The ratio of ages of A & B after 5 years will be 3 : 5. If the sum of ages of A & B at present is 38 years. Find the ages of A & B before 5 years.

Answer

To solve the problem step by step, we will first define the current ages of A and B, and then use the information given to find their ages before 5 years.

### Step 1: Define the current ages of A and B
Let the current age of A be \( a \) years and the current age of B be \( b \) years.

### Step 2: Set up the equations based on the problem statement
According to the problem, we have two pieces of information:
1. The sum of the ages of A and B at present is 38 years:
\[ a + b = 38 \quad \text{(Equation 1)} \]
2. The ratio of their ages after 5 years will be 3:5:
\[ \frac{a + 5}{b + 5} = \frac{3}{5} \]
Cross-multiplying gives:
\[ 5(a + 5) = 3(b + 5) \]
Expanding this:
\[ 5a + 25 = 3b + 15 \]
Rearranging gives us:
\[ 5a - 3b = -10 \quad \text{(Equation 2)} \]

### Step 3: Solve the system of equations
Now we have a system of two equations:
1. \( a + b = 38 \) (Equation 1)
2. \( 5a - 3b = -10 \) (Equation 2)

From Equation 1, we can express \( b \) in terms of \( a \):
\[ b = 38 - a \]

Substituting this expression for \( b \) into Equation 2:
\[ 5a - 3(38 - a) = -10 \]
Expanding this gives:
\[ 5a - 114 + 3a = -10 \]
Combining like terms:
\[ 8a - 114 = -10 \]
Adding 114 to both sides:
\[ 8a = 104 \]
Dividing by 8:
\[ a = 13 \]

### Step 4: Find the age of B
Now substituting \( a = 13 \) back into Equation 1 to find \( b \):
\[ 13 + b = 38 \]
\[ b = 38 - 13 = 25 \]

### Step 5: Find the ages of A and B before 5 years
To find the ages of A and B before 5 years, we subtract 5 from their current ages:
- Age of A before 5 years:
\[ a - 5 = 13 - 5 = 8 \]
- Age of B before 5 years:
\[ b - 5 = 25 - 5 = 20 \]

### Final Answer
The ages of A and B before 5 years were:
- Age of A: 8 years
- Age of B: 20 years