Introduction

In mathematical problem-solving, age problems often require the use of algebraic equations to determine unknown values. This article will explore a specific age problem involving a mother and her son, where the ages of the mother and the son are related both in the past and in the future. By using algebraic equations, we can determine their present ages and the ratio between them. This process can be particularly useful for students and educators alike who are studying algebra and its applications.

The Problem

Let's define:

Current age of the mother: M Current age of the son: S

According to the problem:

Eight years ago, the mother's age was three times the son's age. In ten years, the mother's age will be twice the son's age.

Formulating Equations

Next, let's formulate the equations based on the given information. We'll use the same variables for the current ages of the mother and the son as defined earlier.

1. Finding the mother's current age (M)

Eight years ago, the ages of the mother and the son were M - 8 and S - 8, respectively. According to the problem, the mother's age was three times the son's age at that time:

M - 8 3(S - 8)

2. Finding the son's current age (S)

In ten years, the ages of the mother and the son will be M 10 and S 10, respectively. According to the problem, the mother's age will be twice the son's age at that time:

M 10 2(S 10)

Solving the Equations

Let's solve these two equations to find the current ages of the mother and the son.

3. Equation from Step 1

M - 8 3(S - 8)

By expanding this equation, we get:

M - 8 3S - 24

4. Equation from Step 2

M 10 2(S 10)

By expanding this equation, we get:

M 10 2S 20

5. Simplifying the Equations

Let's simplify each equation:

M - 3S -16

M - 2S 10

Solving for S

Subtract the second simplified equation from the first:

(M - 3S) - (M - 2S) -16 - 10

-S -26

S 26

Solving for M

Using the value of S in the second simplified equation:

M - 2*26 10

M - 52 10

M 62

The Solution

Thus, the current age of the mother (M) is 62 years and the current age of the son (S) is 26 years.

The Ratio of Their Ages

Now, let's find the ratio of their current ages:

M:S 62:26

By simplifying this ratio, we get:

M:S 31:13

Conclusion

Through solving the given age problem using algebraic equations, we determined the current ages of a mother and her son. The mother is currently 62 years old, and the son is 26 years old, resulting in a ratio of 31:13. This example illustrates the usefulness of algebraic equations in solving practical problems involving age relationships.