# Converting repeating decimals to fractions

When converting repeating decimals to fractions, just follow the two steps below carefully.

Step 1:

Let x equal the repeating decimal you are trying to convert to a fraction

Step 2:

Examine the repeating decimal to find the repeating digit(s)

Step 3:

Place the repeating digit(s) to the left of the decimal point

Step 4:

Place the repeating digit(s) to the right of the decimal point

Step 5:

Subtract the left sides of the two equations.Then, subtract the right sides of the two equations

As you subtract, just make sure that the difference is positive for both sides

Now let's practice converting repeating decimals to fractions with two good examples

Example #1:

What rational number or fraction is equal to 0.55555555555

Step 1:

x = 0.5555555555

Step 2:

After examination, the repeating digit is 5

Step 3:

To place the repeating digit ( 5 ) to the left of the decimal point, you need to move the decimal point 1 place to the right

Technically, moving a decimal point one place to the right is done by multiplying the decimal number by 10.

When you multiply one side by a number, you have to multiply the other side by the same number to keep the equation balanced

Thus, 10x = 5.555555555

Step 4:

Place the repeating digit(s) to the right of the decimal point

Look at the equation in step 1 again. In this example, the repeating digit is already to the right, so there is nothing else to do.

x = 0.5555555555

Step 5:

10x = 5.555555555

x = 0.5555555555

10x - x = 5.555555555 − 0.555555555555

9x = 5

Divide both sides by 9

x = 5/9

Example #2:

What rational number or fraction is equal to 1.04242424242

Step 1:

x = 1.04242424242

Step 2:

After examination, the repeating digit is 42

Step 3:

To place the repeating digit ( 42 ) to the left of the decimal point, you need to move the decimal point 3 place to the right

Again, moving a decimal point three place to the right is done by multiplying the decimal number by 1000.

When you multiply one side by a number, you have to multiply the other side by the same number to keep the equation balanced

Thus, 1000x = 1042.42424242

Step 4:

Place the repeating digit(s) to the right of the decimal point

In this example, the repeating digit is not immediately to the right of the decimal point.

Look at the equation in step 1 one more time and you will see that there is a zero between the repeating digit and the decimal point

To accomplish this, you have to move the decimal point 1 place to the right

This is done by multiplying both sides by 10

10x = 10.4242424242

Step 5:

1000x = 1042.42424242

10x = 10.42424242

1000x - 10x = 1042.42424242 − 10.42424242

990x = 1032

Divide both sides by 990

x = 1032/990

To master this lesson about converting repeating decimals to fractions, you will need to study the two examples above carefully and practice with other examples

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