A sum of X 300 is divided among P, Q and R in such a way that Q gets X 30 more than P and R gets X 60 more than Q. Then , ratio of their shares is

Options

A. `2: 3 : 5`
B. `3 : 2: 5`
C. `2 : 5: 3`
D. `5 : 3 :2`

Answer

To solve the problem step by step, we will break down the information given and use algebra to find the ratio of the shares of P, Q, and R.

### Step-by-Step Solution:

1. **Define Variables**:
Let the share of P be \( x \).

2. **Express Q's Share**:
According to the problem, Q gets \( 30 \) more than P. Therefore, Q's share can be expressed as:
\[ Q = x + 30 \]

3. **Express R's Share**:
R gets \( 60 \) more than Q. Thus, R's share can be expressed as:
\[ R = Q + 60 = (x + 30) + 60 = x + 90 \]

4. **Set Up the Equation**:
The total sum of money divided among P, Q, and R is \( 300 \). Therefore, we can write the equation:
\[ P + Q + R = 300 \]
Substituting the expressions for Q and R:
\[ x + (x + 30) + (x + 90) = 300 \]

5. **Combine Like Terms**:
Simplifying the equation:
\[ x + x + 30 + x + 90 = 300 \]
\[ 3x + 120 = 300 \]

6. **Solve for x**:
Now, isolate \( x \):
\[ 3x = 300 - 120 \]
\[ 3x = 180 \]
\[ x = \frac{180}{3} = 60 \]

7. **Calculate Each Share**:
Now that we have \( x \), we can find the shares of P, Q, and R:
- P's share:
\[ P = x = 60 \]
- Q's share:
\[ Q = x + 30 = 60 + 30 = 90 \]
- R's share:
\[ R = x + 90 = 60 + 90 = 150 \]

8. **Find the Ratio**:
The shares of P, Q, and R are \( 60, 90, \) and \( 150 \), respectively. The ratio of their shares can be expressed as:
\[ P : Q : R = 60 : 90 : 150 \]

9. **Simplify the Ratio**:
To simplify the ratio, we can divide each term by \( 30 \):
\[ \frac{60}{30} : \frac{90}{30} : \frac{150}{30} = 2 : 3 : 5 \]

### Final Answer:
The ratio of their shares is \( 2 : 3 : 5 \).