How do you convert 0.54 (54 repeating) to a fraction?

To convert the repeating decimal 0.54 (where 54 repeats indefinitely) into a fraction, follow these steps:

Step 1: Set Up the Equation

Let x equal the repeating decimal:

x = 0.545454...

Step 2: Eliminate the Repeating Part

To eliminate the repeating part, multiply both sides of the equation by 100 (since the repeat is two digits long):

100x = 54.545454...

Step 3: Subtract the Original Equation

Now, subtract the original equation from this new equation:

100x - x = 54.545454... - 0.545454...

This simplifies to:

99x = 54

Step 4: Solve for x

Now, divide both sides by 99:

x = 54/99

Step 5: Simplify the Fraction

Next, simplify the fraction. Both 54 and 99 can be divided by their greatest common divisor, which is 9:

• 54 ÷ 9 = 6
• 99 ÷ 9 = 11

Thus, the simplified fraction is:

x = 6/11

Final Result

The repeating decimal 0.54 (54 repeating) can be expressed as the fraction 6/11.