Problem 7 - College Algebra

Question

For the following exercises, use the information to find a linear algebraic equation model to use to answer the question being asked. Beth and Ann are joking that their combined ages equal Sam's age. If Beth is twice Ann's age and Sam is 69 yr old, what are Beth and Ann's ages?

Step-by-Step Solution

Answer

Ann is 23 years old, and Beth is 46 years old.

Step 1: Define the Variables

First, let's define the variables to represent the unknowns in the problem. Let \( A \) represent Ann's age and \( B \) represent Beth's age. From the problem statement, we can identify that Beth is twice Ann's age, so \( B = 2A \).

Step 2: Formulate the Equation

According to the problem, Beth and Ann's combined ages equal Sam's age. Sam's age is given as 69 years. This can be written as the equation: \( A + B = 69 \).

Step 3: Substitute the Known Relationship

Replace \( B \) in the equation from Step 2 with \( 2A \) (since \( B = 2A \)). The equation becomes: \( A + 2A = 69 \).

Step 4: Simplify the Equation

Combine like terms in the equation: \( 3A = 69 \).

Step 5: Solve for Ann's Age

Divide both sides of the equation by 3 to solve for Ann's age: \( A = \frac{69}{3} = 23 \). So, Ann is 23 years old.

Step 6: Solve for Beth's Age

Use the relationship \( B = 2A \) to find Beth's age. Substitute \( A = 23 \) into the equation: \( B = 2 \times 23 = 46 \). So, Beth is 46 years old.

Key Concepts

Algebraic Modeling

When dealing with linear equations, algebraic modeling can be a helpful approach to solve problems involving unknown quantities. This technique involves representing real-life situations through mathematical equations in order to find solutions. The given exercise revolves around forming and solving a linear equation based on ages. In algebraic modeling, the first step is to convert the problem's description into mathematical expressions or equations.

In this problem, the ages of Beth and Ann are related to each other and to Sam's age. Therefore, we define variables: let \( A \) be Ann's age, and \( B \) be Beth's age. Importantly, Beth's age is expressed as double Ann's age, leading to the equation \( B = 2A \).

Algebraic models like these are particularly valuable because they turn verbal descriptions into equations that are solvable using basic algebraic principles. Thus, they simplify complex word problems by breaking them into clear, mathematical representations.

Age Problems

Age-related problems are a common topic in algebra as they often require solving linear equations. These problems usually involve determining unknown ages based on given relationships and conditions. Understanding these relationships is the key to solving such problems efficiently.

In the given exercise, we know:

• Beth and Ann's combined ages equals Sam's age.
• Beth is twice as old as Ann.
• Sam is 69 years old.

Given these points, the aim is to find Beth and Ann's ages. An important aspect is to express all necessary conditions as algebraic equations. In this scenario, it's crucial to write an equation for the sum of Beth and Ann's ages. Connecting this with Sam's known age helps form the equation \( A + B = 69 \), leading to the solution.

Problem-Solving Strategies

Adopting the right problem-solving strategies is essential for solving equations effectively. First and foremost, understanding the problem thoroughly sets the stage for everything that follows. Using suitable techniques and methods makes solving equations less daunting.

A vital strategy is to define variables correctly, as we did by assigning \( A \) for Ann and \( B \) for Beth. Then, identify the relationships between these variables and any other given numbers. Substituting known relationships simplifies the equation, in this case substituting \( B = 2A \) into \( A + B = 69 \) resulted in a straightforward equation: \( 3A = 69 \).

Lastly, solve the equation step-by-step. Simplifying by combining like terms, as seen here, allows us to solve for \( A \) directly: \( A = 23 \). The entire process highlights the importance of systematic problem-solving strategies to make algebra more approachable and manageable for students.