Rs 3400 Is divided among A, B, C, D in such a way that the share of A and B, B and C, C and D may be as `2 : 3, 4 : 3` and `2 : 3` respectively. The sum of shares of B and D is

Options

A. Rs 2040
B. Rs 1680
C. Rs 2000
D. Rs 1720

Answer

To solve the problem of dividing Rs 3400 among A, B, C, and D based on the given ratios, we can follow these steps:

### Step 1: Set up the ratios
The ratios given are:
- A : B = 2 : 3
- B : C = 4 : 3
- C : D = 2 : 3

### Step 2: Express each variable in terms of a common variable
Let’s express A, B, C, and D in terms of a common variable \( k \).

From the first ratio \( A : B = 2 : 3 \):
- Let \( A = 2k \)
- Let \( B = 3k \)

From the second ratio \( B : C = 4 : 3 \):
- Since \( B = 3k \), we can set up the equation:
\[ 3k : C = 4 : 3 \implies C = \frac{3}{4} \times 3k = \frac{9k}{4} \]

From the third ratio \( C : D = 2 : 3 \):
- Since \( C = \frac{9k}{4} \), we can set up the equation:
\[ \frac{9k}{4} : D = 2 : 3 \implies D = \frac{3}{2} \times \frac{9k}{4} = \frac{27k}{8} \]

### Step 3: Find a common base for all ratios
Now we have:
- \( A = 2k \)
- \( B = 3k \)
- \( C = \frac{9k}{4} \)
- \( D = \frac{27k}{8} \)

To combine these ratios, we need to express them with a common denominator. The least common multiple of the denominators (1, 1, 4, 8) is 8.

So, we can multiply each term by 8 to eliminate the fractions:
- \( A = 16k \)
- \( B = 24k \)
- \( C = 18k \)
- \( D = 27k \)

### Step 4: Sum of the parts
Now, we can sum these parts:
\[ A + B + C + D = 16k + 24k + 18k + 27k = 85k \]

### Step 5: Set the total equal to Rs 3400
Since the total amount is Rs 3400, we set up the equation:
\[ 85k = 3400 \]

### Step 6: Solve for \( k \)
Now, we solve for \( k \):
\[ k = \frac{3400}{85} = 40 \]

### Step 7: Calculate individual shares
Now we can calculate the shares:
- \( A = 16k = 16 \times 40 = 640 \)
- \( B = 24k = 24 \times 40 = 960 \)
- \( C = 18k = 18 \times 40 = 720 \)
- \( D = 27k = 27 \times 40 = 1080 \)

### Step 8: Find the sum of shares of B and D
Now, we find the sum of the shares of B and D:
\[ B + D = 960 + 1080 = 2040 \]

### Final Answer
The sum of the shares of B and D is Rs 2040.

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