What is .54 repeating as a fraction

What is .54 repeating as a fraction

The Correct Answer and Explanation is :

To express ( 0.54\overline{54} ) (where the “54” repeats indefinitely) as a fraction, follow these steps:

Step 1: Represent the repeating decimal as a variable

Let ( x = 0.54\overline{54} ), meaning ( x = 0.5454545454\ldots ).

Step 2: Eliminate the repeating part

Since the decimal repeats every two digits, multiply both sides of the equation by 100 to shift the decimal point two places to the right:

[
100x = 54.545454\ldots
]

Step 3: Subtract the original equation from the new equation

Now subtract the original equation ( x = 0.545454\ldots ) from this new equation:

[
100x – x = 54.545454\ldots – 0.545454\ldots
]

Simplifying both sides:

[
99x = 54
]

Step 4: Solve for ( x )

Now solve for ( x ) by dividing both sides of the equation by 99:

[
x = \frac{54}{99}
]

Step 5: Simplify the fraction

The fraction ( \frac{54}{99} ) can be simplified by dividing both the numerator and denominator by their greatest common divisor (GCD), which is 9:

[
\frac{54 \div 9}{99 \div 9} = \frac{6}{11}
]

Thus, ( 0.54\overline{54} ) as a fraction is ( \frac{6}{11} ).

Explanation:

This method works because we set up the problem by using algebra to eliminate the repeating decimal. By multiplying the original equation by a power of 10 (in this case, 100) that corresponds to the number of repeating digits (2 digits repeating), we were able to subtract the two equations to isolate the repeating part. The result is a simple equation that can be solved and simplified into a fraction.

Therefore, ( 0.54\overline{54} ) is equivalent to ( \frac{6}{11} ).