Shubham work for 5 days and remaining work was completed by Harvinder in 9 days. If Harvinder work for 12 days then remaining work was completed by Shubham in 3 days, then find how much time Harvinder will take to complete the work alone.

Options

A. A)11 days
B. B)`16(1)/(2)`days
C. C)`16(2)/(3)`days
D. D)`11(1)/(2)`days

Answer

To solve the problem step by step, we will denote Shubham's work efficiency as S and Harvinder's work efficiency as H.

### Step 1: Establish the equations based on the work done
1. Shubham works for 5 days, and then Harvinder completes the remaining work in 9 days.
- The equation for this scenario is:
\[ 5S + 9H = \text{Total Work} \quad (1) \]

2. If Harvinder works for 12 days, then Shubham completes the remaining work in 3 days.
- The equation for this scenario is:
\[ 12H + 3S = \text{Total Work} \quad (2) \]

### Step 2: Set the two equations equal to each other
Since both equations represent the total work, we can set them equal to each other:
\[ 5S + 9H = 12H + 3S \]

### Step 3: Rearranging the equation
Rearranging the equation gives:
\[ 5S - 3S = 12H - 9H \]
\[ 2S = 3H \]

### Step 4: Find the ratio of efficiencies
From the equation \(2S = 3H\), we can express the ratio of Shubham's efficiency to Harvinder's efficiency:
\[ \frac{S}{H} = \frac{3}{2} \]
This means if Shubham's efficiency is 3 units, then Harvinder's efficiency is 2 units.

### Step 5: Substitute efficiencies to find total work
Let’s assume Shubham's efficiency \(S = 3\) and Harvinder's efficiency \(H = 2\). Now we can substitute these values into either equation (1) or (2) to find the total work.

Using equation (1):
\[ 5S + 9H = 5(3) + 9(2) = 15 + 18 = 33 \]
So, the total work is 33 units.

### Step 6: Calculate the time taken by Harvinder to complete the work alone
To find out how long it would take Harvinder to complete the work alone, we use the formula:
\[ \text{Time} = \frac{\text{Total Work}}{\text{Efficiency of Harvinder}} = \frac{33}{2} = 16.5 \text{ days} \]

### Final Answer
Harvinder will take **16.5 days** to complete the work alone.

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