A and B undertook a work for Rs. 350. A got Rs. 150 more than that of B, when they worked together. B takes 9 days more than A, when they work individually. In how many days A and B working together can do the whole work :

Options

A. 5
B. `4(2)/(7)`
C. `4(5)/(7)`
D. `5 (4)/(7)`

Answer

To solve the problem step by step, we will break down the information provided and find the required solution.

### Step 1: Understand the given information
- Total payment for the work = Rs. 350
- A receives Rs. 150 more than B.
- B takes 9 days more than A to complete the work individually.

### Step 2: Set up equations based on the information
Let the amount B receives be \( x \). Then, the amount A receives will be \( x + 150 \).

From the total payment, we can write:
\[ x + (x + 150) = 350 \]
This simplifies to:
\[ 2x + 150 = 350 \]

### Step 3: Solve for \( x \)
Subtract 150 from both sides:
\[ 2x = 350 - 150 \]
\[ 2x = 200 \]
Now, divide by 2:
\[ x = 100 \]

### Step 4: Find the amounts received by A and B
Now we can find the amounts:
- Amount received by B = \( x = 100 \)
- Amount received by A = \( x + 150 = 100 + 150 = 250 \)

### Step 5: Establish the efficiency ratio
The wages are divided based on their efficiency. Therefore, the efficiency ratio \( E_A : E_B \) can be expressed as:
\[ E_A : E_B = 250 : 100 = 5 : 2 \]

### Step 6: Determine the time taken by A and B
Let the time taken by A be \( 2x \) days and by B be \( 5x \) days. According to the problem, B takes 9 days more than A:
\[ 5x = 2x + 9 \]

### Step 7: Solve for \( x \)
Subtract \( 2x \) from both sides:
\[ 3x = 9 \]
Now, divide by 3:
\[ x = 3 \]

### Step 8: Calculate the actual days taken by A and B
Now substituting \( x \) back:
- Time taken by A = \( 2x = 2 \times 3 = 6 \) days
- Time taken by B = \( 5x = 5 \times 3 = 15 \) days

### Step 9: Calculate the work done per day by A and B
The work done by A in one day is:
\[ \frac{1}{6} \text{ (since A completes the work in 6 days)} \]
The work done by B in one day is:
\[ \frac{1}{15} \text{ (since B completes the work in 15 days)} \]

### Step 10: Find the total work done when A and B work together
The total work done by A and B together in one day is:
\[ \frac{1}{6} + \frac{1}{15} \]
To add these fractions, find a common denominator:
The LCM of 6 and 15 is 30.
\[ \frac{1}{6} = \frac{5}{30}, \quad \frac{1}{15} = \frac{2}{30} \]
Thus,
\[ \frac{5}{30} + \frac{2}{30} = \frac{7}{30} \]

### Step 11: Calculate the total time taken to complete the work together
If A and B together complete \( \frac{7}{30} \) of the work in one day, the total time to complete the work is the reciprocal of their combined work rate:
\[ \text{Total time} = \frac{30}{7} \text{ days} \]

### Final Answer
Thus, A and B working together can complete the whole work in \( \frac{30}{7} \) days, which is approximately \( 4 \frac{2}{7} \) days.

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