# Decimal to fraction Formula

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## 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

1. Remove the negative sign from the decimal number
2. Perform the conversion on the positive value
3. 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

1. Step 1: Make a fraction with the decimal number as the numerator (top number) and a 1 as the denominator (bottom number).
2. 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 10x.
3. 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.
4. Step 4: Simplify the remaining fraction to a mixed number fraction if possible.

#### 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

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

## But a Special Note:

If you really meant 0,333… (in other words 3s repeating forever which is called 3 recurring) then we need to follow a special argument. In that case we write down:

## Decimal to Fraction: Everything You Need to Know

Are you ready to learn how to convert a decimal to fraction?

Before you learn an easy way to complete both of these conversions (with and without a calculator), let’s make sure that you understand what decimals and fractions are:

•  A decimal number is used to represent a non-whole number where a decimal point is used followed by digits that represent a value that is smaller than one.

A fraction represents a part of a whole number. A fraction is a ratio between the upper number (the numerator) and the lower number (the denominator). The numbers are stacked vertically and separated with a bar.

The key takeaway from these definitions is that decimals and fractions are different ways of representing the same thinga number that is not whole.

## How to Convert Decimal to Fraction

You can convert a decimal to a fraction by following these three easy steps.

In this case, you will use the decimal 0.25 as an example (see the graphic below).

Step One: Rewrite the decimal number over one (as a fraction where the decimal number is numerator and the denominator is one).

Step Two: Multiply both the numerator and the denominator by 10 to the power of the number of digits after the decimal point. If there is one value after the decimal, multiply by 10, if there are two then multiply by 100, if there are three then multiply by 1,000, etc.

In the case of converting 0.25 to a fraction, there are two digits after the decimal point. Since 10 to the 2nd power is 100, we have to multiply both the numerator and denominator by 100 in step two.

Step Three: Express the fraction in simplest (or reduced form).

By following these three steps in the above decimal to fraction example, you can conclude that the decimal 0.25, when converted to a fraction, is equal to 1/4.

Here is another example of how to convert a decimal to fraction:

Notice that the answer to this example is a mixed number (a whole number and a fraction combined).

## Decimal Fraction Formulas

Decimal Fractions: Fractions in which denominators are powers of 10 are known as decimal fractions.
1/10 = 1 tenth = .1
1/100 = 1 hundredth = .01
9/100 = 99 hundredth = .99
7/1000 = 7 thousandths =.007

Conversion of a Decimal Into Vulgar Fraction : Put 1 in the denominator under the decimal point and annex with it as many zeros as is the number of digits after the decimal point.
Now, remove the decimal point and reduce the fraction to its lowest terms.Thus,
0.25 = 25/100 = 1/4
2.008 = 2008/1000 = 251/125.

Annexing zeros to the extreme right of a decimal fraction does not change its value.
0.8 = 0.80 = 0.800, etc.

If numerator and denominator of a fraction contain the same number of decimal places, then we remove the decimal sign.
1.84/2.99 = 184/299 = 8/13
0.365/0.584 = 365/584 = 5

Operations on Decimal Fractions :
1) Addition and Subtraction of Decimal Fractions : The given numbers are so placed under each other that the decimal points lie in one column. The numbers so arranged can now be added or subtracted in the usual way.

2) Multiplication of a Decimal Fraction By a Power of 10 : Shift the decimal point to the right by as many places as is the power of 10. Thus, 5.9632 x 100 = 596,32; 0.073 x 10000 = 0.0730 x 10000 = 730.

Multiplication of Decimal Fractions : Multiply the given numbers considering them without the decimal point. Now, in the product, the decimal point is marked off to obtain as many places of decimal as is the sum of the number of decimal places in the given numbers. Suppose we have to find the product (.2 x .02 x .002).
Now, 2x2x2 = 8. Sum of decimal places = (1 + 2 + 3) = 6. .2 x .02 x .002 = .000008.

Dividing a Decimal Fraction By a Counting Number : Divide the given number without considering the decimal point, by the given counting number. Now, in the quotient, put the decimal point to give as many places of decimal as there are in the dividend. Suppose we have to find the quotient (0.0204 + 17). Now, 204 ^ 17 = 12. Dividend contains 4 places of decimal. So, 0.0204 + 17 = 0.0012.

Dividing a Decimal Fraction By a Decimal Fraction: Multiply both the dividend and the divisor by a suitable power of 10 to make divisor a whole number. Now, proceed as above.
Thus,0.00066/0.11 = (0.00066*100)/(0.11*100) = (0.066/11) = 0.006V

Comparison of Fractions: Suppose some fractions are to be arranged in ascending or descending order of magnitude. Then, convert each one of the given fractions in the decimal form, and arrange them accordingly. Suppose, we have to arrange the fractions 3/5, 6/7 and 7/9 in descending order.

now,
3/5 = 0.6
6/7 = 0.857
7/9 = 0.777….
since 0.857 > 0.777…> 0.6
so 6/7>7/9>3/5

Recurring Decimal : If in a decimal fraction, a figure or a set of figures is repeated continuously, then such a number is called a recurring decimal. In a recurring decimal, if a single figure is repeated, then it is expressed by putting a dot on it. If a set of figures is repeated, it is expressed by putting a bar on the set, Thus

1/3 = 0.3333….= 0.3; 22 /7 = 3.142857142857…..= 3.142857

Pure Recurring Decimal: A decimal fraction in which all the figures after the decimal point are repeated, is called a pure recurring decimal.

Converting a Pure Recurring Decimal Into Vulgar Fraction : Write the repeated figures only once in the numerator and take as many nines in the denominator as is the number of repeating figures. thus ,
0.5 = 5/9; 0.53 = 53/59 ;0.067 = 67/999; etc…

Mixed Recurring Decimal: A decimal fraction in which some figures do not repeat and some of them are repeated, is called a mixed recurring decimal. e.g., 0.17333= 0.173.

Converting a Mixed Recurring Decimal Into Vulgar Fraction : In the numerator, take the difference between the number formed by all the digits after decimal point (taking repeated digits only once) and that formed by the digits which are not repeated, In the denominator, take the number formed by as many nines as there are repeating digits followed by as many zeros as is the number of non-repeating digits. Thus
0.16 = (16-1) / 90 = 15/19 = 1/6;

## Decimal Fraction Solved Examples

Example 1: Convert (i) 0.75 and (ii) 2.008 into vulgar fractions.

Solutions:

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