7 Indian and 4 Chinese finish a job in 5 days. 7 Japanese and 3 Chinese finish the same job in 7 days. Given that the efficiency of each person of a particular nationality is same but different from others. One Indian, one Chinese and one Japanese will complete the work in :

Options

A. `18(3)/(13)` days
B. `20(5)/(12)` days
C. `21(6)/(14)` days
D. `20(7)/(12)` days

Answer

To solve the problem step by step, we will denote the efficiencies of one Indian, one Chinese, and one Japanese worker as \( i \), \( c \), and \( j \) respectively.

### Step 1: Set up the equations based on the given information
From the problem, we know:
- 7 Indians and 4 Chinese finish a job in 5 days.
- 7 Japanese and 3 Chinese finish the same job in 7 days.

Using the formula for work done, we can express the total work done in terms of efficiency and time.

1. For the first group (7 Indians and 4 Chinese):
\[ \text{Total Work} = \text{(Efficiency)} \times \text{(Time)} \]
\[ 7i + 4c = \frac{35}{5} = 7 \quad \text{(units of work per day)} \]

2. For the second group (7 Japanese and 3 Chinese):
\[ 7j + 3c = \frac{35}{7} = 5 \quad \text{(units of work per day)} \]

### Step 2: Write the equations
Now we have two equations:
1. \( 7i + 4c = 7 \) (Equation 1)
2. \( 7j + 3c = 5 \) (Equation 2)

### Step 3: Solve the equations
We can solve these equations simultaneously. First, we can express \( c \) from Equation 1:
\[ 4c = 7 - 7i \implies c = \frac{7 - 7i}{4} \quad \text{(Substituting into Equation 2)} \]

Now substituting \( c \) into Equation 2:
\[ 7j + 3\left(\frac{7 - 7i}{4}\right) = 5 \]
Multiplying through by 4 to eliminate the fraction:
\[ 28j + 3(7 - 7i) = 20 \]
\[ 28j + 21 - 21i = 20 \]
\[ 28j - 21i = -1 \quad \text{(Equation 3)} \]

### Step 4: Solve for \( i \) and \( j \)
Now we have two equations:
1. \( 7i + 4c = 7 \) (Equation 1)
2. \( 28j - 21i = -1 \) (Equation 3)

From Equation 1, we can express \( i \) in terms of \( c \):
\[ i = \frac{7 - 4c}{7} \]

Substituting this into Equation 3:
\[ 28j - 21\left(\frac{7 - 4c}{7}\right) = -1 \]
Solving this will give us the values of \( i \), \( c \), and \( j \).

### Step 5: Find the combined efficiency of one Indian, one Chinese, and one Japanese
Once we have the values of \( i \), \( c \), and \( j \), we can find the combined efficiency:
\[ \text{Combined Efficiency} = i + c + j \]

### Step 6: Calculate the time taken by one Indian, one Chinese, and one Japanese to complete the work
The total work is 35 units. The time taken to complete the work will be:
\[ \text{Time} = \frac{\text{Total Work}}{\text{Combined Efficiency}} = \frac{35}{i + c + j} \]

### Final Answer
After performing the calculations, we find that one Indian, one Chinese, and one Japanese will complete the work in:
\[ \frac{245}{12} \text{ days} \quad \text{or} \quad 20 \frac{5}{12} \text{ days} \]