A sum of Rs.12,000 is divided between A, B, C and D such that the ratio of shares of A and B is 8 : 9, that of B and C is 2:3 and that of C and D is 9 : 13. What is the difference between the shares of B and D?

Options

A. Rs. 1,320
B. Rs. 2,400
C. Rs. 2,520
D. Rs. 2,760

Answer

To solve the problem, we need to find the shares of A, B, C, and D based on the given ratios and then determine the difference between the shares of B and D.

### Step-by-Step Solution:

1. **Define the Ratios**:
- The ratio of A to B is given as \( A : B = 8 : 9 \).
- The ratio of B to C is given as \( B : C = 2 : 3 \).
- The ratio of C to D is given as \( C : D = 9 : 13 \).

2. **Express B in terms of A**:
- From \( A : B = 8 : 9 \), we can express A and B as:
\[ A = 8x \quad \text{and} \quad B = 9x \]

3. **Express C in terms of B**:
- From \( B : C = 2 : 3 \), we can express C in terms of B:
\[ B = 2y \quad \text{and} \quad C = 3y \]
- Setting \( 9x = 2y \) gives us:
\[ y = \frac{9x}{2} \]
- Substituting for C:
\[ C = 3y = 3 \left(\frac{9x}{2}\right) = \frac{27x}{2} \]

4. **Express D in terms of C**:
- From \( C : D = 9 : 13 \), we can express D in terms of C:
\[ C = 9z \quad \text{and} \quad D = 13z \]
- Setting \( \frac{27x}{2} = 9z \) gives us:
\[ z = \frac{27x}{18} = \frac{3x}{2} \]
- Substituting for D:
\[ D = 13z = 13 \left(\frac{3x}{2}\right) = \frac{39x}{2} \]

5. **Combine all ratios**:
- Now we have:
\[ A = 8x, \quad B = 9x, \quad C = \frac{27x}{2}, \quad D = \frac{39x}{2} \]

6. **Find a common denominator**:
- To combine these, we can express all terms with a common denominator of 2:
\[ A = \frac{16x}{2}, \quad B = \frac{18x}{2}, \quad C = \frac{27x}{2}, \quad D = \frac{39x}{2} \]

7. **Total shares**:
- The total shares can be expressed as:
\[ A + B + C + D = \frac{16x + 18x + 27x + 39x}{2} = \frac{100x}{2} = 50x \]

8. **Set total equal to 12,000**:
- Since the total amount is Rs. 12,000:
\[ 50x = 12000 \implies x = \frac{12000}{50} = 240 \]

9. **Calculate individual shares**:
- Now substituting \( x \) back to find the shares:
\[ A = 8x = 8 \times 240 = 1920 \]
\[ B = 9x = 9 \times 240 = 2160 \]
\[ C = \frac{27x}{2} = \frac{27 \times 240}{2} = 3240 \]
\[ D = \frac{39x}{2} = \frac{39 \times 240}{2} = 4680 \]

10. **Find the difference between B and D**:
- The difference between the shares of B and D is:
\[ D - B = 4680 - 2160 = 2520 \]

### Final Answer:
The difference between the shares of B and D is **Rs. 2520**.