Three friends P,Q and R share an apartment and share the rent equally. The monthly income of R is 25 % less than that of Q and Rs . 8000 less than that of P. After meeting the expenses on rent and food , they save amounts in the ratio `6:7:4`
If Q saves `62 1/2 `% of his total monthly income , then how much percent does R save out of his monthly income

Options

A. `47 13/21`%
B. `48 12/21`%
C. `45 5/21 `%
D. `49 11/21`%

Answer

To solve the problem step by step, we will first define the incomes of P, Q, and R based on the information provided and then calculate R's savings percentage.

### Step 1: Define the incomes of P, Q, and R
Let the income of Q be \( x \).
Since R's income is 25% less than Q's, we can express R's income as:
\[ R = x - 0.25x = 0.75x \]

Also, we know that R's income is Rs. 8000 less than P's income:
\[ R = P - 8000 \]
Substituting R's income:
\[ 0.75x = P - 8000 \]
Thus, we can express P's income as:
\[ P = 0.75x + 8000 \]

### Step 2: Define the savings of P, Q, and R
According to the problem, their savings are in the ratio of 6:7:4.
Let the savings of P, Q, and R be represented as:
- P's savings = \( 6y \)
- Q's savings = \( 7y \)
- R's savings = \( 4y \)

### Step 3: Relate Q's savings to his income
We know that Q saves \( 62.5\% \) of his total monthly income. Therefore, we can express this as:
\[ 7y = 0.625x \]
From this, we can find \( y \):
\[ y = \frac{0.625x}{7} \]

### Step 4: Calculate R's savings
Now we need to find R's savings in terms of \( x \):
R's savings = \( 4y \)
Substituting the value of \( y \):
\[ R's savings = 4 \left( \frac{0.625x}{7} \right) = \frac{2.5x}{7} \]

### Step 5: Calculate R's income
We already defined R's income as:
\[ R = 0.75x \]

### Step 6: Calculate R's savings percentage
Now, we can find the percentage of R's savings out of his income:
\[ \text{Savings Percentage} = \left( \frac{R's savings}{R's income} \right) \times 100 \]
Substituting the values:
\[ \text{Savings Percentage} = \left( \frac{\frac{2.5x}{7}}{0.75x} \right) \times 100 \]

### Step 7: Simplify the expression
Cancelling \( x \) from the numerator and denominator:
\[ \text{Savings Percentage} = \left( \frac{2.5}{7 \times 0.75} \right) \times 100 \]
Calculating \( 7 \times 0.75 = 5.25 \):
\[ \text{Savings Percentage} = \left( \frac{2.5}{5.25} \right) \times 100 \]

### Step 8: Final calculation
Calculating \( \frac{2.5}{5.25} \):
\[ \frac{2.5}{5.25} = \frac{250}{525} = \frac{50}{105} \approx 0.4762 \]
Thus,
\[ \text{Savings Percentage} \approx 0.4762 \times 100 \approx 47.62\% \]

### Conclusion
R saves approximately **47.62%** of his monthly income.

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