The ratio of the present ages of a father and his son is ` 3 : 1` . The sum of their ages after five years is 58. Find the ratio of their ages 3 years ago.

Answer

To solve the problem step by step, we will follow these instructions:

### Step 1: Define the Present Ages
Let the present age of the father be \(3x\) and the present age of the son be \(x\). This is based on the given ratio of their ages, which is \(3:1\).

### Step 2: Set Up the Equation for Future Ages
In 5 years, the father's age will be \(3x + 5\) and the son's age will be \(x + 5\). According to the problem, the sum of their ages after 5 years is 58. Therefore, we can set up the equation:
\[ (3x + 5) + (x + 5) = 58 \]

### Step 3: Simplify the Equation
Combine like terms in the equation:
\[ 3x + x + 5 + 5 = 58 \]
This simplifies to:
\[ 4x + 10 = 58 \]

### Step 4: Solve for \(x\)
Now, isolate \(x\) by subtracting 10 from both sides:
\[ 4x = 58 - 10 \]
\[ 4x = 48 \]
Now, divide both sides by 4:
\[ x = 12 \]

### Step 5: Calculate Present Ages
Now that we have \(x\), we can find the present ages:
- Father's age: \(3x = 3 \times 12 = 36\)
- Son's age: \(x = 12\)

### Step 6: Calculate Ages 3 Years Ago
To find their ages 3 years ago:
- Father's age 3 years ago: \(36 - 3 = 33\)
- Son's age 3 years ago: \(12 - 3 = 9\)

### Step 7: Find the Ratio of Their Ages 3 Years Ago
Now, we can find the ratio of their ages 3 years ago:
\[ \text{Ratio} = \frac{33}{9} \]
To simplify this ratio, we can divide both numbers by 3:
\[ \frac{33 \div 3}{9 \div 3} = \frac{11}{3} \]

### Final Answer
The ratio of their ages 3 years ago is \(11:3\).

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