A can do a work in 36 days while B can do the same in 48 days. If A works for $ x $ days while B works for $ x+2 $ days, then one-third of the work is complete. Find the value of $ x $.

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**Solution:**

1. Calculate the efficiency of A and B:
   - A can complete the work in 36 days, so efficiency of A = $ \frac{1}{36} $ units/day.
   - B can complete the work in 48 days, so efficiency of B = $ \frac{1}{48} $ units/day.

2. To work with whole numbers, consider the total work as the LCM of 36 and 48, which is 144 units.
   - Efficiency of A = $ \frac{144}{36} = 4 $ units/day.
   - Efficiency of B = $ \frac{144}{48} = 3 $ units/day.

3. One-third of the work is completed, so work done = $ \frac{1}{3} \times 144 = 48 $ units.

4. Let A work for $ x $ days and B work for $ x + 2 $ days.

5. Total work done by A and B:
   $$
   4x + 3(x + 2) = 48
   $$

6. Simplify the equation:
   $$
   4x + 3x + 6 = 48 \ 
   7x + 6 = 48 \ 
   7x = 42 \ 
   x = 6
   $$

**Answer:** $6$

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