Recurring Decimals
What is a Recurring Decimal?
A recurring decimal is a decimal number that repeats forever.
Example:
0.777... means 7 keeps repeating.
0.0777... means 7 keeps repeating, but it starts after a 0.
We show repeating digits with a dot or bar above them:
0.777... = 0.7̇
0.0777... = 0.07̇
Examples:
Example 1: Turn 0.7̇ into a fraction
Let’s call the number x: x = 0.777...
Step 1: Multiply x by 10 to move the decimal point:
10x = 7.777...
Step 2: Subtract the two equations:
10x = 7.777...
- x = 0.777...
----------------
9x = 7
Solve for x:
x = 7 ÷ 9
Answer; 0.7̇ = 7/9
Example 2: Turn 0.07̇ into a fraction
Let’s call the number x: x = 0.07̇
Step 1: Multiply by 10: 10x = 0.7̇
Multiply by 10 again: 100x = 7.7̇
Step 2: Subtract the two equations:
100x = 7.777...
-10x = 0.777...
----------------
90x = 7
x = 7 ÷ 90
Answer: 0.07̇ = 7⁄90
🎯 Top Tips:
⭐ Use x to name the decimal first.
⭐ Multiply by 10, 100, or 1000 — whatever helps line up the repeating parts.
⭐ Subtract the equations to get rid of the repeating decimals.
⭐ Always simplify the fraction if needed.
⭐ If the repeating digits start after a 0, you’ll probably need to multiply twice.