Rohan and Priya together can finish a project in 'T' days, and their individual work rates are in the ratio 5:6 respectively. Rohan began working on the project alone, and after 11 days Priya joined him; together they finished the remaining portion of the project in 31 days. What is the value of 'T'?

The correct option is 36.

Step 1: Understand the ratio of work rates
Let Rohan's daily work rate be $5x$ units per day and Priya's daily work rate be $6x$ units per day.
When Rohan and Priya work together, their combined daily work rate is:

$\text{Combined Rate}=5x+6x=11x\text{units/day}$

Step 2: Express total work in terms of T
Since Rohan and Priya together can finish the entire project in $T$ days, the total work $W$ required for the project is given by:

$W=11x\times T$

Step 3: Calculate the work completed in stages
First, Rohan worked alone for 11 days:

$\text{Work done by Rohan in 11 days}=11\times 5x=55x\text{units}$

Next, Priya joined Rohan, and together they worked for 31 days to complete the remaining portion of the project:

$\text{Work done together in 31 days}=31\times 11x=341x\text{units}$

Step 4: Find the total work done
Adding the work done in both phases gives the total work $W$:

$W=55x+341x=396x\text{units}$

Step 5: Solve for T
Equating the two expressions for the total work $W$:

$11x\times T=396x$

Dividing both sides by $11x$:

$T=\frac{396x}{11x}=36$

Therefore, the value of 'T' is 36.