Convert the decimal $0.929292 \ldots$ to a fraction.
$\frac{92}{99}$
$\frac{92}{999}$
$\frac{92}{100}$
$\frac{92}{1000}$
Answer (1)
Let x = 0.929292 … .
Multiply by 100: 100 x = 92.929292 … .
Subtract x from 100 x : 99 x = 92 .
Solve for x : x = 99 92 .
99 92
Explanation
Understanding the Problem We are asked to convert the repeating decimal 0.929292 … to a fraction. This means we need to find a fraction that is equal to this repeating decimal. Let's denote the repeating decimal as x .
Multiplying by 100 Let x = 0.929292 … . Since the repeating block is '92', which has two digits, we multiply x by 100 to shift the decimal point two places to the right: 100 x = 92.929292 …
Subtracting x from 100x Now, we subtract x from 100 x to eliminate the repeating part: 100 x − x = 92.929292 … − 0.929292 … This simplifies to: 99 x = 92
Solving for x To solve for x , we divide both sides of the equation by 99: x = 99 92 Thus, the fraction equivalent to the repeating decimal 0.929292 … is 99 92 .
Final Answer Therefore, the decimal 0.929292 … is equal to the fraction 99 92 .
Examples
Converting repeating decimals to fractions is useful in various real-life scenarios, such as financial calculations or measurements where precision is important. For example, if you are calculating the exact repayment amount for a loan with a repeating decimal interest rate, converting it to a fraction ensures accuracy. Similarly, in engineering or construction, converting repeating decimal measurements to fractions can help in precise material calculations and avoid errors in the final product. This conversion provides a more manageable and accurate representation of the value.
