Deepak and Shivam together can finish a task in $T$ days. The ratio of their efficiencies is $5:6$. Deepak starts the work alone and after $11$ days, Shivam joins him. Together, they complete the remaining work in $31$ days. Find the value of $T$.

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**Solution:**

1. Let the efficiency of Deepak be $5x$ and that of Shivam be $6x$.

2. Together, their combined efficiency is:
   $$5x + 6x = 11x$$

3. Since they can complete the work together in $T$ days, the total work can be expressed as:
   $$	ext{Total Work} = 11x 	imes T$$

4. Deepak works alone for $11$ days, so work done by Deepak in $11$ days is:
   $$5x 	imes 11 = 55x$$

5. Remaining work after $11$ days is:
   $$11xT - 55x = x(11T - 55)$$

6. Deepak and Shivam together complete the remaining work in $31$ days, so:
   $$11x 	imes 31 = 341x$$

7. Equate the remaining work to the work done together in $31$ days:
   $$x(11T - 55) = 341x$$

8. Cancel $x$ from both sides:
   $$11T - 55 = 341$$

9. Solve for $T$:
   $$11T = 341 + 55 = 396$$
   $$T = \frac{396}{11} = 36$$

**Answer:** $36$

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