Problem 32 - 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: Define Variables

Let \( x \) be the number of home runs Babe Ruth hit. Since Hank Aaron hit 41 more home runs than Babe Ruth, Hank Aaron hit \( x + 41 \) home runs.

Step 2: Set Up the Equation

According to the problem, together they hit 1469 home runs. Therefore, we can set up the equation: \[ x + (x + 41) = 1469 \]

Step 3: Simplify the Equation

Combine like terms in the equation: \[ 2x + 41 = 1469 \]

Step 4: Solve for x

Subtract 41 from both sides to isolate the term with \( x \): \[ 2x = 1469 - 41 \] \[ 2x = 1428 \]

Step 5: Divide to Find x

Divide both sides by 2 to solve for \( x \): \[ x = \frac{1428}{2} \] \[ x = 714 \]

Step 6: Interpret the Result

Babe Ruth hit 714 home runs during his career.

Key Concepts

The Art of Problem-Solving in Algebra

Problem-solving in algebra involves breaking down the given information into smaller, manageable parts. This makes understanding and solving equations straightforward. In this exercise, we have an interesting problem involving the calculation of home runs by two famous baseball players. Here's the simple process to tackle such problems:

1. **Read and Understand:** Comprehend the scenario or story behind the problem first. Know what is being asked and what information is given.
2. **Identify Key Quantities:** In this question, the total home runs are 1469, with Hank Aaron having 41 more than Babe Ruth.

3. **Define Variables:** Use variables like 'x' to represent unknown quantities. This forms the basis of your equation.

Follow these steps, and problem-solving in algebra becomes a clear and logical process. Breaking down complex scenarios into simpler forms helps in finding effective solutions.

Understanding Variables in Algebra

Variables play a crucial role in algebra. They are symbols, usually letters, that stand for numbers we need to find. In the given exercise, we use the variable \( x \) to represent the unknown quantity, which is the number of home runs Babe Ruth hit.

Here's what makes variables so handy:

• **Flexibility:** They can represent any number, allowing you to work with unknowns until they are finally solved.
• **Simplification:** Use variables to simplify sentences into shorter and easier-to-handle algebraic expressions or equations.
• **Translation:** Translate real-world situations into mathematical language using variables.
  

For instance, when Hank Aaron's home runs are described as 41 more than Babe Ruth's, the expression \( x + 41 \) uses a variable to efficiently capture this relationship. Understanding and using variables correctly is foundational to mastering algebra.

Solving Linear Equations

Linear equations are equations where variables like \( x \) are raised to the power of one. They have a straightforward solution path. In our problem, the equation formed is \( x + (x + 41) = 1469 \). Here's how we solve such equations:

1. **Combine Like Terms:** Add or subtract terms with the same variable to simplify the equation. In our step-by-step solution, this simplifies to \( 2x + 41 = 1469 \).

2. **Isolate the Variable:** Move terms from one side of the equation to the other using addition or subtraction, keeping operations balanced on both sides.
Example: Subtract 41 to bring all constant terms to one side, leading to \( 2x = 1428 \).

3. **Solve for the Variable:** Once the variable is isolated, perform operations to solve for it. Here, dividing both sides by 2 helps us find \( x = 714 \).

Understanding each of these steps helps solve linear equations effortlessly, building confidence in tackling various algebraic problems.