How do we convert 0.23 (3 repeating) to a fraction?

To convert the repeating decimal 0.23 (with 3 repeating) into a fraction, follow these steps:

Step 1: Define the Decimal

Let x equal the repeating decimal:

x = 0.233333...

Step 2: Eliminate the Repeating Part

To isolate the repeating part, multiply x by 10:

10x = 2.33333...

Step 3: Set Up an Equation

Now, subtract the original x from this new equation:

10x - x = 2.33333... - 0.23333...

This simplifies to:

9x = 2.1

Step 4: Solve for x

Next, divide both sides by 9:

x = 2.1 / 9

Step 5: Convert 2.1 to a Fraction

Rewrite 2.1 as a fraction:

2.1 = 21/10

Now substitute this back into the equation:

x = (21/10) / 9

Step 6: Simplify the Fraction

This can be rewritten as:

x = 21 / 90

Now, simplify the fraction:

x = 7 / 30

Final Result

The repeating decimal 0.23 (3 repeating) can be expressed as the fraction:

7/30