Manoj is 25% less efficient than Hemant. Vikash and Hemant working together can complete a task in 16 days and Vikash is half efficient as Hemant, then find in how many days Manoj alone can complete the task with 150% of his original efficiency?

Options

A. `21(1)/(3)`days
B. 21 days
C. `22(2)/(3)`days
D. 24 days

Answer

To solve the problem step by step, we will follow the information given in the question and derive the required values systematically.

### Step 1: Determine the efficiency of Manoj and Hemant
Manoj is 25% less efficient than Hemant. If we assume Hemant's efficiency is 4 units, then Manoj's efficiency can be calculated as follows:

\[ \text{Efficiency of Manoj} = 4 - (25\% \text{ of } 4) = 4 - 1 = 3 \text{ units} \]

**Hint:** Remember that 25% of a number is calculated by multiplying the number by 0.25.

### Step 2: Determine Vikash's efficiency
Vikash is half as efficient as Hemant. Therefore, if Hemant's efficiency is 4 units, Vikash's efficiency is:

\[ \text{Efficiency of Vikash} = \frac{4}{2} = 2 \text{ units} \]

**Hint:** To find half of a number, simply divide it by 2.

### Step 3: Calculate the total efficiency of Vikash and Hemant
When Vikash and Hemant work together, their combined efficiency is:

\[ \text{Total efficiency (Vikash + Hemant)} = 2 + 4 = 6 \text{ units} \]

**Hint:** When combining efficiencies, simply add the individual efficiencies together.

### Step 4: Calculate the total work done
We know that Vikash and Hemant can complete the task in 16 days. The total work can be calculated using the formula:

\[ \text{Total Work} = \text{Total Time} \times \text{Total Efficiency} \]

Substituting the values:

\[ \text{Total Work} = 16 \text{ days} \times 6 \text{ units} = 96 \text{ units} \]

**Hint:** The total work is the product of the time taken and the efficiency of the workers.

### Step 5: Calculate Manoj's new efficiency
Manoj will work at 150% of his original efficiency. His original efficiency is 3 units, so:

\[ \text{New Efficiency of Manoj} = 150\% \text{ of } 3 = \frac{150}{100} \times 3 = 1.5 \times 3 = 4.5 \text{ units} = \frac{9}{2} \text{ units} \]

**Hint:** To find a percentage of a number, convert the percentage to a fraction and multiply.

### Step 6: Calculate the time taken by Manoj to complete the work
Now we can find out how many days Manoj will take to complete the total work of 96 units with his new efficiency of \( \frac{9}{2} \) units:

\[ \text{Time taken by Manoj} = \frac{\text{Total Work}}{\text{New Efficiency}} = \frac{96}{\frac{9}{2}} = 96 \times \frac{2}{9} = \frac{192}{9} = 21 \frac{1}{3} \text{ days} \]

**Hint:** When dividing by a fraction, multiply by its reciprocal.

### Final Answer
Manoj alone can complete the task in \( 21 \frac{1}{3} \) days.