Age Puzzle Solved Using Algebra: An Interesting Mathematical Challenge

Mathematics often presents us with intriguing problems that can challenge our minds and test our algebraic skills. One such problem involves Ann and her son, a classic age puzzle that can be solved using algebra and systems of equations. Let's delve into this problem and see how we can find the solution.

The Problem

Ann is eighteen years older than her son. One year ago, Ann was three times as old as her son. How old are they now?

A) Method 1: Traditional Algebraic Approach

Let's denote Ann's current age as A and her son's current age as S. We can then represent the problem using the following equations:

Ann is eighteen years older than her son: A S 18 One year ago, Ann was three times as old as her son: A - 1 3(S - 1)

First, we substitute the first equation into the second equation:

S 18 - 1 3S - 3

This simplifies to:

S 17 3S - 3

Now, let's solve for S by combining like terms:

S - 3S -3 - 17

-2S -20

S 10

Now that we have the son's age, we can find Ann's age using the first equation:

A S 18 10 18 28

Thus, Ann is 28 years old, and her son is 10 years old.

B) Method 2: Revised Scenario

Let's consider another scenario where Ann is twice as old as her son now, and five years ago, she was three times as old as her son. Let's denote Ann's current age as M and her son's current age as S.

Ann's current age is twice her son's age: M 2S Five years ago, Ann was three times as old as her son: M - 5 3(S - 5)

We can use the second equation and simplify it:

M - 5 3S - 15

Since M 2S, we can substitute M in the equation:

2S - 5 3S - 15

Now, let's solve for S by combining like terms:

2S - 3S -15 5

-S -10

S 10

Substituting S 10 back into M 2S:

M 2 × 10 20

Thus, Ann is 20 years old, and her son is 10 years old.

C) Alternative Method: Direct Substitution

Another approach involves setting up a system of simultaneous equations and solving them:

M 2S

M - 5 3(S - 5)

If we substitute M 2S into the second equation:

2S - 5 3(S - 5)

This simplifies to:

2S - 5 3S - 15

Now, let's solve for S by isolating it:

2S - 3S -15 5

S 10

Substituting S 10 back into M 2S:

M 2 × 10 20

Therefore, Ann is 20 years old, and her son is 10 years old.

Proof and Validation

To validate our solution, we can check both the initial conditions:

Ann is 18 years older than her son: 20 - 10 18 One year ago, Ann was three times older than her son: (20 - 1) 19 and (3 × (10 - 1) 27 - 1 19

The solution satisfies both conditions, confirming its correctness.