A sum of Rs. 8,235 is divided between A,B, C and D such that the ratio of the shares of A and B is 3 : 4 that of B and C is 8 : 5 , and that of C and D is 7 : 10 .What is the difference between the shares of A and C ?

Options

A. Rs. 270
B. Rs. 310
C. Rs. 405
D. Rs.315

Answer

To solve the problem, we need to find the shares of A, B, C, and D based on the given ratios and then calculate the difference between the shares of A and C.

### Step-by-Step Solution:

1. **Establish Ratios**:
- The ratio of A to B is \(3:4\).
- The ratio of B to C is \(8:5\).
- The ratio of C to D is \(7:10\).

2. **Express Ratios in Terms of a Common Variable**:
- Let the share of A be \(3x\) and the share of B be \(4x\) (from the ratio \(3:4\)).
- From the ratio \(B:C = 8:5\), we can express B in terms of a new variable \(y\). Let \(B = 8y\) and \(C = 5y\).
- Since \(B\) is the same in both ratios, we can equate \(4x = 8y\) which gives us \(x = 2y\).
- Now substituting \(x\) back, we have:
- \(A = 3x = 6y\)
- \(B = 8y\)
- \(C = 5y\)

3. **Express C and D**:
- From the ratio \(C:D = 7:10\), let \(C = 7z\) and \(D = 10z\).
- Since \(C = 5y\), we equate \(5y = 7z\) which gives us \(y = \frac{7z}{5}\).
- Now substituting \(y\) back, we have:
- \(A = 6y = 6 \times \frac{7z}{5} = \frac{42z}{5}\)
- \(B = 8y = 8 \times \frac{7z}{5} = \frac{56z}{5}\)
- \(C = 5y = 7z\)
- \(D = 10z\)

4. **Total Shares**:
- The total sum of shares is:
\[ A + B + C + D = \frac{42z}{5} + \frac{56z}{5} + 7z + 10z \]
- Converting \(7z\) and \(10z\) to a common denominator:
\[ 7z = \frac{35z}{5}, \quad 10z = \frac{50z}{5} \]
- Now adding them together:
\[ \frac{42z}{5} + \frac{56z}{5} + \frac{35z}{5} + \frac{50z}{5} = \frac{183z}{5} \]

5. **Set Total Equal to Rs. 8235**:
- We know the total is Rs. 8235, so:
\[ \frac{183z}{5} = 8235 \]
- Multiplying both sides by 5:
\[ 183z = 41175 \]
- Solving for \(z\):
\[ z = \frac{41175}{183} = 225 \]

6. **Calculate Shares**:
- Now substituting \(z\) back to find A, B, C, and D:
- \(A = \frac{42 \times 225}{5} = 945\)
- \(B = \frac{56 \times 225}{5} = 2520\)
- \(C = 7 \times 225 = 1575\)
- \(D = 10 \times 225 = 2250\)

7. **Find the Difference Between A and C**:
- The difference between the shares of A and C is:
\[ A - C = 945 - 1575 = -630 \]
- Since we are interested in the absolute difference:
\[ |A - C| = |945 - 1575| = 630 \]

### Final Answer:
The difference between the shares of A and C is **630**.