Age Riddle Solution: A Mathematical Approach

Mathematics can be a fascinating tool for solving age-related riddles. This article explores a particular age problem using both algebraic and graphical methods. We will solve the equation step-by-step to find the ages of a man and his son.

The Problem Recap

One year ago, a man was 8 times as old as his son. Now, his age is equal to the square of his son's age. The question is, what are their present ages?

Solving the Problem: Algebraic Method

Let's denote the son's present age as s years. Then, one year ago, the son's age was s - 1 years, and the man's age was 8s - 1.

The man's present age can be expressed as:

8s - 1 1 8s - 8 1 8s - 7

According to the problem, the man's present age is equal to the square of his son's present age:

8s - 7 s^2

Rearranging this equation:

s^2 - 8s 7 0

This is a quadratic equation. We can solve it using the quadratic formula s frac{-b pm sqrt{b^2 - 4ac}}{2a}.

Here, a 1, b -8, and c 7;

The discriminant is:

b^2 - 4ac (-8)^2 - 4 cdot 1 cdot 7 64 - 28 36

Applying the quadratic formula:

s frac{8 pm sqrt{36}}{2 cdot 1} frac{8 pm 6}{2}

This gives us two possible values for s:

s frac{14}{2} 7

s frac{2}{2} 1

Since s 1 implies the man was 8 years old a year ago, making him only 9 currently, which doesn't satisfy the condition of his age being the square of his son's age, we discard s 1 and take s 7.

Substituting s 7 back into the equation for the man's age:

Man's age 8s - 7 8(7) - 7 49 - 7 49

Thus, the present ages are:

- Son: 7 years old

- Man: 49 years old

Verification

To verify, we check if the man was 8 times the son's age one year ago:

49 - 1 48

8 times 7 - 1 56 - 1 55

Since 49 - 1 8 times (7 - 1), the solution satisfies the given conditions.

Alternative Approach: Systems of Equations

From the premises and assumptions, we have:

F - 8 4S - 8

F 8 2S 8

Let F the father's current age

Let S the son's current age

Let F - 8 the father's age 8 years ago

Let S - 8 the son's age 8 years ago

Let F 8 the father's age in 8 years

Let S 8 the son's age in 8 years

A system of two equations in two variables:

F - 4S - 8 and F - 2S - 8

F - 4S - (-32) and -2S - (-16)

S -32 / -2 16

Therefore, if S 16, then:

F 4S - 24 4(16) - 24 64 - 24 40

Hence, F S 40 16

Conclusion

This problem involves using mathematical tools like quadratic equations and systems of equations to solve an age riddle. The solution reveals that the son is 7 years old, and the man is 49 years old, satisfying all given conditions.

Key Takeaways

Age problems can be effectively solved using algebraic methods. Quadratic equations help find solutions to problems where an age relationship is non-linear. Systems of equations are useful for solving problems with multiple conditions.

Understanding these methods can help in solving a wide range of similar problems in various contexts.