Write 8/11 as a recurring decimal

Write 8/11 as a recurring decimal

The Correct Answer and Explanation is :

To express ( \frac{8}{11} ) as a recurring decimal, divide 8 by 11. The answer is:

[
\frac{8}{11} = 0.\overline{72}
]

where the “72” repeats indefinitely.

Explanation

1. Understanding the Division:
   To find the decimal form of ( \frac{8}{11} ), we perform long division. Dividing 8 by 11, we see that 11 doesn’t go into 8, so we place a decimal point and add a zero, making it 80. Now, 11 goes into 80 seven times (since (11 \times 7 = 77)). Subtracting 77 from 80 leaves a remainder of 3.
2. Continuing the Process:
   Bringing down another zero makes it 30. Dividing 30 by 11 gives us 2 (since (11 \times 2 = 22)). Subtracting 22 from 30 leaves a remainder of 8, bringing us back to the beginning. Now, 80 divided by 11 gives 7, repeating the process.
3. Establishing the Repeating Pattern:
   This cycle — where 7 and 2 keep appearing — will continue indefinitely. As such, the decimal representation of ( \frac{8}{11} ) is ( 0.\overline{72} ), with “72” as the repeating block.
4. Why It’s a Recurring Decimal:
   Since 11 is a prime number and does not divide evenly into 100, the division produces a repeating decimal rather than terminating. For a fraction to have a terminating decimal, its denominator (in simplest form) must contain only the prime factors 2 or 5. Because 11 doesn’t meet this condition, ( \frac{8}{11} ) results in a recurring decimal.
5. Conclusion:
   The result ( 0.\overline{72} ) accurately represents ( \frac{8}{11} ) in decimal form.