Convert the decimal 0.929292 to a fraction.
Understand the Problem
The question is asking us to convert the repeating decimal 0.929292... into a fraction. This typically involves identifying the repeating part and using algebraic techniques to express it as a fraction.
Answer
The repeating decimal $0.929292\ldots$ is expressed as the fraction $\frac{92}{99}$.
Answer for screen readers
The fraction representation of the repeating decimal $0.929292\ldots$ is $\frac{92}{99}$.
Steps to Solve
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Set up the equation with a variable
Let $x$ represent the repeating decimal:
$$ x = 0.929292\ldots $$ -
Multiply to shift the decimal
Since the repeating part (92) has a length of 2, multiply both sides of the equation by 100 to move the decimal point two places right:
$$ 100x = 92.929292\ldots $$ -
Subtract the original equation from this new equation
Now, subtract the original equation ($x = 0.929292\ldots$) from this new equation:
$$ 100x - x = 92.929292\ldots - 0.929292\ldots $$
This simplifies to:
$$ 99x = 92 $$ -
Solve for x
Now, divide both sides by 99 to solve for $x$:
$$ x = \frac{92}{99} $$ -
Simplify the fraction if possible
Check if the fraction $\frac{92}{99}$ can be simplified. The greatest common divisor (GCD) of 92 and 99 is 1, so it is already in its simplest form.
The fraction representation of the repeating decimal $0.929292\ldots$ is $\frac{92}{99}$.
More Information
The fraction $\frac{92}{99}$ can be evaluated to confirm it equals the repeating decimal. In addition, repeating decimals can often be expressed as fractions by using similar algebraic methods.
Tips
- Forgetting to subtract the original equation from the multiplied equation, which can lead to incorrect simplifications.
- Not simplifying the fraction at the end, leaving the answer in a form that could be reduced.
