The ratio of incomes of P and Q is `5 : 3`. Ratio of income of Q and R is `5 : 2`. If one-fourth of P's income is Rs 500 more than the R's income, then what is the income (in Rs) of Q ?

Options

A. 7500
B. 15000
C. 30000
D. 18000

Answer

To solve the problem step by step, let's break down the information given and use it to find the income of Q.

### Step 1: Set up the ratios
We know the following ratios:
- The ratio of incomes of P and Q is \(5:3\).
- The ratio of incomes of Q and R is \(5:2\).

Let:
- Income of P = \(5x\)
- Income of Q = \(3x\)
- Income of Q = \(5y\)
- Income of R = \(2y\)

### Step 2: Relate the two ratios
Since Q has two expressions based on the ratios, we can set them equal to each other:
\[ 3x = 5y \]
From this, we can express \(y\) in terms of \(x\):
\[ y = \frac{3x}{5} \]

### Step 3: Express R's income in terms of x
Now, we can express R's income using the value of \(y\):
\[ R = 2y = 2 \left(\frac{3x}{5}\right) = \frac{6x}{5} \]

### Step 4: Use the condition given in the problem
According to the problem, one-fourth of P's income is Rs 500 more than R's income:
\[ \frac{1}{4} \times 5x = R + 500 \]
Substituting R's income:
\[ \frac{5x}{4} = \frac{6x}{5} + 500 \]

### Step 5: Clear the fractions
To eliminate the fractions, multiply the entire equation by 20 (the least common multiple of 4 and 5):
\[ 20 \times \frac{5x}{4} = 20 \times \left(\frac{6x}{5} + 500\right) \]
This simplifies to:
\[ 25x = 24x + 10000 \]

### Step 6: Solve for x
Subtract \(24x\) from both sides:
\[ 25x - 24x = 10000 \]
\[ x = 10000 \]

### Step 7: Calculate the income of Q
Now that we have \(x\), we can find the income of Q:
\[ Q = 3x = 3 \times 10000 = 30000 \]

### Final Answer
The income of Q is Rs 30,000.

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