A sum of x is divided among A, B and C such that the ratio of the shares of A and B is `7:12` and that of B and C is `8: 5`. If the difference in the shares of A and C is 219, then the value of x is:

Options

A. 17231
B. 15321
C. 11607
D. 21901

Answer

To solve the problem step by step, we will first express the shares of A, B, and C in terms of a common variable based on the given ratios.

### Step 1: Establish the Ratios
Given the ratios:
- The ratio of A to B is \( 7:12 \).
- The ratio of B to C is \( 8:5 \).

Let’s express A, B, and C in terms of a common variable.

Let:
- A's share = \( 7k \)
- B's share = \( 12k \)

Now, we need to express C's share. Since B's share is also related to C's share, we can use the second ratio \( 8:5 \).

### Step 2: Express C in Terms of B
From the ratio \( B:C = 8:5 \), we can express C in terms of B:
- Let B's share = \( 8m \)
- C's share = \( 5m \)

Since we have two expressions for B, we can equate them:
\[ 12k = 8m \]

### Step 3: Solve for k in Terms of m
From the equation \( 12k = 8m \), we can solve for \( k \):
\[ k = \frac{8m}{12} = \frac{2m}{3} \]

### Step 4: Substitute k into A and C
Now, we can express A and C in terms of m:
- A's share = \( 7k = 7 \times \frac{2m}{3} = \frac{14m}{3} \)
- C's share = \( 5m \)

### Step 5: Set Up the Equation for the Difference
According to the problem, the difference between the shares of A and C is 219:
\[ A - C = 219 \]
Substituting the expressions we derived:
\[ \frac{14m}{3} - 5m = 219 \]

### Step 6: Solve for m
To solve for m, we first convert \( 5m \) to have a common denominator:
\[ \frac{14m}{3} - \frac{15m}{3} = 219 \]
This simplifies to:
\[ \frac{-m}{3} = 219 \]
Multiplying both sides by -3 gives:
\[ m = -657 \]

### Step 7: Find A, B, and C
Now we can find the shares:
- B's share = \( 8m = 8 \times -657 = -5256 \)
- A's share = \( \frac{14m}{3} = \frac{14 \times -657}{3} = -3078 \)
- C's share = \( 5m = 5 \times -657 = -3285 \)

### Step 8: Calculate Total Sum x
The total sum \( x \) is the sum of A, B, and C:
\[ x = A + B + C = -3078 - 5256 - 3285 \]

### Step 9: Final Calculation
Calculating the total:
\[ x = -3078 - 5256 - 3285 = -11619 \]

Since we are looking for a positive value, we take the absolute value:
\[ x = 11619 \]

### Conclusion
The value of \( x \) is \( 11619 \).