A man is now 8 times as old as his son. In eight years the man will be 4 times as old as his son. Find the present age of the man and his son.
Understand the Problem
The question is asking us to determine the current ages of a man and his son based on the given conditions about their ages now and in the future. We can set up a system of equations to represent the relationships described and then solve for the present ages.
Answer
The man is $40$ years old and the son is $10$ years old.
Answer for screen readers
The current age of the man is $40$ years, and the current age of the son is $10$ years.
Steps to Solve
- Define Variables
Let $m$ be the current age of the man and $s$ be the current age of the son.
- Set Up the Equations
Based on the problem, we can formulate the following equations:
-
The problem states that in 5 years, the man will be 3 times as old as his son. So, we can express this as:
$$ m + 5 = 3(s + 5) $$ -
We also know that the man is currently 4 times as old as his son. This can be expressed as:
$$ m = 4s $$
- Substitute and Simplify
Now we will substitute the second equation into the first one. Replace $m$ in the first equation with $4s$:
$$ 4s + 5 = 3(s + 5) $$
Now simplify this equation:
$$ 4s + 5 = 3s + 15 $$
- Solve for s
To solve for $s$, first, isolate $s$ on one side of the equation:
$$ 4s - 3s = 15 - 5 $$
This simplifies to:
$$ s = 10 $$
- Find m
Now that we have $s$, we can find $m$ using the second equation:
$$ m = 4s = 4(10) = 40 $$
- Conclusion
Thus, the current ages are:
- Man's age: $m = 40$
- Son's age: $s = 10$
The current age of the man is $40$ years, and the current age of the son is $10$ years.
More Information
This problem illustrates how to set up and solve a system of equations based on real-world age relationships. In many scenarios, age-related problems involve linear equations.
Tips
- Incorrect substitution: Make sure the variables are substituted correctly in each equation. Double-check the initial conditions.
- Sign errors: Be careful with signs when moving terms from one side of the equation to another. Small mistakes can lead to incorrect age calculations.
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