Final answer:
Vinay can paint the house alone in 10 days, which is determined by setting up an algebraic equation to solve for his work rate and factoring the resulting quadratic equation.
Step-by-step explanation:
If Vinay and Praveen together can paint a house in 6 days, and Vinay can finish the job 5 days faster than Praveen, we can find out how long it will take Vinay to paint the house alone using algebra. Let's represent the amount of work Vinay and Praveen can do in one day as 1/V and 1/P respectively. If Vinay finishes 5 days faster than Praveen, then it takes Praveen V + 5 days to do the job alone.
Since they can finish the job together in 6 days, their combined work rate is 1/6 of the house per day. Therefore, 1/V + 1/(V + 5) = 1/6. To find the value of V, we multiply through by 6V(V + 5) to clear the denominators: 6(V + 5) + 6V = V(V + 5). Simplify this to get 6V + 30 + 6V = V^2 + 5V. This simplifies further to V^2 - 7V - 30 = 0.
Factoring the quadratic equation, we get (V - 10)(V + 3) = 0. Ignoring the negative root because time cannot be negative, we find that V = 10. Therefore, it takes Vinay 10 days to paint the house alone. The correct answer is option b) 10 days.