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 2: 3. that of B and C is 8:9 and that of C and D is 3:4. If C's share is Rs 4,347. then the value of x is:

Options

A. A)`16,583`
B. B)`16,788`
C. C)`16,686`
D. D)`16,422`

Answer

To solve the problem step by step, we will first establish the ratios and then find the total sum \( x \).

### Step 1: Establish Ratios
We are given the following ratios:
- Ratio of A and B: \( A : B = 2 : 3 \)
- Ratio of B and C: \( B : C = 8 : 9 \)
- Ratio of C and D: \( C : D = 3 : 4 \)

### Step 2: Express All Shares in Terms of a Common Variable
Let’s express the shares of A, B, C, and D in terms of a common variable.

1. From \( A : B = 2 : 3 \), we can write:
- \( A = 2k \)
- \( B = 3k \)

2. From \( B : C = 8 : 9 \), we can express C in terms of B:
- Let \( B = 8m \) and \( C = 9m \)
- Since \( B = 3k \), we can equate:
\[ 3k = 8m \implies k = \frac{8m}{3} \]
- Therefore, substituting \( k \) into \( A \) and \( B \):
- \( A = 2k = 2 \times \frac{8m}{3} = \frac{16m}{3} \)
- \( B = 3k = 3 \times \frac{8m}{3} = 8m \)
- \( C = 9m \)

3. From \( C : D = 3 : 4 \), we can express D in terms of C:
- Let \( C = 3n \) and \( D = 4n \)
- Since \( C = 9m \), we can equate:
\[ 9m = 3n \implies n = 3m \]
- Therefore, substituting \( n \) into \( D \):
- \( D = 4n = 4 \times 3m = 12m \)

### Step 3: Combine All Shares
Now we have:
- \( A = \frac{16m}{3} \)
- \( B = 8m \)
- \( C = 9m \)
- \( D = 12m \)

To express all shares in a common ratio, we can multiply each term by 3 to eliminate the fraction:
- \( A = 16m \)
- \( B = 24m \)
- \( C = 27m \)
- \( D = 36m \)

### Step 4: Find Total Shares
Now, we can find the total shares:
\[ A + B + C + D = 16m + 24m + 27m + 36m = 103m \]

### Step 5: Use C's Share to Find m
We know that C's share is Rs 4347. Therefore:
\[ C = 27m = 4347 \implies m = \frac{4347}{27} = 161 \]

### Step 6: Calculate Total Sum x
Now substituting \( m \) back into the total shares:
\[ x = 103m = 103 \times 161 = 16543 \]

### Final Answer
The value of \( x \) is Rs 16543.