0.81 repeating as a fraction

0.81 repeating as a fraction

The Correct Answer and Explanation is:

To convert the repeating decimal 0.81 repeating (written as ( 0.\overline{81} )) into a fraction, follow these steps:

Step 1: Represent the repeating decimal as a variable

Let ( x ) represent the repeating decimal:

[
x = 0.\overline{81}
]

This means that the digits “81” repeat infinitely.

Step 2: Eliminate the repeating part by multiplying

To remove the repeating decimal, multiply both sides of the equation by 100 (because the repeating part starts after two decimal places, which corresponds to multiplying by ( 10^2 ) or 100):

[
100x = 81.\overline{81}
]

Now, you have two equations:

1. ( x = 0.\overline{81} )
2. ( 100x = 81.\overline{81} )

Step 3: Subtract the original equation from the new equation

Subtract the first equation from the second:

[
100x – x = 81.\overline{81} – 0.\overline{81}
]

This simplifies to:

[
99x = 81
]

Step 4: Solve for ( x )

To find ( x ), divide both sides of the equation by 99:

[
x = \frac{81}{99}
]

Step 5: Simplify the fraction

Next, simplify ( \frac{81}{99} ) by dividing both the numerator and the denominator by their greatest common divisor (GCD), which is 9:

[
\frac{81}{99} = \frac{81 \div 9}{99 \div 9} = \frac{9}{11}
]

Final Answer:

Thus, the repeating decimal ( 0.\overline{81} ) as a fraction is ( \frac{9}{11} ).

Explanation:

The process of converting a repeating decimal into a fraction involves using algebraic methods to eliminate the repeating part. By multiplying the decimal by a power of 10 that moves the decimal point to a position where the repeating part aligns with the original decimal, we can set up an equation and solve for the unknown. This technique of subtracting the two equations allows us to express the repeating decimal as a fraction. Simplifying the resulting fraction ensures that the answer is in its simplest form.