Two numbers are in the ratio `3 : 5`. If `8` is added to each number, the ratio becomes `2 : 3`. Find the numbers.

Answer

To solve the problem step by step, we will denote the two numbers as \( x \) and \( y \).

### Step 1: Set up the ratio
Given that the two numbers are in the ratio \( 3:5 \), we can express this as:
\[ \frac{x}{y} = \frac{3}{5} \]
From this, we can derive the first equation:
\[ 5x = 3y \quad \text{(Equation 1)} \]

### Step 2: Set up the second condition
We are also given that if \( 8 \) is added to each number, the ratio becomes \( 2:3 \). This can be expressed as:
\[ \frac{x + 8}{y + 8} = \frac{2}{3} \]
Cross-multiplying gives us:
\[ 3(x + 8) = 2(y + 8) \]
Expanding this, we have:
\[ 3x + 24 = 2y + 16 \]
Rearranging gives us the second equation:
\[ 3x - 2y = -8 \quad \text{(Equation 2)} \]

### Step 3: Solve the equations
Now we have two equations:
1. \( 5x - 3y = 0 \) (Equation 1)
2. \( 3x - 2y = -8 \) (Equation 2)

We can solve these equations simultaneously. From Equation 1, we can express \( y \) in terms of \( x \):
\[ 3y = 5x \implies y = \frac{5}{3}x \]

### Step 4: Substitute \( y \) into Equation 2
Now, substitute \( y = \frac{5}{3}x \) into Equation 2:
\[ 3x - 2\left(\frac{5}{3}x\right) = -8 \]
This simplifies to:
\[ 3x - \frac{10}{3}x = -8 \]
To combine the terms, we convert \( 3x \) to a fraction:
\[ \frac{9}{3}x - \frac{10}{3}x = -8 \]
This gives:
\[ -\frac{1}{3}x = -8 \]
Multiplying both sides by \( -3 \):
\[ x = 24 \]

### Step 5: Find \( y \)
Now that we have \( x \), we can find \( y \) using Equation 1:
\[ 5x = 3y \implies 5(24) = 3y \implies 120 = 3y \implies y = 40 \]

### Final Answer
The two numbers are:
\[ x = 24, \quad y = 40 \]