Manish is four times as efficient as Puneet. Manish takes 90 days less than Puneet to complete a book. If both of them work together, then in how many days the book will be completed?

Options

A. 28
B. 25
C. 24
D. 30

Answer

To solve the problem step by step, we can follow these steps:

### Step 1: Define the efficiencies
Let Puneet's efficiency be \( E \). Since Manish is four times as efficient as Puneet, Manish's efficiency will be \( 4E \).

### Step 2: Define the time taken to complete the book
Let Puneet take \( D \) days to complete the book. Therefore, Manish will take \( D - 90 \) days to complete the same book.

### Step 3: Relate efficiency to time
The relationship between efficiency and time is given by:
\[ \text{Efficiency} = \frac{\text{Total Work}}{\text{Time Taken}} \]
Since both are completing the same work (let's assume the total work is 1 book), we can express their efficiencies in terms of days:
- For Puneet:
\[ E = \frac{1}{D} \]
- For Manish:
\[ 4E = \frac{1}{D - 90} \]

### Step 4: Set up the equation
From the efficiencies, we can set up the equation:
\[ 4 \left(\frac{1}{D}\right) = \frac{1}{D - 90} \]
Multiplying both sides by \( D(D - 90) \) to eliminate the denominators gives:
\[ 4(D - 90) = D \]

### Step 5: Solve for \( D \)
Expanding the equation:
\[ 4D - 360 = D \]
Rearranging gives:
\[ 4D - D = 360 \]
\[ 3D = 360 \]
\[ D = 120 \]

### Step 6: Calculate Manish's time
Now that we have \( D \), we can find the time taken by Manish:
\[ \text{Time taken by Manish} = D - 90 = 120 - 90 = 30 \text{ days} \]

### Step 7: Calculate the combined work rate
Now, we can find their combined work rate.
- Manish's work rate is \( \frac{1}{30} \) (books per day).
- Puneet's work rate is \( \frac{1}{120} \) (books per day).

The combined work rate when both work together is:
\[ \text{Combined Work Rate} = \frac{1}{30} + \frac{1}{120} \]
To add these fractions, we need a common denominator, which is 120:
\[ \frac{1}{30} = \frac{4}{120} \]
Thus,
\[ \text{Combined Work Rate} = \frac{4}{120} + \frac{1}{120} = \frac{5}{120} = \frac{1}{24} \]

### Step 8: Calculate the time taken when working together
If their combined work rate is \( \frac{1}{24} \), it means together they can complete the book in:
\[ \text{Time} = \frac{1 \text{ book}}{\frac{1}{24} \text{ books per day}} = 24 \text{ days} \]

### Final Answer
Thus, the book will be completed in **24 days** when both Manish and Puneet work together.

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