Deepak and Shivam together can finish a job in $T$ days. The efficiency ratio of Deepak to Shivam is $5:6$. Deepak starts the work alone and works for 11 days. Then Shivam joins him, and 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 $5x$ units/day and Shivam be $6x$ units/day.

2. Together, their combined efficiency is:
   $$5x + 6x = 11x 	ext{ units/day}$$

3. Since they can complete the work together in $T$ days, total work =
   $$11x 	imes T = 11xT 	ext{ units}$$

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

5. Remaining work after 11 days:
   $$11xT - 55x = x(11T - 55) 	ext{ units}$$

6. Deepak and Shivam together complete the remaining work in 31 days:
   $$11x 	imes 31 = 341x 	ext{ units}$$

7. Equate remaining work to 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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