Solving Age Problems with Algebra: A Step-by-Step Guide

Algebra is a powerful tool for solving problems, especially those involving relationships between variables. One common type of problem involves determining the ages of individuals based on given conditions. For instance, let's explore the problem of finding Andy's age in four years, given that Barbara is six years old and Andy is twice her age.

Problem: Barbara is 6 years old. Andy is double Barbararsquo;s age. How old will Andy be in four years?

Setting Up the Equations

To solve this, we can use algebraic variables. Let's denote:

x as Andy's current age, 2x - 6 as Andy's age four years ago (since Barbara is currently 6 and Andy is double that age).

Step-by-Step Solution:

Let the boy's present age be (x). In four years, the boy's age will be (x 4). Six years ago, the boy's age was (x - 6). We know that in four years, Andy's age will be double his age six years ago. Therefore, we can write the equation: x 4 2(x - 6). Solving the equation: (x 4 2x - 12) Subtract (x) from both sides: (4 x - 12) Add 12 to both sides: (x 16) This means Andy is currently 16 years old. In four years, Andy will be (16 4 20). Check the solution by verifying that Andy's age in four years (20) is indeed double his age six years ago (10): (2 times 10 20).

Solution:

Andy will be 20 years old in four years.

Understanding Linear Equations in Age Problems

Linear equations are fundamental in solving age problems. They help represent the relationship between two variables based on given conditions. In this problem, the condition that Andy's age in four years is double his age six years ago was key to setting up the equation:

x 4 2(x - 6)

This type of problem requires translating the given information into a mathematical equation and then solving it step by step. The key steps include:

Identifying the variables and what they represent. Setting up the equation based on the given conditions. Solving the equation to find the unknown variable. Verifying the solution by substituting it back into the original problem.

Practice Problems

Now you can try your hand at solving similar age problems:

If Sarah is 8 years old now and her sister Emily is 4 years older, how old will Emily be in three years? James is 12 years old. His brother Tommy is three times his age. Will Tommy be four times as old as James' age six years ago in five years? If Peter is 5 years older than his sister Lily and Lily is 9 years old, how old will Peter be in two years?

By practicing these types of problems, you can improve your algebraic problem-solving skills and enhance your understanding of how linear equations can be applied to real-life scenarios.