A and B together can complete a work in $2T$ days, while A alone can finish the same work in $10$ days. If B alone can complete the work in $5T$ days, find the value of $T$.

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**Solution:**

1. Given:
   - A and B together complete the work in $2T$ days.
   - A alone completes the work in $10$ days.
   - B alone completes the work in $5T$ days.

2. Assume total work = 1 unit.

3. Calculate daily work rates:
   - A's one day work = $\frac{1}{10}$
   - B's one day work = $\frac{1}{5T}$
   - (A + B)'s one day work = $\frac{1}{2T}$

4. Since A and B working together complete the work in $2T$ days, their combined daily work is:
   $$\frac{1}{10} + \frac{1}{5T} = \frac{1}{2T}$$

5. Find a common denominator on the left side:
   $$\frac{1}{10} + \frac{1}{5T} = \frac{T}{10T} + \frac{2}{10T} = \frac{T + 2}{10T}$$

6. Equate and solve:
   $$\frac{T + 2}{10T} = \frac{1}{2T}$$

7. Multiply both sides by $10T$:
   $$T + 2 = 5$$

8. Solve for $T$:
   $$T = 5 - 2 = 3$$

**Answer:** $3$

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