Express the decimal 0.82(68) as a vulgar fraction Interpret 0.82(68) to mean 0.82686868… with 68 repeating. Convert this recurring decimal to a simplified fraction.

Introduction / Context:
The task is to convert a mixed recurring decimal, where some digits do not repeat and a block of digits repeats, into a fraction. The notation 0.82(68) means the digits “68” repeat indefinitely after the initial “82”.

Given Data / Assumptions:

• Number: x = 0.82686868…
• Non-repeating part: 82 (two digits).
• Repeating block: 68 (two digits).
• Goal: reduce to lowest terms.

Concept / Approach:
For a decimal with k non-repeating digits and r repeating digits, use place-value shifts by 10^k and 10^(k+r). Subtract the two to eliminate the repeating tail. The closed-form shortcut is: x = (integer formed by “nonrepeat+repeat” − integer formed by “nonrepeat”) / (99…00… with r nines and k zeros). Here k = 2 and r = 2, so denominator is 9900.

Step-by-Step Solution:
Write x = 0.82686868…Take the 4-digit chunk “8268” (nonrepeat “82” followed by repeat “68”).Compute numerator: 8268 − 82 = 8186.Denominator: 99 for the repeating block and 00 for the two non-repeating digits ⇒ 9900.Thus x = 8186/9900 = 4093/4950 after simplification.

Verification / Alternative check:
Divide 4093 by 4950 to obtain approximately 0.82686868…, confirming the repeating pattern 68.

Why Other Options Are Wrong:
3093/4950, 3043/4850, and 3039/4950 yield different decimals; “8268/9990” uses the wrong denominator (should be 9900 for 2+2 digits), giving an incorrect value.

Common Pitfalls:
Using 99 for any repeating case without appending zeros for the non-repeating digits; forgetting to subtract the non-repeating integer before forming the fraction.

Final Answer:
4093/4950