To solve the problem, we need to find the shares of A, B, C, and D based on the given ratios and then determine the difference between the shares of B and D.
### Step 1: Set up the ratios
Given the ratios:
- A : B = 1 : 3
- B : C = 2 : 5
- C : D = 2 : 3
We can express the shares of A, B, C, and D in terms of a common variable.
Let:
- A = x
- B = 3x (from A:B = 1:3)
Now, since B = 3x, we can find C using the ratio B:C = 2:5.
- B = 2y and C = 5y
From B = 3x, we have:
\[ 2y = 3x \]
\[ y = \frac{3x}{2} \]
Thus, C can be expressed as:
\[ C = 5y = 5 \left(\frac{3x}{2}\right) = \frac{15x}{2} \]
Next, we can find D using the ratio C:D = 2:3.
- C = 2z and D = 3z
From C = \(\frac{15x}{2}\), we have:
\[ 2z = \frac{15x}{2} \]
\[ z = \frac{15x}{4} \]
Thus, D can be expressed as:
\[ D = 3z = 3 \left(\frac{15x}{4}\right) = \frac{45x}{4} \]
### Step 2: Express the total sum
Now we can express the total sum of their shares:
\[ A + B + C + D = x + 3x + \frac{15x}{2} + \frac{45x}{4} \]
To combine these, we need a common denominator. The least common multiple of 1, 1, 2, and 4 is 4. So we convert each term:
- A = \( \frac{4x}{4} \)
- B = \( \frac{12x}{4} \)
- C = \( \frac{30x}{4} \)
- D = \( \frac{45x}{4} \)
Now, adding these together:
\[ A + B + C + D = \frac{4x + 12x + 30x + 45x}{4} = \frac{91x}{4} \]
### Step 3: Set the total equal to Rs 4095
Now we set the total equal to Rs 4095:
\[ \frac{91x}{4} = 4095 \]
To solve for x, multiply both sides by 4:
\[ 91x = 4095 \times 4 \]
\[ 91x = 16380 \]
Now divide by 91:
\[ x = \frac{16380}{91} = 180 \]
### Step 4: Calculate the shares of A, B, C, and D
Now we can find the shares:
- A = x = 180
- B = 3x = 3(180) = 540
- C = \(\frac{15x}{2} = \frac{15(180)}{2} = 1350\)
- D = \(\frac{45x}{4} = \frac{45(180)}{4} = 2025\)
### Step 5: Find the difference between B and D
Now we calculate the difference between the shares of B and D:
\[ \text{Difference} = D - B = 2025 - 540 = 1485 \]
Thus, the difference between the shares of B and D is Rs 1485.
### Final Answer
The difference between the shares of B and D is Rs 1485.
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