Rahul and Rajeev can complete a work in 17 days while Rahul, Rajeev & Raj together can complete the same work in 12 days. If Rahul is 60% more efficient than Raj, then find the time taken by Rajeev to complete the whole work.

Correct Answer: The correct option is 51 days.

Step-by-Step Explanation:

Step 1: Understand the given data

Let the efficiency (work done per day) of Rahul, Rajeev, and Raj be represented by $\mathrm{Rahul}$, $\mathrm{Rajeev}$, and $\mathrm{Raj}$ respectively.

We are given:

1. Time taken by (Rahul + Rajeev) to complete the work = 17 days.

2. Time taken by (Rahul + Rajeev + Raj) to complete the work = 12 days.

Step 2: Find total work and individual combined efficiencies

Let the total unit of work be the Least Common Multiple (LCM) of 17 and 12.

$\text{Total Work}=17\times 12=204\text{units}$

Efficiency of (Rahul + Rajeev):

$\text{Efficiency of (Rahul + Rajeev)}=\frac{204}{17}=12\text{units/day}$

Efficiency of (Rahul + Rajeev + Raj):

$\text{Efficiency of (Rahul + Rajeev + Raj)}=\frac{204}{12}=17\text{units/day}$

Step 3: Calculate Raj's efficiency

Subtracting the combined efficiency of (Rahul + Rajeev) from (Rahul + Rajeev + Raj):

$\text{Efficiency of Raj}=17-12=5\text{units/day}$

Step 4: Calculate Rahul's efficiency using the efficiency ratio

We are given that Rahul is 60% more efficient than Raj.

$\text{Efficiency of Rahul}=\text{Efficiency of Raj}\times (1+\frac{60}{100})$

$\text{Efficiency of Rahul}=5\times 1.6=8\text{units/day}$

Step 5: Calculate Rajeev's efficiency

We know that the combined efficiency of Rahul and Rajeev is 12 units/day:

$\text{Efficiency of Rahul}+\text{Efficiency of Rajeev}=12$

$8+\text{Efficiency of Rajeev}=12$

$\text{Efficiency of Rajeev}=12-8=4\text{units/day}$

Step 6: Find the time taken by Rajeev to complete the whole work alone

$\text{Time taken by Rajeev}=\frac{\text{Total Work}}{\text{Efficiency of Rajeev}}$

$\text{Time taken by Rajeev}=\frac{204}{4}=51\text{days}$

Hence, Rajeev alone can complete the work in 51 days.