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:
Your two equations are:
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:
Your two equations are:
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