9 men finish one-third work in 10 days. The number of additional men required for finishing the remaining work in 2 more days will be

Options

A. 78
B. 81
C. 55
D. 30

Answer

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

### Step 1: Determine the total work done by 9 men in 10 days.
Given that 9 men finish one-third of the work in 10 days, we can denote the total work as W.

- Work done in 10 days = \( \frac{1}{3}W \)

### Step 2: Calculate the total work (W).
If 9 men complete \( \frac{1}{3}W \) in 10 days, we can find the total work W as follows:

- Total work \( W = 3 \times \frac{1}{3}W = W \)

### Step 3: Find the work done by one man in one day.
To find out how much work one man can do in one day, we can use the following formula:

- Work done by 9 men in 10 days = \( \frac{1}{3}W \)
- Work done by 1 man in 10 days = \( \frac{1}{3}W \div 9 = \frac{1}{27}W \)
- Work done by 1 man in 1 day = \( \frac{1}{27}W \div 10 = \frac{1}{270}W \)

### Step 4: Calculate the remaining work.
Since one-third of the work is completed, the remaining work is:

- Remaining work = \( W - \frac{1}{3}W = \frac{2}{3}W \)

### Step 5: Determine how many men are needed to finish the remaining work in 2 days.
Let M be the number of men required to finish the remaining work in 2 days. The total work done by M men in 2 days can be expressed as:

- Work done by M men in 2 days = \( M \times \text{(work done by 1 man in 1 day)} \times 2 \)
- Work done by M men in 2 days = \( M \times \frac{1}{270}W \times 2 = \frac{2M}{270}W = \frac{M}{135}W \)

### Step 6: Set up the equation for the remaining work.
To finish the remaining work \( \frac{2}{3}W \) in 2 days, we set up the equation:

\[ \frac{M}{135}W = \frac{2}{3}W \]

### Step 7: Solve for M.
We can cancel W from both sides (assuming W ≠ 0):

\[ \frac{M}{135} = \frac{2}{3} \]

Now, cross-multiply to solve for M:

\[ M = 135 \times \frac{2}{3} = 90 \]

### Step 8: Calculate the additional men required.
We already have 9 men working, so the additional men required will be:

\[ \text{Additional men} = M - 9 = 90 - 9 = 81 \]

### Final Answer:
The number of additional men required to finish the remaining work in 2 more days is **81**.

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