40 men could have finished the whole project in 28 days but due to the inclusion of a few more men, work got done in 3/4 of the time. Find the ratio of number of new men to number of old men.

Options

A. `2 : 9`
B. ` 2 : 7`
C. `1 :3`
D. None of these

Answer

To solve the problem step by step, we will follow these calculations:

### Step 1: Understand the given information
- 40 men can complete a project in 28 days.
- Due to the inclusion of more men, the project was completed in \( \frac{3}{4} \) of the time.

### Step 2: Calculate the new time taken
The new time taken to complete the project is:
\[ d_2 = \frac{3}{4} \times 28 = 21 \text{ days} \]

### Step 3: Set up the equation using the formula
We will use the formula for work, which states:
\[ \frac{m_1 \cdot d_1}{w_1} = \frac{m_2 \cdot d_2}{w_2} \]
Where:
- \( m_1 = 40 \) (initial number of men)
- \( d_1 = 28 \) (initial days)
- \( m_2 = 40 + x \) (new number of men, where \( x \) is the number of new men)
- \( d_2 = 21 \) (new days)

Since the work \( w \) is the same in both cases, we can cancel it out:
\[ 40 \cdot 28 = (40 + x) \cdot 21 \]

### Step 4: Simplify the equation
Now we simplify:
\[ 1120 = (40 + x) \cdot 21 \]
Expanding the right side:
\[ 1120 = 840 + 21x \]

### Step 5: Solve for \( x \)
Rearranging the equation to isolate \( x \):
\[ 1120 - 840 = 21x \]
\[ 280 = 21x \]
\[ x = \frac{280}{21} = \frac{40}{3} \]

### Step 6: Find the ratio of new men to old men
Now we know the number of new men \( x = \frac{40}{3} \) and the number of old men is 40. The ratio of new men to old men is:
\[ \text{Ratio} = \frac{x}{40} = \frac{\frac{40}{3}}{40} = \frac{1}{3} \]
This can also be expressed as \( 1:3 \).

### Final Answer
The ratio of the number of new men to the number of old men is \( 1:3 \).

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