What is .83 repeating expressed as a fraction in simplest form???
The Correct Answer and Explanation is:
To express the repeating decimal ( 0.\overline{83} ) (where 83 repeats indefinitely) as a fraction in its simplest form, follow these steps:
Step 1: Set up an equation
Let ( x = 0.\overline{83} ). This means:
[
x = 0.8383838383\ldots
]
Step 2: Eliminate the repeating part
To eliminate the repeating part, we need to shift the decimal point two places to the right, so we multiply both sides of the equation by 100:
[
100x = 83.8383838383\ldots
]
Step 3: Subtract the original equation from the new equation
Now subtract the original equation (( x = 0.\overline{83} )) from the new equation (( 100x = 83.\overline{83} )) to eliminate the repeating part:
[
100x – x = 83.8383838383\ldots – 0.8383838383\ldots
]
This simplifies to:
[
99x = 83
]
Step 4: Solve for ( x )
Now solve for ( x ) by dividing both sides of the equation by 99:
[
x = \frac{83}{99}
]
Step 5: Simplify the fraction
The fraction ( \frac{83}{99} ) is already in its simplest form because 83 is a prime number and does not share any common factors with 99.
Final Answer:
Thus, the repeating decimal ( 0.\overline{83} ) as a fraction is:
[
\boxed{\frac{83}{99}}
]
Explanation:
This process relies on a standard technique for converting repeating decimals into fractions. By multiplying the equation by a power of 10 that matches the number of repeating digits, we align the decimals and subtract to eliminate the repeating part. The result is a simple linear equation that can be solved for the fraction form. In this case, because the repeating part is two digits long (83), we multiplied by 100. This method can be applied to other repeating decimals as well.