Study the pattern produced by division of numbers ending with 1 by 9.
$\frac{1}{9}=0.1111111111 \ldots$
$\frac{11}{9}=1.2222222222 \ldots$
$\frac{21}{9}=2.3333333333 \ldots$
$\frac{31}{9}=3.4444444444 \ldots$
$\frac{101}{9}=11.222222222 \ldots$
What is the answer to $\frac{100001}{9}$?
Answer (1)
Divide 100001 by 9.
Obtain the result: 11111.222222222223 .
Identify the repeating decimal pattern.
Express the final answer as 11111. 2 .
Explanation
Problem Analysis We are asked to find the decimal representation of 9 100001 based on the pattern observed in the given examples. The pattern suggests that the digits preceding the final '1' in the numerator form the non-repeating part of the decimal, while the final '1' contributes to the repeating decimal '.11111...'.
Direct Calculation To find the decimal representation of 9 100001 , we can perform the division directly. The result of the division is 11111.222222222223 .
Identifying the Repeating Decimal Based on the pattern and the calculation, we can express the answer as 11111. 2 , where the overline indicates that the digit 2 repeats indefinitely.
Final Answer Therefore, the answer to 9 100001 is 11111. 2 .
Examples
Understanding repeating decimals is useful in various real-life situations, such as when dealing with recurring billing cycles or calculating fractional parts of measurements. For instance, if a subscription service charges 3 1 of the monthly fee daily, expressing this fraction as a repeating decimal (0.333...) helps in accurately calculating the daily charge and forecasting revenue over time. Similarly, in engineering, repeating decimals can arise when converting units or calculating ratios, ensuring precision in designs and measurements.
