Convert the decimal $0.9 \overline{2}$ to a fraction.
A. $\frac{83}{100}$
B. $\frac{92}{99}$
C. $\frac{83}{90}$
D. $\frac{83}{9}$
Answer (2)
Let x = 0.9 2 .
Multiply by 10 and 100 to get 10 x = 9.2 2 and 100 x = 92. 2 .
Subtract the equations: 100 x − 10 x = 92. 2 − 9.2 2 , which simplifies to 90 x = 83 .
Solve for x : x = 90 83 . The final answer is 90 83 .
Explanation
Understanding the Problem We are asked to convert the decimal 0.9 2 to a fraction. This means we need to express the repeating decimal 0.922222... as a fraction in the form q p , where p and q are integers.
Setting up Equations Let x = 0.9 2 = 0.92222... . To eliminate the repeating part, we can multiply x by powers of 10. First, multiply x by 10: 10 x = 9.2222... Next, multiply x by 100: 100 x = 92.2222...
Eliminating the Repeating Decimal Now, subtract the first equation from the second equation to eliminate the repeating decimal part: 100 x − 10 x = 92.2222... − 9.2222... This simplifies to: 90 x = 83
Solving for x Finally, solve for x by dividing both sides by 90: x = 90 83 Therefore, the decimal 0.9 2 is equal to the fraction 90 83 .
Final Answer The fraction representation of the decimal 0.9 2 is 90 83 .
Examples
Converting repeating decimals to fractions is useful in various real-life scenarios, such as precise measurements in engineering or finance. For example, if a machine part needs to be manufactured with a precision of 0.9 2 cm, it's easier to work with the fractional equivalent 90 83 cm for calculations and measurements. This ensures accuracy and avoids rounding errors that can occur with decimals.
The decimal 0.9 2 can be converted to the fraction 90 83 by setting up an equation to eliminate the repeating part. After performing the necessary algebraic operations, we find that x = 90 83 . Therefore, the correct answer from the options provided is 90 83 .
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