Pawan and Qureshi working together can do a piece of work in 10 days whereas Qureshi and Rohit working together can do the same work in 12 days. All three work together to do a job for which they are paid Rs. 300. If Qureshi’s share is Rs. 140, then what is Pawan’s share?

Options

A. Rs. 100
B. Rs. 60
C. Rs. 80
D. cannot be determined

Answer

To solve the problem step by step, we need to determine the work rates of Pawan, Qureshi, and Rohit based on the information given, and then calculate Pawan's share of the payment.

### Step 1: Determine the work rates of Pawan and Qureshi
Pawan and Qureshi together can complete the work in 10 days. Therefore, their combined work rate is:
\[ \text{Work rate of (Pawan + Qureshi)} = \frac{1}{10} \text{ work/day} \]

### Step 2: Determine the work rates of Qureshi and Rohit
Qureshi and Rohit together can complete the work in 12 days. Therefore, their combined work rate is:
\[ \text{Work rate of (Qureshi + Rohit)} = \frac{1}{12} \text{ work/day} \]

### Step 3: Let the work rates of Pawan, Qureshi, and Rohit be represented as P, Q, and R respectively
From the above information, we can write:
\[ P + Q = \frac{1}{10} \quad \text{(1)} \]
\[ Q + R = \frac{1}{12} \quad \text{(2)} \]

### Step 4: Express R in terms of Q
From equation (2), we can express R as:
\[ R = \frac{1}{12} - Q \quad \text{(3)} \]

### Step 5: Substitute R into the equation for P + Q
We need to find a relationship involving P, Q, and R. We can add equations (1) and (2):
\[ (P + Q) + (Q + R) = \frac{1}{10} + \frac{1}{12} \]
This simplifies to:
\[ P + 2Q + R = \frac{1}{10} + \frac{1}{12} \]

### Step 6: Find a common denominator and simplify
The common denominator for 10 and 12 is 60. Thus:
\[ \frac{1}{10} = \frac{6}{60}, \quad \frac{1}{12} = \frac{5}{60} \]
Adding these gives:
\[ P + 2Q + R = \frac{6}{60} + \frac{5}{60} = \frac{11}{60} \]

### Step 7: Substitute R from equation (3)
Substituting R from equation (3) into the equation gives:
\[ P + 2Q + \left(\frac{1}{12} - Q\right) = \frac{11}{60} \]
This simplifies to:
\[ P + Q + \frac{1}{12} = \frac{11}{60} \]

### Step 8: Solve for P + Q
Subtract \(\frac{1}{12}\) from both sides:
\[ P + Q = \frac{11}{60} - \frac{5}{60} = \frac{6}{60} = \frac{1}{10} \]

### Step 9: Find the individual shares based on the payment
We know that Qureshi's share is Rs. 140. The total payment is Rs. 300. Thus, the remaining amount for Pawan and Rohit is:
\[ 300 - 140 = 160 \]

### Step 10: Determine Pawan's share
Let Pawan's share be X and Rohit's share be \(160 - X\). The work done by each person is proportional to their share of the payment.

Using the work rates:
\[ \frac{P}{Q} = \frac{P + Q}{Q + R} = \frac{1/10}{1/12} = \frac{12}{10} = \frac{6}{5} \]
Letting Q = 140, we can find P:
\[ \frac{P}{140} = \frac{6}{5} \implies P = \frac{6}{5} \times 140 = 168 \]

### Step 11: Calculate Pawan's share
Now we have:
\[ P + Q + R = 300 \]
Substituting Q and R:
\[ P + 140 + (160 - X) = 300 \]
Solving gives:
\[ P = 100 \]

Thus, Pawan's share is Rs. 100.

### Final Answer:
Pawan's share is Rs. 100.

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