Rs.1200 divided among P, Q and R. P gets half of the total amount received by Q and R. Q gets one-third of the total amount received by P and R. Find the amount received by R ?

Options

A. A) Rs. 1100
B. B) Rs. 500
C. C) Rs. 1200
D. D) Rs. 700

Answer

To solve the problem, we need to set up equations based on the information given about how the money is divided among P, Q, and R.

1. **Let the amounts received by P, Q, and R be represented as follows:**
- Let P = x
- Let Q = y
- Let R = z

2. **From the problem statement, we have the following information:**
- The total amount is Rs. 1200, so:
\[ x + y + z = 1200 \quad \text{(1)} \]
- P gets half of the total amount received by Q and R:
\[ x = \frac{1}{2}(y + z) \quad \text{(2)} \]
- Q gets one-third of the total amount received by P and R:
\[ y = \frac{1}{3}(x + z) \quad \text{(3)} \]

3. **Now, let's solve these equations step by step.**

### Step 1: Substitute Equation (2) into Equation (1)

From Equation (2):
\[ x = \frac{1}{2}(y + z) \]
Substituting this into Equation (1):
\[ \frac{1}{2}(y + z) + y + z = 1200 \]
Multiplying the entire equation by 2 to eliminate the fraction:
\[ y + z + 2y + 2z = 2400 \]
This simplifies to:
\[ 3y + 3z = 2400 \]
Dividing by 3:
\[ y + z = 800 \quad \text{(4)} \]

### Step 2: Substitute Equation (4) into Equation (2)

Now we can substitute Equation (4) back into Equation (2):
\[ x = \frac{1}{2}(800) = 400 \quad \text{(5)} \]

### Step 3: Substitute Equation (5) into Equation (3)

Now we substitute \( x = 400 \) into Equation (3):
\[ y = \frac{1}{3}(400 + z) \]
Multiplying both sides by 3:
\[ 3y = 400 + z \quad \text{(6)} \]

### Step 4: Substitute Equation (4) into Equation (6)

From Equation (4), we know \( z = 800 - y \). Substitute this into Equation (6):
\[ 3y = 400 + (800 - y) \]
This simplifies to:
\[ 3y = 400 + 800 - y \]
Combining like terms:
\[ 3y + y = 1200 \]
\[ 4y = 1200 \]
Dividing by 4:
\[ y = 300 \quad \text{(7)} \]

### Step 5: Find the value of z

Now substitute \( y = 300 \) back into Equation (4):
\[ 300 + z = 800 \]
Solving for z:
\[ z = 800 - 300 = 500 \quad \text{(8)} \]

### Conclusion

Thus, the amount received by R is:
\[ \text{Amount received by R} = z = 500 \]