Let's solve the problem step by step.

Step 1: Define the Variables

• Let the present age of the son be $x$ years.
• Let the present age of the father be $y$ years.

Step 2: Set Up the Equations

We have two conditions based on the problem statement:

1. Condition 1 (3 years ago): $y - 3 = 4(x - 3)$ This means that 3 years ago, the father's age was 4 times the son's age.

2. Condition 2 (10 years in the future): $y + 10 = 2(x + 10) + 5$ This means that 10 years from now, the father's age will be 5 more than 2 times the son's age.

Step 3: Solve the Equations

Simplify the first equation:

$y - 3 = 4x - 12$ Add 3 to both sides: $y = 4x - 9 \quad \text{(Equation 1)}$

Simplify the second equation:

$y + 10 = 2x + 20 + 5$ Simplify further: $y + 10 = 2x + 25$ Subtract 10 from both sides: $y = 2x + 15 \quad \text{(Equation 2)}$

Step 4: Solve the System of Equations

Now, we'll solve the two equations:

1. $y = 4x - 9$
2. $y = 2x + 15$

Since both expressions equal $y$, set them equal to each other: $4x - 9 = 2x + 15$ Subtract $2x$ from both sides: $2x - 9 = 15$ Add 9 to both sides: $2x = 24$ Divide by 2: $x = 12$

Now that we have $x = 12$ (the son's current age), substitute it back into either equation to find $y$.

Using $y = 2x + 15$: $y = 2(12) + 15 = 24 + 15 = 39$

Final Answer

• The son's present age is $12$ years.
• The father's present age is $39$ years.

Would you like more details on any of the steps or have any other questions?

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Here are five related questions you might be interested in:

1. How would the problem change if the father's age was 3 times the son's age 3 years ago?
2. What if the father's age in 10 years will be exactly 2 times the son's age, without any extra years added?
3. Can you derive a general formula for these types of age problems?
4. How does changing the time frames (like 5 years ago or 20 years in the future) affect the solution?
5. How would the solution change if both father and son had the same age difference but different ratios?

Tip: When dealing with age problems, carefully define your variables and set up equations based on the relationships given at different times.