A contractor has the target of completing a work in 40 days . He employed 20 persons who completed (1/4) of the work in 10 days and left. The number of persons he has to employ to finish the remaining part as per target is :

Options

A. 20
B. 40
C. 30
D. 10

Answer

To solve the problem step by step, we will use the concept of work done, which can be expressed in terms of the number of men, the number of days they work, and the fraction of work completed.

### Step 1: Determine the total work done by the initial group of workers.
The contractor employed 20 persons who completed \( \frac{1}{4} \) of the work in 10 days.

**Work done (W1) = \( \frac{1}{4} \)**

### Step 2: Calculate the total work in terms of man-days.
Since 20 persons worked for 10 days, the total man-days used is:
\[ \text{Man-days} = \text{Number of persons} \times \text{Number of days} = 20 \times 10 = 200 \text{ man-days} \]

### Step 3: Calculate the total work.
If \( \frac{1}{4} \) of the work corresponds to 200 man-days, then the total work (W) can be calculated as follows:
\[ W = \text{Total work} = 200 \text{ man-days} \times 4 = 800 \text{ man-days} \]

### Step 4: Determine the remaining work.
Since \( \frac{1}{4} \) of the work is completed, the remaining work is:
\[ \text{Remaining work} = W - \frac{1}{4}W = \frac{3}{4}W = \frac{3}{4} \times 800 = 600 \text{ man-days} \]

### Step 5: Calculate the number of days left to complete the work.
The total target time is 40 days, and 10 days have already been used, so the remaining time (D2) is:
\[ D2 = 40 - 10 = 30 \text{ days} \]

### Step 6: Use the work formula to find the number of workers needed.
Using the relationship:
\[ M1 \times D1 = M2 \times D2 \]
Where:
- \( M1 = 20 \) (initial workers)
- \( D1 = 10 \) (days worked)
- \( W1 = \frac{1}{4}W \) (work done)
- \( M2 \) is the number of workers needed to finish the remaining work in 30 days.

We can express this as:
\[ 20 \times 10 = M2 \times 30 \]

### Step 7: Solve for \( M2 \).
\[ 200 = M2 \times 30 \]
\[ M2 = \frac{200}{30} = \frac{20}{3} \approx 6.67 \]

Since we cannot have a fraction of a worker, we round up to the nearest whole number:
\[ M2 = 7 \]

### Final Answer:
The contractor needs to employ **7 persons** to finish the remaining work in the allotted time.