The income of P is 50% more than Q's Income and the income of Q is 50% more than R's income. P's income is how much per cent more than R's income?

Options

A. A) 225
B. B) 150
C. C) 125
D. D) 100

Answer

To solve the problem, we will break it down step by step.

### Step 1: Define Q's Income
Let's assume Q's income is \( Q = 100 \).

### Step 2: Calculate P's Income
Since P's income is 50% more than Q's income, we can calculate P's income as follows:
\[ P = Q + 0.5 \times Q = 100 + 0.5 \times 100 = 100 + 50 = 150 \]

### Step 3: Define R's Income
Now, we know that Q's income is 50% more than R's income. Let's denote R's income as \( R \). Therefore, we can express Q's income in terms of R's income:
\[ Q = R + 0.5 \times R = 1.5R \]
Since we assumed \( Q = 100 \), we can set up the equation:
\[ 1.5R = 100 \]
To find R's income, we solve for \( R \):
\[ R = \frac{100}{1.5} = \frac{1000}{15} = \frac{200}{3} \approx 66.67 \]

### Step 4: Compare P's Income to R's Income
Now we have:
- \( P = 150 \)
- \( R \approx 66.67 \)

To find out how much more P's income is than R's income, we calculate the difference:
\[ P - R = 150 - \frac{200}{3} = 150 - 66.67 \approx 83.33 \]

### Step 5: Calculate the Percentage Increase
To find out how much percentage P's income is more than R's income, we use the formula:
\[ \text{Percentage Increase} = \left( \frac{P - R}{R} \right) \times 100 \]
Substituting the values:
\[ \text{Percentage Increase} = \left( \frac{150 - \frac{200}{3}}{\frac{200}{3}} \right) \times 100 \]
Calculating the numerator:
\[ 150 - \frac{200}{3} = \frac{450}{3} - \frac{200}{3} = \frac{250}{3} \]
Now substituting back:
\[ \text{Percentage Increase} = \left( \frac{\frac{250}{3}}{\frac{200}{3}} \right) \times 100 = \left( \frac{250}{200} \right) \times 100 = 1.25 \times 100 = 125\% \]

### Conclusion
P's income is 125% more than R's income.