Find the value of :

Step-by-Step Solution

To solve this, we convert each recurring decimal into a fraction and then sum them up.

1. Convert each term to a fraction:

   • 8.31 (bar on 1) = 8 + 31/90 = 8 + (31 - 3)/90 = 8 + 28/90 = 8 + 14/45 = 374/45

   • 0.6 (bar on 6) = 6/9 = 2/3

   • 0.002 (bar on 2) = 2/900 = 1/450

2. Sum the fractions: Find a common denominator, which is 900.

   • 374/45 = 7480/900

   • 2/3 = 600/900

   • 1/450 = 2/900

   • Total = (7480 + 600 + 2) / 900 = 8082 / 900

3. Convert back to decimal:

   • 8082 / 900 = 8.98

   • Wait, let's re-evaluate 8.31 (bar on 1) as 8 + 31/90 = 8 + 28/90 = 8.3111...

   • 0.6 (bar on 6) = 0.6666...

   • 0.002 (bar on 2) = 0.0022...

   Summing: 8.3111... 0.6666... 0.0022...

   8.9799...

This results in 8.979 (bar on 9).