Method

Converting a Repeating Decimal Number Into a Fraction

Terminating decimal numbers can be converted to fractions by dividing the number by 10^n, where n is the number of decimal places. Terminating Decimal & & Fraction 0.56 & & 56/100 However, it is impossible to count the number of decimal places for repeating decimals because they have an infinite number of digits after the decimal point. Repeating Decimal & & Fraction 0.083 & & ? Even though they have an infinite number of digits, these numbers can be expressed as fractions. Follow these five steps to do so.

Assign a Variable for the Repeating Decimal

Let x be a variable to represent the given repeating decimal number. An equation can be written by setting x equal to the number. Consider 0.083 as an example. x = 0.083

Count the Number of Repeating Digits and Let It Be n

For the given decimal, the repeating digit is 3. x = 0.083 There is only one digit with a bar above it. This means n= 1.

Multiply the Repeating Decimal by 10^n

Since there is one repeating digit, multiply each side of the equation by 10^1, or 10.

x = 0.083

MultEqn

LHS * 10=RHS* 10

x * 10 = 0.083 * 10

Multiply

10x = 0.83

Subtract the Equation in Step 1 From the Equation in Step 3

Now subtract the original equation from the equation obtained in Step 3. cl & 10x = 0.83 - & x = 0.083 Writing the decimals without using bar notation can make subtraction easier to understand. cl & 10x = 0.83333 ... - & x = 0.08333 ... & 9x = 0.75000 ... The zeros can be ignored. It is found that 9x is equal to 0.75.

Solve for the Variable and Express It as a Fraction in Its Simplest Form

Finally, solve the equation for x.

9x = 0.75

DivEqn

.LHS /9.=.RHS /9.

9x/9 = 0.75/9

CancelCommonFac

Cancel out common factors

9x/9 = 0.75/9

SimpQuot

Simplify quotient

x = 0.75/9

Start by eliminating the decimal in the numerator to simplify the fraction. This can be done by multiplying the numerator and denominator by 10^2 because the numerator has 2 decimal places.

x = 0.75/9

ExpandFrac

a/b=a * 100/b * 100

x = 0.75 * 100/9* 100

Multiply

x = 75/900

Since the greatest common factor of 900 and 75 is 75, simplify the fraction by dividing both the numerator and denominator by 75.

x = 75/900

ReduceFrac

a/b=.a /75./.b /75.

x = 75/ 75/900 / 75

CalcQuot

Calculate quotient

x = 1/12

It is found that x is equal to 112. Remember that x is also equal to 0.083. x = 0.83 and x = 1/12 Therefore, by the Transitive Property of Equality, the fraction is equal to the given repeating decimal number. 0.083=1/12

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