The ratio of the sum of the salaries of A and B to the difference of their salaries is `11: 1` and the ratio of the sum of the salaries of B and C to the difference of their salarles is also `11:1`. If A's salary Is the highest and C's is the lowest then what is B's salary (in Rs.) given the total of all their salaries Is Rs. 1,82,000 ?

Options

A. 72000
B. 6000
C. 50000
D. 86400

Answer

To solve the problem step-by-step, we need to establish the relationships between the salaries of A, B, and C based on the given ratios and the total salary.

### Step 1: Set up the equations based on the ratios
We know that:
1. The ratio of the sum of the salaries of A and B to the difference of their salaries is \(11:1\).
\[ \frac{A + B}{A - B} = 11 \]
This can be rearranged to:
\[ A + B = 11(A - B) \]

2. The ratio of the sum of the salaries of B and C to the difference of their salaries is also \(11:1\).
\[ \frac{B + C}{B - C} = 11 \]
This can be rearranged to:
\[ B + C = 11(B - C) \]

### Step 2: Simplify the equations
From the first equation:
\[ A + B = 11A - 11B \]
Rearranging gives:
\[ A + B + 11B = 11A \]
\[ A + 12B = 11A \]
\[ 10A = 12B \quad \Rightarrow \quad \frac{A}{B} = \frac{12}{10} = \frac{6}{5} \]

From the second equation:
\[ B + C = 11B - 11C \]
Rearranging gives:
\[ B + C + 11C = 11B \]
\[ B + 12C = 11B \]
\[ 10B = 12C \quad \Rightarrow \quad \frac{B}{C} = \frac{12}{10} = \frac{6}{5} \]

### Step 3: Establish the ratios of A, B, and C
From the ratios we derived:
- \(A : B = 6 : 5\)
- \(B : C = 6 : 5\)

Now we can express A, B, and C in terms of a common variable \(k\):
- Let \(B = 5k\).
- Then \(A = 6k\).
- From \(B : C = 6 : 5\), we have \(C = \frac{5}{6}B = \frac{5}{6}(5k) = \frac{25}{6}k\).

### Step 4: Write the total salary equation
The total salary of A, B, and C is given as Rs. 1,82,000:
\[ A + B + C = 1,82,000 \]
Substituting the values:
\[ 6k + 5k + \frac{25}{6}k = 1,82,000 \]
Combining the terms:
\[ 11k + \frac{25}{6}k = 1,82,000 \]
To combine these, convert \(11k\) into sixths:
\[ \frac{66}{6}k + \frac{25}{6}k = 1,82,000 \]
\[ \frac{91}{6}k = 1,82,000 \]

### Step 5: Solve for k
Multiplying both sides by 6:
\[ 91k = 1,09,20,000 \]
Now, divide by 91:
\[ k = \frac{1,09,20,000}{91} = 1,20,000 \]

### Step 6: Calculate B's salary
Now we can find B's salary:
\[ B = 5k = 5 \times 1,20,000 = 60,000 \]

### Final Answer
B's salary is Rs. 60,000.

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