cr X &= &\hfill0.6...\cr \hline 9X &= &6\cr} \]We get\[ 9 X = 6 \]Solve for X
\[ X = \frac{6}{9} \]Find the Greatest Common Factor (GCF) of 6 and 9, if it exists, and reduce the fraction by dividing both numerator and denominator by GCF = 3,\[ \frac{6 \div 3}{9 \div 3}= \frac{2}{3} \]Therefore\[ X = \frac{2}{3} \]In conclusion,\[ 0.\overline{6} = \frac{2}{3} \]

Share this Answer Link: help
Paste this link in email, text or social media.

Get a Widget for this Calculator

Calculator Use

This calculator converts a decimal number to a fraction or a decimal number to a mixed number. For repeating decimals enter how many decimal places in your decimal number repeat.

Entering Repeating Decimals

For a repeating decimal such as 0.66666... where the 6 repeats forever, enter 0.6 and since the 6 is the only one trailing decimal place that repeats, enter 1 for decimal places to repeat. The answer is 2/3
For a repeating decimal such as 0.363636... where the 36 repeats forever, enter 0.36 and since the 36 are the only two trailing decimal places that repeat, enter 2 for decimal places to repeat. The answer is 4/11
For a repeating decimal such as 1.8333... where the 3 repeats forever, enter 1.83 and since the 3 is the only one trailing decimal place that repeats, enter 1 for decimal places to repeat. The answer is 1 5/6
For the repeating decimal 0.857142857142857142..... where the 857142 repeats forever, enter 0.857142 and since the 857142 are the 6 trailing decimal places that repeat, enter 6 for decimal places to repeat. The answer is 6/7

How to Convert a Negative Decimal to a Fraction

Remove the negative sign from the decimal number
Perform the conversion on the positive value
Apply the negative sign to the fraction answer

If a = b then it is true that -a = -b.

How to Convert a Decimal to a Fraction

Step 1: Make a fraction with the decimal number as the numerator (top number) and a 1 as the denominator (bottom number).
Step 2: Remove the decimal places by multiplication. First, count how many places are to the right of the decimal. Next, given that you have x decimal places, multiply numerator and denominator by 10^x.
Step 3: Reduce the fraction. Find the Greatest Common Factor (GCF) of the numerator and denominator and divide both numerator and denominator by the GCF.
Step 4: Simplify the remaining fraction to a mixed number fraction if possible.

Example: Convert 2.625 to a fraction

1. Rewrite the decimal number number as a fraction (over 1)

\( 2.625 = \dfrac{2.625}{1} \)

2. Multiply numerator and denominator by by 10³ = 1000 to eliminate 3 decimal places

\( \dfrac{2.625}{1}\times \dfrac{1000}{1000}= \dfrac{2625}{1000} \)

3. Find the Greatest Common Factor (GCF) of 2625 and 1000 and reduce the fraction, dividing both numerator and denominator by GCF = 125

\( \dfrac{2625 \div 125}{1000 \div 125}= \dfrac{21}{8} \)

4. Simplify the improper fraction

\( = 2 \dfrac{5}{8} \)

Therefore,

\( 2.625 = 2 \dfrac{5}{8} \)

Decimal to Fraction

For another example, convert 0.625 to a fraction.
Multiply 0.625/1 by 1000/1000 to get 625/1000.
Reducing we get 5/8.

Convert a Repeating Decimal to a Fraction

Create an equation such that x equals the decimal number.
Count the number of decimal places, y. Create a second equation multiplying both sides of the first equation by 10^y.
Subtract the second equation from the first equation.
Solve for x
Reduce the fraction.

Example: Convert repeating decimal 2.666 to a fraction

1. Create an equation such that x equals the decimal number
Equation 1:

\( x = 2.\overline{666} \)

2. Count the number of decimal places, y. There are 3 digits in the repeating decimal group, so y = 3. Ceate a second equation by multiplying both sides of the first equation by 10³ = 1000
Equation 2:

\( 1000 x = 2666.\overline{666} \)

3. Subtract equation (1) from equation (2)

\( \eqalign{1000 x &= &\hfill2666.666...\cr x &= &\hfill2.666...\cr \hline 999x &= &2664\cr} \)

We get

\( 999 x = 2664 \)

4. Solve for x

\( x = \dfrac{2664}{999} \)

5. Reduce the fraction. Find the Greatest Common Factor (GCF) of 2664 and 999 and reduce the fraction, dividing both numerator and denominator by GCF = 333

\( \dfrac{2664 \div 333}{999 \div 333}= \dfrac{8}{3} \)

Simplify the improper fraction

\( = 2 \dfrac{2}{3} \)

Therefore,

\( 2.\overline{666} = 2 \dfrac{2}{3} \)

Repeating Decimal to Fraction

For another example, convert repeating decimal 0.333 to a fraction.
Create the first equation with x equal to the repeating decimal number:
x = 0.333
There are 3 repeating decimals. Create the second equation by multiplying both sides of (1) by 10³ = 1000:
1000X = 333.333 (2)
Subtract equation (1) from (2) to get 999x = 333 and solve for x
x = 333/999
Reducing the fraction we get x = 1/3
Answer: x = 0.333 = 1/3

Related Calculators

To convert a fraction to a decimal see the Fraction to Decimal Calculator.

References

Wikipedia contributors. "Repeating Decimal," Wikipedia, The Free Encyclopedia. Last visited 18 July, 2016.