When 5 is added to the numerator and denominator both of a (positive) fraction , then the new ratio of numerator to denominator becomes 11:15. What is the original ratio?

Options

A. `17:25`
B. `3:5`
C. `28:40`
D. None of these

Answer

To solve the problem, we need to find the original ratio of a fraction given that when 5 is added to both the numerator and the denominator, the new ratio becomes 11:15.

Let's denote the original numerator as \( x \) and the original denominator as \( y \).

### Step-by-step Solution:

1. **Set up the equation based on the problem statement:**
When 5 is added to both the numerator and the denominator, the new fraction becomes:
\[ \frac{x + 5}{y + 5} = \frac{11}{15} \]

2. **Cross-multiply to eliminate the fraction:**
Cross-multiplying gives us:
\[ 15(x + 5) = 11(y + 5) \]

3. **Expand both sides:**
Expanding both sides results in:
\[ 15x + 75 = 11y + 55 \]

4. **Rearrange the equation:**
Rearranging the equation to isolate terms gives us:
\[ 15x - 11y = 55 - 75 \]
\[ 15x - 11y = -20 \]

5. **Express one variable in terms of the other:**
We can express \( y \) in terms of \( x \):
\[ 15x + 20 = 11y \]
\[ y = \frac{15x + 20}{11} \]

6. **Find integer values for \( x \) and \( y \):**
Since \( x \) and \( y \) must be integers, we can test integer values for \( x \) to find a corresponding integer \( y \).

Let's try \( x = 1 \):
\[ y = \frac{15(1) + 20}{11} = \frac{35}{11} \quad \text{(not an integer)} \]
Try \( x = 2 \):
\[ y = \frac{15(2) + 20}{11} = \frac{50}{11} \quad \text{(not an integer)} \]
Try \( x = 3 \):
\[ y = \frac{15(3) + 20}{11} = \frac{65}{11} \quad \text{(not an integer)} \]
Try \( x = 4 \):
\[ y = \frac{15(4) + 20}{11} = \frac{80}{11} \quad \text{(not an integer)} \]
Try \( x = 5 \):
\[ y = \frac{15(5) + 20}{11} = \frac{95}{11} \quad \text{(not an integer)} \]
Try \( x = 6 \):
\[ y = \frac{15(6) + 20}{11} = \frac{110}{11} = 10 \quad \text{(is an integer)} \]

7. **Determine the original ratio:**
Now we have \( x = 6 \) and \( y = 10 \). Therefore, the original ratio of the fraction is:
\[ \frac{x}{y} = \frac{6}{10} = \frac{3}{5} \]

### Final Answer:
The original ratio of the fraction is \( 3:5 \).