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 4 : 5 , that of B and C is 3 : 4 and that of C and D is 5 : 7 . If the difference between the shares of B and D is Rs. 3,276 , then the value of x is
(a)17,500
(b)18,900
(c)17,400
(d)18,800

Options

A. `17,500`
B. `18,900`
C. `17,400`
D. `18,800`

Answer

To solve the problem, we will break it down step by step.

### Step 1: Establish the Ratios
We are given the following ratios:
- A : B = 4 : 5
- B : C = 3 : 4
- C : D = 5 : 7

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

Let:
- B = 5k (from A : B = 4 : 5)
- A = 4k
- C = (4/3)B = (4/3)(5k) = (20/3)k
- D = (7/5)C = (7/5)((20/3)k) = (28/3)k

### Step 3: Find the Shares
Now we have:
- A = 4k
- B = 5k
- C = (20/3)k
- D = (28/3)k

### Step 4: Calculate the Difference Between B and D
We are given that the difference between the shares of B and D is Rs. 3,276. Therefore, we can set up the equation:
\[ B - D = 3,276 \]
Substituting the values we have:
\[ 5k - \frac{28}{3}k = 3,276 \]

### Step 5: Solve for k
To solve for k, we first express both terms with a common denominator:
\[ \frac{15k}{3} - \frac{28k}{3} = 3,276 \]
This simplifies to:
\[ \frac{-13k}{3} = 3,276 \]
Multiplying both sides by -3 gives:
\[ 13k = -3 \times 3,276 \]
Calculating the right side:
\[ 13k = 9,828 \]
Now, divide by 13:
\[ k = \frac{9,828}{13} = 757.54 \]

### Step 6: Calculate Total Shares
Now we can calculate the total amount X:
Total shares = A + B + C + D
\[ = 4k + 5k + \frac{20}{3}k + \frac{28}{3}k \]
Combining terms:
\[ = 9k + \left(\frac{20 + 28}{3}\right)k \]
\[ = 9k + \frac{48}{3}k \]
\[ = 9k + 16k = 25k \]

### Step 7: Substitute k to Find X
Now we substitute k back to find X:
\[ X = 25 \times 757.54 \]
Calculating:
\[ X = 18,938.5 \]
Since we are looking for the closest option, we round it to Rs. 18,900.

### Final Answer
Thus, the value of X is Rs. 18,900.