Convert the decimal $0.929292 \ldots$ to a fraction.
A. $\frac{92}{99}$
B. $\frac{92}{999}$
C. $\frac{92}{100}$
D. $\frac{92}{1000}$
Answer (2)
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 .
Explanation
Understanding the Problem We are asked to convert the repeating decimal 0.929292 … to a fraction. This type of decimal, where a sequence of digits repeats indefinitely, can be expressed as a fraction.
Setting up the Equation Let x = 0.929292 … . Since the digits '92' repeat, we multiply x by 100 to shift the repeating block to the left of the decimal point: 100 x = 92.929292 …
Eliminating the Repeating Decimal Now, we subtract x from 100 x to eliminate the repeating decimal part: 100 x − x = 92.929292 … − 0.929292 … This simplifies to: 99 x = 92
Solving for x and Stating the Answer Finally, we solve for x by dividing both sides of the equation by 99: x = 99 92 Thus, the repeating decimal 0.929292 … is equal to the fraction 99 92 .
Examples
Repeating decimals often appear when dealing with measurements or ratios that don't result in whole numbers. For instance, when dividing a length into equal parts, you might encounter a repeating decimal representing one of the parts. Converting this decimal to a fraction allows for more precise calculations and easier comparison with other fractions or ratios. This is particularly useful in fields like engineering, finance, and physics, where accuracy is paramount.
The repeating decimal 0.929292 … converts to the fraction 99 92 . This is derived by shifting the decimal to eliminate repetition and solving for the fraction. Therefore, the correct answer is 99 92 .
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