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.

To find the value of T, we can break down the problem by determining the daily work rates of Rohan and Priya, calculating the work done in each phase of the project, and setting up an equation for the total work completed.

Step 1: Express the daily work rates.

The ratio of Rohan's individual work rate to Priya's individual work rate is given as 5:6.

Let Rohan's daily work rate = $5x$ units per day.

Let Priya's daily work rate = $6x$ units per day.

When Rohan and Priya work together, their combined daily work rate is:

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

Step 2: Express the total work in terms of T.

We are given that Rohan and Priya together can finish the entire project in T days.
Therefore, the total work required for the project is:

$\text{Total Work}=\text{Combined Rate}\times T$

$\text{Total Work}=11x\times T=11xT$ ... (Equation 1)

Step 3: Calculate the work completed in each phase.

The project is completed in two distinct phases:

1. Phase 1: Rohan works alone for the first 11 days.
Work completed in Phase 1 = Rohan's rate × 11 days

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

2. Phase 2: Priya joins Rohan, and together they finish the remaining portion in 31 days.
Work completed in Phase 2 = Combined rate × 31 days

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

Adding the work done in both phases gives the total work of the project:

$\text{Total Work}=55x+341x=396x$ units ... (Equation 2)

Step 4: Equate the expressions to solve for T.

Since Equation 1 and Equation 2 both represent the total work required to complete the project, we can set them equal to each other:

$11xT=396x$

Divide both sides of the equation by $11x$ to isolate T:

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

$T=36$

Thus, the value of T is 36 days.