Rs. 960 were distributed among A,B,C and D in such a way that C and D together gets half of what A and B together gets and C gets one third amount of B. Also D gets `5/3` times as much as C. What is the amount of A?

Options

A. Rs. 240
B. Rs. 280
C. Rs. 320
D. Data insufficient

Answer

To solve the problem step by step, we will define the variables for the amounts received by A, B, C, and D, and then set up equations based on the information given in the question.

### Step 1: Define the Variables
Let:
- A = amount received by A
- B = amount received by B
- C = amount received by C
- D = amount received by D

### Step 2: Set Up the Equations
According to the problem:
1. The total amount distributed is Rs. 960:
\[ A + B + C + D = 960 \quad (1) \]

2. C and D together get half of what A and B together get:
\[ C + D = \frac{1}{2}(A + B) \quad (2) \]

3. C gets one third of the amount of B:
\[ C = \frac{1}{3}B \quad (3) \]

4. D gets \( \frac{5}{3} \) times as much as C:
\[ D = \frac{5}{3}C \quad (4) \]

### Step 3: Substitute C in terms of B
From equation (3), we can substitute C in equation (4):
\[ D = \frac{5}{3} \left(\frac{1}{3}B\right) = \frac{5}{9}B \quad (5) \]

### Step 4: Substitute C and D in Equation (2)
Now substitute equations (3) and (5) into equation (2):
\[ \frac{1}{3}B + \frac{5}{9}B = \frac{1}{2}(A + B) \]

To combine the left side, find a common denominator (which is 9):
\[ \frac{3}{9}B + \frac{5}{9}B = \frac{1}{2}(A + B) \]
\[ \frac{8}{9}B = \frac{1}{2}(A + B) \quad (6) \]

### Step 5: Substitute A + B from Equation (1)
From equation (1), we can express A + B as:
\[ A + B = 960 - (C + D) = 960 - \frac{1}{2}(A + B) \]
Let \( x = A + B \):
\[ x = 960 - \frac{1}{2}x \]
Multiply through by 2 to eliminate the fraction:
\[ 2x = 1920 - x \]
\[ 3x = 1920 \]
\[ x = 640 \quad (7) \]
Thus, \( A + B = 640 \).

### Step 6: Substitute A + B in Equation (6)
Now substitute \( A + B = 640 \) into equation (6):
\[ \frac{8}{9}B = \frac{1}{2}(640) \]
\[ \frac{8}{9}B = 320 \]
Multiply both sides by \( \frac{9}{8} \):
\[ B = 320 \times \frac{9}{8} = 360 \]

### Step 7: Find C and D
Now substitute B back into equations (3) and (5) to find C and D:
\[ C = \frac{1}{3}B = \frac{1}{3}(360) = 120 \]
\[ D = \frac{5}{9}B = \frac{5}{9}(360) = 200 \]

### Step 8: Find A
Now substitute C and D back into equation (1) to find A:
\[ A + B + C + D = 960 \]
\[ A + 360 + 120 + 200 = 960 \]
\[ A + 680 = 960 \]
\[ A = 960 - 680 = 280 \]

### Final Answer
The amount received by A is Rs. 280.