A sum of Rs x, is divided between A, B,C and D such that the ratio of the shares of A and B is 3:2 , that B and C is 9 :8 and that of C and D is 4:3. If the difference between that shares of B and D is Rs708, then what is the value of x ?

Options

A. 8732
B. 8496
C. 8850
D. 8614

Answer

To solve the problem, we need to find the value of \( x \) given the ratios of shares among A, B, C, and D, and the difference between the shares of B and D.

### Step-by-Step Solution:

1. **Define the Ratios**:
- Let the share of A be \( 3k \).
- Let the share of B be \( 2k \).
- From the ratio of B and C (9:8), we can express C in terms of B:
\[ B : C = 9 : 8 \implies C = \frac{8}{9}B = \frac{8}{9}(2k) = \frac{16k}{9} \]
- From the ratio of C and D (4:3), we can express D in terms of C:
\[ C : D = 4 : 3 \implies D = \frac{3}{4}C = \frac{3}{4}\left(\frac{16k}{9}\right) = \frac{12k}{9} = \frac{4k}{3} \]

2. **Express Shares in Terms of k**:
- Now we have:
- A's share = \( 3k \)
- B's share = \( 2k \)
- C's share = \( \frac{16k}{9} \)
- D's share = \( \frac{4k}{3} \)

3. **Set Up the Equation for the Difference**:
- According to the problem, the difference between the shares of B and D is Rs 708:
\[ B - D = 708 \]
- Substitute the values of B and D:
\[ 2k - \frac{4k}{3} = 708 \]

4. **Solve for k**:
- To solve the equation, first find a common denominator:
\[ 2k = \frac{6k}{3} \]
- Now, substitute:
\[ \frac{6k}{3} - \frac{4k}{3} = 708 \]
\[ \frac{2k}{3} = 708 \]
- Multiply both sides by 3:
\[ 2k = 2124 \]
- Divide by 2:
\[ k = 1062 \]

5. **Calculate the Shares**:
- Now we can find the shares:
- A's share = \( 3k = 3 \times 1062 = 3186 \)
- B's share = \( 2k = 2 \times 1062 = 2124 \)
- C's share = \( \frac{16k}{9} = \frac{16 \times 1062}{9} = 1888 \)
- D's share = \( \frac{4k}{3} = \frac{4 \times 1062}{3} = 1416 \)

6. **Calculate the Total Sum (x)**:
- The total sum \( x \) is the sum of all shares:
\[ x = A + B + C + D = 3186 + 2124 + 1888 + 1416 \]
- Calculate:
\[ x = 3186 + 2124 + 1888 + 1416 = 8624 \]

### Final Answer:
The value of \( x \) is Rs 8624.