Problem 24 - Algebra and Trigonometry

Question

Career Home Runs During his major league career, Hank Aaron hit 41 more home runs than Babe Ruth hit during his career. Together they hit 1469 home runs. How many home runs did Babe Ruth hit?

Step-by-Step Solution

Answer

Babe Ruth hit 714 home runs.

Step 1: Introduction to Variables

Let's denote the number of home runs Babe Ruth hit during his career as \( x \). According to the problem, Hank Aaron hit 41 more home runs than Babe Ruth, so Hank Aaron hit \( x + 41 \) home runs.

Step 2: Equation Formation

Together, both players hit 1469 home runs. Therefore, we can form the equation \( x + (x + 41) = 1469 \) to model the total number of home runs hit by both players.

Step 3: Simplification of the Equation

Simplify the equation from the previous step by combining like terms: \( 2x + 41 = 1469 \).

Step 4: Solving for x

Subtract 41 from both sides to get \( 2x = 1428 \). Then, divide both sides by 2 to find \( x \): \( x = \frac{1428}{2} = 714 \).

Step 5: Conclusion

Therefore, Babe Ruth hit 714 home runs during his career.

Key Concepts

Variables

In the realm of linear equations, the use of variables is crucial for representing unknown quantities. In our exercise, the variable is represented by the letter \(x\). It stands for the number of home runs Babe Ruth hit in his career. By using \(x\), we create a placeholder that may later be replaced with the actual number.

Variables make it simple to express relationships in mathematical sentences. For example, when the exercise mentions that Hank Aaron hit 41 more home runs than Babe Ruth, the expression \(x + 41\) accurately conveys this relationship using the variable \(x\).

Recognizing and defining variables is like setting up the stage for solving the problem. If you can properly identify what each variable represents, it simplifies the process of finding the solution.

Equation Formation

Equation formation involves translating word problems into mathematical equations. It is a skill that allows you to construct a problem-solving framework. In our exercise, the total number of home runs hit by both players is given as 1469.

To create the equation, note two components: Babe Ruth's home runs expressed as \(x\), and Hank Aaron's home runs as \(x + 41\). Together, they form the equation \(x + (x + 41) = 1469\) because we are adding the home runs of both players.

Using equations, we convert a problem written in words into an algebraic language. Equations systematically define relationships and dependencies. Most importantly, forming equations helps to lay out the road map to solutions.

Problem Solving

Problem solving involves breaking down complex issues into manageable steps. In the given exercise, once the equation \(x + (x + 41) = 1469\) is formed, the next task is to simplify and solve it.

Simplification begins by combining like terms to condense equations. For example, \(2x + 41 = 1469\) results from simplifying the initial equation. Tackling equations step by step not only makes them less daunting but also clarifies each operation's purpose.

To find the solution, the method of isolating \(x\) through operations like subtraction and division is employed. Subtract 41 from both sides, leading to \(2x = 1428\), and then divide by 2 to get \(x = 714\). This orderly approach determines that Babe Ruth hit 714 home runs. Problem solving in algebra is about applying logical steps to reach a reliable solution.