Converting Recurring Decimals to Fractions

The Big Idea: Recurring decimals like \(0.777...\) are hard to work with because they never end. Our goal is to use algebra to subtract the infinite part away, leaving us with a simple whole number.

The 4-Step Universal Method

Step 1: Assign 'x'

Write down your decimal and call it \(x\). Write out at least 4 or 5 of the repeating digits so you can see the pattern clearly.

\(x = 0.77777...\)

Step 2: Multiply to Shift the Pattern

Multiply \(x\) by 10, 100, or 1000 to "shift" the decimal point until the digits after the decimal point match your original equation.

If 1 digit repeats: Multiply by 10
If 2 digits repeat: Multiply by 100
If 3 digits repeat: Multiply by 1000

\(10x = 7.77777...\)

Step 3: Subtract the Equations

Subtract the smaller equation from the larger one. Because the decimals are identical, they cancel each other out!

10x = 7.77777...\) \(-\quad x = 0.77777...\) \(9x = 7.00000...

Step 4: Solve for x

Now just divide to find the fraction.

x = \frac{7}{9}

A More Difficult Example: \(0.2\dot{5}\)

In this case, only the 5 repeats (it looks like \(0.25555...\)). We need to be careful to make the decimals match.

1. Let \(x = 0.25555...\)
2. Multiply by 10: \(10x = 2.5555...\) (This doesn't match \(x\) yet).
3. Multiply by 100: \(100x = 25.5555...\)

Look! The decimals for \(100x\) and \(10x\) match perfectly now! Both have infinite 5s after the point.

100x = 25.555...\) \(- 10x = 2.555...\) \(90x = 23

Final Step: \(x = \frac{23}{90}\)

Summary Table

Decimal	Multiply by...	Subtraction Result	Fraction
\(0.\dot{4}\)	\(10x\) and \(x\)	\(9x = 4\)	\(\frac{4}{9}\)
\(0.\dot{1}\dot{2}\)	\(100x\) and \(x\)	\(99x = 12\)	\(\frac{12}{99} = \frac{4}{33}\)