It was decided that a sum of Rs. 780 should be divided among P, Q and R In the ratio `1:2:3`. But by mistake, it was divided In the ratio `1/2:1/3:1/4`. What was the loss of Q in share due to this mistake ?

Options

A. Rs 30
B. Rs 40
C. Rs 20
D. Rs 10

Answer

To solve the problem step by step, we will calculate the shares of P, Q, and R in both scenarios: the intended ratio and the mistaken ratio. Finally, we will find the loss incurred by Q due to the mistake.

### Step 1: Calculate the shares in the intended ratio (1:2:3)

1. **Define the ratio parts:**
- Let the shares of P, Q, and R be represented as:
- P's share = \(x\)
- Q's share = \(2x\)
- R's share = \(3x\)

2. **Set up the equation:**
- According to the problem, the total amount is Rs. 780.
- Therefore, we can write the equation:
\[ x + 2x + 3x = 780 \]

3. **Combine like terms:**
- This simplifies to:
\[ 6x = 780 \]

4. **Solve for \(x\):**
- Divide both sides by 6:
\[ x = \frac{780}{6} = 130 \]

5. **Calculate Q's share:**
- Q's share in the intended ratio is:
\[ 2x = 2 \times 130 = 260 \]

### Step 2: Calculate the shares in the mistaken ratio (1/2:1/3:1/4)

1. **Define the shares in the mistaken ratio:**
- Let the shares of P, Q, and R be represented as:
- P's share = \( \frac{x}{2} \)
- Q's share = \( \frac{x}{3} \)
- R's share = \( \frac{x}{4} \)

2. **Set up the equation:**
- According to the problem, the total amount is still Rs. 780.
- Therefore, we can write the equation:
\[ \frac{x}{2} + \frac{x}{3} + \frac{x}{4} = 780 \]

3. **Find a common denominator:**
- The least common multiple (LCM) of 2, 3, and 4 is 12.
- Rewrite the equation using the common denominator:
\[ \frac{6x}{12} + \frac{4x}{12} + \frac{3x}{12} = 780 \]

4. **Combine like terms:**
- This simplifies to:
\[ \frac{13x}{12} = 780 \]

5. **Solve for \(x\):**
- Multiply both sides by 12:
\[ 13x = 780 \times 12 \]
- Divide both sides by 13:
\[ x = \frac{780 \times 12}{13} = 720 \]

6. **Calculate Q's share in the mistaken ratio:**
- Q's share in the mistaken ratio is:
\[ \frac{x}{3} = \frac{720}{3} = 240 \]

### Step 3: Calculate the loss of Q due to the mistake

1. **Determine the loss:**
- The loss incurred by Q is:
\[ \text{Loss} = \text{Q's share in intended ratio} - \text{Q's share in mistaken ratio} \]
- Substitute the values:
\[ \text{Loss} = 260 - 240 = 20 \]

### Final Answer:
The loss of Q in share due to this mistake is Rs. 20.

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