## 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 10^{3} = 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 10^{3} = 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 10
^{3}= 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.