Solving Age Ratio Problems with Algebraic Methods: A Comprehensive Guide

Introduction

Mathematics, particularly algebra, is a powerful tool in solving real-life problems, including those related to age ratios. This article explores how to solve age ratio problems using algebraic methods, focusing on scenarios where the ratio of ages of a mother and son is given and asks for their current ages. Let's dive into several examples and the methods to solve them step-by-step.

The Ratio of Ages: 5:2 for a Mother and Son

Suppose the ratio of the present ages of a mother and her son is 5:2. One straightforward method to find their current ages involves using variables and setting up algebraic equations based on the given ratio and additional information.

Example: Initial Ratio Method

Given that the ratio of the mother's age to the son's age is 5:2, let the mother's age be represented as 5x and the son's age as 2x.

Given that the mother's present age is 16, we can set up the equation: 5x 16.

Solving for x: x 16 / 5 3.2.

Substituting x back to find the son's age: 2x 2 × 3.2 6.4.

Hence, the son's age is 6.4 years.

Alternative Method

Another method to solve the same problem involves setting up a system of equations based on the given information.

Let MA be the mother's age and SA be the son's age. From the problem statement, we know:

The ratio of their ages is 5:2, so MA : SA 5 : 2.

After six years, the mother's age will be twice the age of her son: MA 6 2(SA 6).

Using these equations:

Let's solve these equations step-by-step:

From the first equation, we can write: MA 5k and SA 2k for some constant k.

Substitute MA and SA into the second equation: 5k 6 2(2k 6).

Expanding and simplifying: 5k 6 4k 12.

Solving for k: 5k - 4k 12 - 6, thus k 6.

Therefore, the mother's age is 5k 5 × 6 30 years and the son’s age is 2k 2 × 6 12 years.

Another Approach

Consider a third method involving the sum of their present ages:

Let x be the sum of their present ages. Then the mother’s age is (5/7)x and the son’s age is (2/7)x.

After six years:

The mother's age is (5/7)x 6.

The son's age is (2/7)x 6.

According to the problem, the mother's age after six years is twice the son's age after six years: (5/7)x 6 2((2/7)x 6).

Solving for x: (5/7)x 6 12/7x 12.

Subtract (12/7x) from both sides: (5/7 - 12/7)x 12 - 6.

Simplify: -7/7x 6, thus x 42.

The mother’s present age is (5/7) * 42 30 years and the son’s present age is (2/7) * 42 12 years.

Conclusion

Algebraic methods provide a structured and systematic way to solve age ratio problems. By setting up and solving equations, one can effectively find the present ages of individuals given a ratio of their ages at any point in time and additional conditions. Whether dealing with the simplest form or more complex scenarios, the principles remain fundamentally the same; employing algebra allows for a clear and accurate solution.