Converting a Repeating Decimal to a Fraction

Students studying pre-algebra learn that fractions can be written as decimals and decimals can be written as fractions. For example, the fraction 1/2 can be written as the decimal 0.5. And the decimal 0.25 can be written as the fraction 1/4.

 

In this lesson we will learn about "repeating decimals", and how we can express them as a fraction.

 

What is a repeating decimal? Here's an example:

 

 0.333333... (the "3" keeps repeating)

 

Here are two more examples:

 

 0.18181818... (the "18" keeps repeating)
 0.1222222... (the "2" keeps repeating)

 

In a repeating decimal one digit, or a series of digits, keeps repeating without end.

 

It turns out that every repeating decimal can be converted to a fraction. And once it is a fraction, it is easier to use if we need to do a calculation.

 

Now let's learn how to convert a repeating decimal to a fraction.

 

The first repeating decimal above is equal to the fraction 1/3. You may have known that, but there is a method we can use to find the fraction. Once you know the method you can convert any repeating decimal to its equivalent fraction.

 

Here are the steps to convert 0.333333... to its equivalent fraction:

 

Step 1 - Identify the repeating part in the decimal 0.333333... . The repeating part is "3".

 

Step 2 - Set the decimal equal to some variable. We'll use "a" for the variable. This is our first equation.

 

 a = 0.333333...

 

Step 3 - Modify the equation so that the repeating part (the "3") appears to the left of the decimal point. Notice that we can do this by multiplying both sides of the equation by 10.

 

 10a = 3.333333...

 

Step 4 - Notice we now have two equations. Let's subtract the first from the second.

 

 10a = 3.333333...
 - a = 0.333333...
               
  9a = 3

 

Since the decimal part of both equations is the same, when we subtract we are just left with 3 - 0 = 3 on the right hand side.

 

Step 5 - Let's solve for a. Dividing both sides by 9 we get:

 

      3   1
  a = - = -
      9   3

 

So we can see that our original decimal of 0.333333... is equal to the fraction 1/3.

---

Let's summarize the steps to follow to convert any repeating decimal to a fraction.

 

(a) Identify the repeating part in the decimal. Then set the decimal equal to a variable. Call this Equation 1.

 

(b) Modify Equation 1 so that the repeating part appears directly to the left of the decimal point. This usually involves multiplying by 10, 100, or another power of ten. Call the result Equation 2.

 

(c) Modify Equation 1 so that the repeating part appears directly to the right of the decimal point. This may involve multiplying by 10, 100, or another power of ten. Sometimes Equation 1 is already in this form. Call the result Equation 3.

 

(d) Subtract Equation 3 from Equation 2. Then solve for the variable. After simplifying you will have the fraction.

 

---

Example

 

Convert 0.18181818... to a fraction

 

Solution

 

Let's use the steps above.

 

(a) The repeating part of the decimal is "18". Our equation is:

 

 a = 0.18181818... (Equation 1)

 

(b) Multiply both sides of Equation 1 by 100 so the "18" appears directly to the left of the decimal point.

 

 100a = 18.18181818... (Equation 2)

 

(c) We want another equation in which the repeating part is to the right of the decimal point. We won't need to modify Equation 1 since it is already in that form.

 

 a = 0.18181818... (Equation 3)

 

(d) Subtract Equation 3 from Equation 2, then solve for a.

 

 100a = 18.18181818...
  - a =  0.18181818...
                   
  99a = 18

 

To solve for a, divide each side by 99 and then simplify:

 

      18    6
  a =    =   
      99   33

 

So we can see that our original decimal of 0.18181818... is equal to the fraction 6/33.