Two numbers are in the ratio of 3 : 5. If 9 be subtracted from each, then ratio between numbers becomes 12 : 23. Second number is :

Question

Accepted Answer

To solve the problem, we need to determine the second number based on the given conditions.

1. **Understanding the Ratio:**
   - The two numbers are in the ratio of 3:5. Let's denote the first number as $3x$ and the second number as $5x$.

2. **Applying the Condition:**
   - According to the problem, if 9 is subtracted from each number, the new ratio becomes 12:23.
   - Therefore, the equation based on this condition is:
     $$
     \frac{3x - 9}{5x - 9} = \frac{12}{23}
     $$

3. **Solving the Equation:**
   - Cross-multiply to solve for $x$:
     $$
     23(3x - 9) = 12(5x - 9)
     $$
   - Simplify both sides:
     $$
     69x - 207 = 60x - 108
     $$
   - Rearrange to find $x$:
     $$
     69x - 60x = 207 - 108
     $$
     $$
     9x = 99
     $$
     $$
     x = 11
     $$

4. **Finding the Second Number:**
   - The second number is $5x$, so substitute $x = 11$:
     $$
     5x = 5 \times 11 = 55
     $$

5. **Verification:**
   - The second number is 55, which matches Option 2.
   - Subtract 9 from both numbers: 
     - First number: $3x - 9 = 33 - 9 = 24$
     - Second number: $5x - 9 = 55 - 9 = 46$
   - Check the new ratio:
     $$
     \frac{24}{46} = \frac{12}{23}
     $$
   - The condition is satisfied, confirming that the calculations are correct.

**Explanation for Incorrect Options:**

- **Option 1 (11):** This is the value of $x$, not the second number. The second number is $5x$, which equals 55.
- **Option 3 (50):** If the second number were 50, the ratio after subtracting 9 would not match the given condition of 12:23.
- **Option 4 (35):** Similarly, if the second number were 35, the resulting ratio after subtracting 9 would not satisfy the condition.

Thus, the correct answer is Option 2, which is 55.