Rs. 5600 is to be divided into A, B, C, and D in such a way that the ratio of share of A: B is 1: 2, B: C is 3: 1, and C: D is 2: 3. Find the sum of (A and C) and (B and C).

Options

A. Rs 2400, Rs 3000
B. Rs 2000, Rs 3000
C. Rs 2400, Rs 3200
D. Rs 2000, Rs 3200

Answer

To solve the problem of dividing Rs. 5600 among A, B, C, and D based on the given ratios, we can follow these steps:

### Step 1: Understand the Ratios
We are given three ratios:
1. A : B = 1 : 2
2. B : C = 3 : 1
3. C : D = 2 : 3

### Step 2: Express All Ratios in Terms of a Common Variable
Let's express each variable in terms of a common variable \( x \).

From A : B = 1 : 2, we can write:
- A = 1x
- B = 2x

From B : C = 3 : 1, we can write:
- B = 3y
- C = 1y

Since B is expressed in two ways (2x and 3y), we can set them equal to each other:
\[ 2x = 3y \]
From this, we can express \( y \) in terms of \( x \):
\[ y = \frac{2x}{3} \]

Now substituting \( y \) back into the expression for C:
- C = 1y = \( \frac{2x}{3} \)

From C : D = 2 : 3, we can write:
- C = 2z
- D = 3z

Again, since C is expressed in two ways (as \( \frac{2x}{3} \) and as \( 2z \)), we set them equal:
\[ \frac{2x}{3} = 2z \]
From this, we can express \( z \) in terms of \( x \):
\[ z = \frac{x}{3} \]

Now substituting \( z \) back into the expression for D:
- D = 3z = \( 3 \times \frac{x}{3} = x \)

### Step 3: Summarize the Values of A, B, C, and D
Now we have:
- A = 1x
- B = 2x
- C = \( \frac{2x}{3} \)
- D = x

### Step 4: Find the Total
Now we can find the total sum of A, B, C, and D:
\[ A + B + C + D = 1x + 2x + \frac{2x}{3} + x \]

To add these, we need a common denominator. The common denominator for the fractions is 3:
- \( 1x = \frac{3x}{3} \)
- \( 2x = \frac{6x}{3} \)
- \( x = \frac{3x}{3} \)

So,
\[ A + B + C + D = \frac{3x + 6x + 2x + 3x}{3} = \frac{14x}{3} \]

### Step 5: Set the Total Equal to Rs. 5600
Now we set this equal to Rs. 5600:
\[ \frac{14x}{3} = 5600 \]

Multiplying both sides by 3:
\[ 14x = 16800 \]

Now divide by 14:
\[ x = 1200 \]

### Step 6: Calculate Individual Shares
Now we can find the individual shares:
- A = 1x = 1200
- B = 2x = 2400
- C = \( \frac{2x}{3} = \frac{2 \times 1200}{3} = 800 \)
- D = x = 1200

### Step 7: Find the Required Sums
Now we need to find the sums of (A + C) and (B + C):
- A + C = 1200 + 800 = 2000
- B + C = 2400 + 800 = 3200

### Final Answer
Thus, the sum of (A and C) and (B and C) is:
- (A + C) + (B + C) = 2000 + 3200 = 5200