Rs 7800 distributed among A,B and C. the share of A is `3/4 th` of the share of B , and share of B is `2/3` of the share of C . Then , find the difference between share of B and C?

Answer

To solve the problem of distributing Rs 7800 among A, B, and C, we can follow these steps:

### Step 1: Define the Variables
Let the shares of A, B, and C be represented as:
- A = share of A
- B = share of B
- C = share of C

### Step 2: Set Up the Equations
From the problem statement, we have the following relationships:
1. The total amount distributed is:
\[ A + B + C = 7800 \quad \text{(Equation 1)} \]
2. The share of A is \( \frac{3}{4} \) of the share of B:
\[ A = \frac{3}{4}B \quad \text{(Equation 2)} \]
3. The share of B is \( \frac{2}{3} \) of the share of C:
\[ B = \frac{2}{3}C \quad \text{(Equation 3)} \]

### Step 3: Substitute Equation 3 into Equation 2
Substituting Equation 3 into Equation 2 gives:
\[ A = \frac{3}{4} \left(\frac{2}{3}C\right) \]
Calculating this:
\[ A = \frac{3 \times 2}{4 \times 3}C = \frac{1}{2}C \]

### Step 4: Substitute A and B into Equation 1
Now substitute the expressions for A and B in terms of C into Equation 1:
\[ \frac{1}{2}C + \frac{2}{3}C + C = 7800 \]

### Step 5: Find a Common Denominator
The common denominator for the fractions is 6. Rewrite the equation:
\[ \frac{3}{6}C + \frac{4}{6}C + \frac{6}{6}C = 7800 \]
Combining the fractions:
\[ \frac{3 + 4 + 6}{6}C = 7800 \]
\[ \frac{13}{6}C = 7800 \]

### Step 6: Solve for C
To isolate C, multiply both sides by \( \frac{6}{13} \):
\[ C = 7800 \times \frac{6}{13} \]
Calculating this:
\[ C = 3600 \]

### Step 7: Find B
Now substitute C back into Equation 3 to find B:
\[ B = \frac{2}{3}C = \frac{2}{3} \times 3600 \]
Calculating this:
\[ B = 2400 \]

### Step 8: Find the Difference Between B and C
Now we need to find the difference between the shares of B and C:
\[ \text{Difference} = C - B = 3600 - 2400 = 1200 \]

### Final Answer
The difference between the share of B and C is **Rs 1200**.

---