20 persons completed `(1)/(3)rd` of the work in 24 days. How many more person are required to finish the rest of the work in next 12 days?

Options

A. 20
B. 12
C. 18
D. 40

Answer

To solve the problem step by step, we will use the concept of work done by a certain number of persons in a given number of days.

### Step 1: Determine the total work
Given that 20 persons completed \( \frac{1}{3} \) of the work in 24 days, we can denote the total work as 1 unit. Therefore, the work done by 20 persons in 24 days is:
\[ W_1 = \frac{1}{3} \]

### Step 2: Calculate the total work done by one person in one day
The work done by 20 persons in 24 days can be expressed as:
\[ \text{Work done by 20 persons in 24 days} = 20 \times 24 = 480 \text{ person-days} \]
Since this is equal to \( \frac{1}{3} \) of the total work, we can find the total work in person-days:
\[ \text{Total work in person-days} = 480 \times 3 = 1440 \text{ person-days} \]

### Step 3: Determine the remaining work
The remaining work, which is \( W_2 \), is:
\[ W_2 = 1 - \frac{1}{3} = \frac{2}{3} \]

### Step 4: Set up the equation for the remaining work
Let \( x \) be the number of additional persons required. The total number of persons working to complete the remaining work will be \( 20 + x \). They need to complete \( \frac{2}{3} \) of the work in 12 days. The equation for work done can be set up as:
\[ (20 + x) \times 12 = \frac{2}{3} \times 1440 \]

### Step 5: Calculate the right-hand side
Calculating the right-hand side:
\[ \frac{2}{3} \times 1440 = 960 \text{ person-days} \]

### Step 6: Substitute and solve for \( x \)
Now substituting back into the equation:
\[ (20 + x) \times 12 = 960 \]
Dividing both sides by 12:
\[ 20 + x = \frac{960}{12} = 80 \]
Now, solving for \( x \):
\[ x = 80 - 20 = 60 \]

### Conclusion
Thus, the number of additional persons required to finish the remaining work in the next 12 days is:
\[ \boxed{60} \]

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