Two numbers are in the ratio 3 : 5. If 9 be subtracted from each, then they are in the ratio 12 : 23. The second number is:

Question

Accepted Answer

To solve the problem, we need to determine the second number based on the given conditions. Let's break down the problem step by step:

1. **Understanding the Ratio**:  
   The two numbers are in the ratio 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, we have:
   $$
   \frac{3x - 9}{5x - 9} = \frac{12}{23}
   $$

3. **Cross-Multiplying to Solve for $x$**:  
   Cross-multiply to eliminate the fractions:
   $$
   23(3x - 9) = 12(5x - 9)
   $$

4. **Expanding Both Sides**:  
   $$
   69x - 207 = 60x - 108
   $$

5. **Solving for $x$**:  
   Rearrange the equation to solve for $x$:
   $$
   69x - 60x = 207 - 108
   $$
   $$
   9x = 99
   $$
   $$
   x = 11
   $$

6. **Finding the Second Number**:  
   The second number is $5x$. Substitute $x = 11$ into the expression for the second number:
   $$
   5x = 5 \times 11 = 55
   $$

Therefore, the second number is 55, which corresponds to Option 4.

**Explanation for Incorrect Options**:

- **Option 1 (52)**: If the second number were 52, the first number would be $\frac{3}{5} \times 52 = 31.2$, which is not an integer. Additionally, subtracting 9 from both numbers would not yield a ratio of 12:23.

- **Option 2 (53)**: Similarly, if the second number were 53, the first number would be $\frac{3}{5} \times 53 = 31.8$, which is not an integer. Subtracting 9 from both numbers would not result in the required ratio.

- **Option 3 (54)**: If the second number were 54, the first number would be $\frac{3}{5} \times 54 = 32.4$, which is not an integer. Again, subtracting 9 from both numbers would not give the ratio 12:23.

Thus, Option 4 is the only correct choice, as it satisfies all the conditions given in the problem.