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

Age problems are a common type of algebraic equation that can be solved through the use of linear equations. These problems often require identifying relationships between the ages of individuals to form the necessary equation. In this article, we will walk you through the process of solving a real example of an age problem and provide additional practice problems for further practice.

Introduction to Age Problems

Age problems can often be complex and require a clear understanding of the relationships between the ages of the individuals mentioned in the problem. The key to solving these problems is setting up the correct equation based on the given information and solving it step by step.

Example Problem

Let's begin with an example problem: The sum of the ages of a man and his wife is 96 years. The man is 6 years older than his wife. How old is his wife?

Step-by-Step Solution

1. Define Variables

Let w represent the wife's age. Since the husband is 6 years older than the wife, his age can be represented as w 6.

2. Set Up the Equation

The problem states that the sum of their ages is 96 years. Therefore, we can write the equation as:

w (w 6) 96

3. Simplify and Solve

Now, let's simplify and solve the equation:

Step 1: Combine like terms: 2w 6 96 Step 2: Subtract 6 from both sides: 2w 90 Step 3: Divide both sides by 2: w 45

Therefore, the wife's age is 45 years.

4. Verify the Solution

To verify the solution, we check the husband's age:

Husband's age 45 6 51 Sum of their ages 45 51 96

The solution is correct as the sum of their ages matches the given information.

Additional Practice Problems

Now, let's explore a few more problems to solidify your understanding:

Problem 1

Description

The sum of a wife's and a husband's age is 79. He is 5 years older than her. How old is the husband?

Step-by-Step Solution

Define the Variables: Wife's age x Husband's age x 5 Equation: x (x 5) 79 Simplify: 2x 5 79 Solve: 2x 74, x 37 Therefore, the wife's age is 37, and the husband's age is: Husband's age 37 5 42

Verification

Wife's age Husband's age 37 42 79

Problem 2

Description

Step-by-step solution to the following problem: The wife's age is 39.5 when rounded, and the husband's age is 42.5 when rounded. Using these rounded values, solve for the exact ages using the given information.

Step-by-Step Solution

Define the Variables: Wife's age 39.5 - 2.5 37 Husband's age 39.5 2.5 42

Verification

Wife's age Husband's age 37 42 79

Conclusion

Solving age problems in algebra helps to improve your understanding of linear equations and their applications. By following the steps provided, you can easily solve these types of problems. Practice is key to mastering these concepts, and with consistent practice, you will become more proficient in solving complex age problems.