Solving Age-Related Word Problems Using Algebra

Mathematics, particularly algebra, offers powerful tools to solve age-related word problems by setting up and solving a system of equations. In this article, we will explore how to determine the present ages of individuals through a series of steps and examples. Let's begin with a classic problem:

The Classic Problem

A few years ago, a person's age was thrice another's. Ten years in the future, the age of the first person will be twice the age of the second. What are the present ages of both, and what is their sum?

Step-by-Step Solution

We start by setting up the equations based on the given conditions.

Five Years Ago

A was thrice as old as B five years ago.

Let the present ages of A and B be x and y.

Five years ago, the ages were:

A: x - 5

B: y - 5

The relationship is given by:

x - 5 3(y - 5)

Expanding and simplifying:

x - 5 3y - 15

x 3y - 10 … Equation 1

Ten Years Later

In ten years, A's age will be twice B's age.

Write the equation for ten years later:

x 10 2(y 10)

Expanding and simplifying:

x 10 2y 20

x 2y 10 … Equation 2

Solving the System of Equations

We now have a system of two linear equations:

x 3y - 10

x 2y 10

Equate the right-hand sides to find y:

3y - 10 2y 10

Solve for y:

3y - 2y 10 10

y 20

Substitute y 20 into Equation 1 to find x:

x 3(20) - 10 60 - 10 50

The present ages of A and B are:

A: 50 years B: 20 years

The sum of their ages:

50 20 70

Another Example

Let's explore a different scenario.

Problem: Four Years Ago

A was three times as old as B four years ago. What are the present ages of both A and B if A is 10 years older than B now?

Solution

Let:

A's present age: 3x

B's present age: 5x

Four years ago:

A: 3x - 4

B: 5x - 4

In ten years:

A: 3x 6 (since A is 10 years older, A's present age is 3x 6)

B: 5x 6

The ratio of their ages in ten years:

(3x 6) / (5x 6) 5/6

Cross multiply:

18x 36 25x 30

7x 6

x 2

Present ages:

A: 3(2) 6 12 6 18 B: 5(2) 6 10 6 16

The sum of their ages:

18 16 34

By setting up and solving these equations, we can effectively find the current ages of individuals and their sums.

Conclusion

Understanding how to set up and solve age-related word problems using algebra is crucial for solving a variety of real-world scenarios. Whether you are dealing with simple or complex situations, algebraic methods offer a clear and precise way to find solutions.