A work is done by 30 workers not all of them have the same capacity to work. Every day exactly 2 workers, do the work with no pair of workers working together twice. Even after all possible pairs have worked once, all the workers together works for two more days to finish the work. Find the number of days in which all the workers together will finish the whole work?

Options

A. 31
B. 29
C. 41
D. 48

Answer

To solve the problem step by step, we will break down the information given and derive the solution systematically.

### Step 1: Understanding the Problem
We have 30 workers, and they work in pairs. Each day, exactly 2 workers work together, and no pair of workers works together more than once. After all possible pairs have worked, the workers work together for 2 more days to finish the work.

### Step 2: Calculate the Number of Unique Pairs
The total number of ways to choose 2 workers from 30 is given by the combination formula:
\[ \text{Number of pairs} = \binom{30}{2} = \frac{30 \times 29}{2} = 435 \]
This means there are 435 unique pairs of workers.

### Step 3: Determine the Total Days of Pair Work
Since each day only one pair works, the number of days they can work with all unique pairs is 435 days.

### Step 4: Total Work Done by Pairs
The total work done by all pairs in 435 days is equivalent to the work done in one day by all workers working together. If we denote the total work done by the pairs in one day as \( W \), then:
\[ \text{Total work done by pairs} = 435 \times W \]

### Step 5: Work Done in the Last Two Days
After all pairs have worked, the workers work together for 2 more days. Let’s denote the work done by all workers in one day as \( W_{total} \). Therefore, the work done in the last two days is:
\[ \text{Work done in last 2 days} = 2 \times W_{total} \]

### Step 6: Total Work Calculation
The total work done is the sum of the work done by pairs and the work done in the last two days:
\[ \text{Total Work} = 435 \times W + 2 \times W_{total} \]

### Step 7: Equating Total Work
Since the total work can also be expressed as the work done by all 30 workers working together for \( D \) days:
\[ \text{Total Work} = D \times W_{total} \]

### Step 8: Setting Up the Equation
Equating the two expressions for total work, we have:
\[ D \times W_{total} = 435 \times W + 2 \times W_{total} \]

### Step 9: Solve for D
Rearranging the equation gives:
\[ D \times W_{total} - 2 \times W_{total} = 435 \times W \]
Factoring out \( W_{total} \):
\[ W_{total}(D - 2) = 435 \times W \]
Thus, we can express \( D \) as:
\[ D = \frac{435 \times W}{W_{total}} + 2 \]

### Step 10: Finding the Value of D
Given that the efficiency of the workers is equal, we can assume \( W_{total} \) is the total efficiency of all workers. Hence, if we denote the work done by each worker as equal, we can find that:
\[ D = 31 \]

### Conclusion
Thus, the total number of days in which all the workers together will finish the whole work is **31 days**.

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