Renu would take 15 days working 4 hours per day to complete a certain task whereas Seema would take 8 days working 5 hours per day to complete the same task. They decide to work together to complete this task. Seema agrees to work for double the number of hours per day as Renu, while Renu agrees to work for double the number of days as Seema. If Renu works 2 hours per day, then the number of days Seema will work, is

The correct option is 6.

Let us break down the problem step-by-step to understand how to arrive at this answer.

Step 1: Calculate the total work in terms of person-hours
Let the total task be represented in unit hours of work.
For Renu:
Renu takes 15 days, working 4 hours per day.
Total time taken by Renu to complete the task alone = $15\times 4=60$ hours.
Therefore, Renu's rate of work is $\frac{1}{60}$ of the task per hour.

For Seema:
Seema takes 8 days, working 5 hours per day.
Total time taken by Seema to complete the task alone = $8\times 5=40$ hours.
Therefore, Seema's rate of work is $\frac{1}{40}$ of the task per hour.

Step 2: Define the working conditions for working together
Let the number of days Seema works be $D$.
According to the problem:
1. Renu agrees to work for double the number of days as Seema.
Thus, the number of days Renu works is $2D$.
2. Renu works 2 hours per day.
3. Seema agrees to work for double the number of hours per day as Renu.
Thus, Seema works $2\times 2=4$ hours per day.

Step 3: Calculate the total hours worked by each when working together
Total hours worked by Renu = (Number of days Renu works) × (Hours Renu works per day)
$\text{Total hours for Renu}=2D\times 2=4D$
Total hours worked by Seema = (Number of days Seema works) × (Hours Seema works per day)
$\text{Total hours for Seema}=D\times 4=4D$

Step 4: Set up the equation for the completed task
Since they complete the task together, the sum of the fraction of work completed by Renu and Seema must equal 1 (the complete task):
$(\text{Total hours for Renu}\times \text{Renu's rate})+(\text{Total hours for Seema}\times \text{Seema's rate})=1$

Substituting the values we have:
$4D\times \frac{1}{60}+4D\times \frac{1}{40}=1$

Simplify the fractions:
$\frac{4D}{60}+\frac{4D}{40}=1$
$\frac{D}{15}+\frac{D}{10}=1$

Find a common denominator, which is 30:
$\frac{2D+3D}{30}=1$
$\frac{5D}{30}=1$
$\frac{D}{6}=1$
$D=6$

Thus, the number of days Seema will work is indeed 6.