Problem 32 - Precalculus Mathematics for Calculus

Question

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's define variables for the unknowns. 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

We know that together they hit a total of 1469 home runs. So, we can write 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 \( x \): \[ 2x = 1469 - 41 \] \[ 2x = 1428 \]Now, divide both sides by 2: \[ x = \frac{1428}{2} \] \[ x = 714 \]

Step 5: Verify the Solution

Babe Ruth hit 714 home runs. Since Hank Aaron hit 41 more, he hit 714 + 41 = 755 home runs. Together, they hit 714 + 755 = 1469 home runs, which matches the given total.

Key Concepts

Equation Solving

Equation solving is the process of finding the value of variables that satisfy the condition of an algebraic equation. In many real-world scenarios, we deal with situations involving unknown quantities that can be represented as variables. In this exercise, equation solving helps us find out how many home runs Babe Ruth hit during his career by utilizing the information about Hank Aaron's performance.

The process begins with setting up an equation based on the problem statement. We then apply algebraic operations such as addition, subtraction, multiplication, and division to isolate the variable. The key is to maintain balance in the equation, ensuring the equality holds true throughout the manipulations. This is typically done in a few clear steps:

• Combine like terms: If there are similar terms on the same side of the equation, add or subtract them to simplify.
• Isolate the variable: Use inverse operations to get the variable by itself on one side of the equation.
• Solve for the variable: Once isolated, perform any necessary calculations to determine the value of the variable.
• Verify the solution: Substitute the found value back into the original equation to ensure it satisfies the condition given.

These steps provide a structured approach to solving equations and finding solutions to word problems.

Defining Variables

Defining variables is a crucial step when translating word problems into algebraic expressions or equations. A variable acts as a placeholder representing the unknown quantity we seek to determine. In our exercise, the home runs hit by Babe Ruth were not known, so we represented this unknown number with the variable \( x \).

To appropriately define variables, here are some helpful tips:

• Read the problem carefully: Identify the quantities that you do not know but need to find.
• Choose clear symbols: Use letters like \( x \), \( y \), or \( z \) to represent unknowns. Make sure the choice is logical and consistent.
• State what each variable represents: Clearly mention what the variable stands for, e.g., "Let \( x \) be the number of home runs Babe Ruth hit."
• Account for relationships: Recognize how different quantities relate to each other, as in "Hank Aaron hit 41 more than \( x \)."

Defining variables properly sets the stage for setting up equations correctly and ensures that you can straightforwardly translate the problem's words into mathematical operations.

System of Equations

While this particular problem involved solving a single equation, understanding systems of equations is essential for more complex problems. A system of equations consists of two or more equations with the same set of unknowns. They are solved simultaneously when each equation is considered at once to find a common solution.

Here's a simple process on how to approach systems of equations:

• Write all equations: Make sure all the conditions of the problem are represented as equations.
• Simplify each equation: Make sure each equation is in its simplest form.
• Use substitution or elimination methods: In substitution, solve one equation for a variable and substitute in others. In elimination, add or subtract equations to cancel out a variable.
• Check your solutions: Substitute back into original equations to verify correctness.

Systems of equations allow us to tackle problems where more than one condition needs to be satisfied, giving a comprehensive solution to complex real-life scenarios. Although in this exercise we dealt with a single equation, using similar logic and techniques can help solve more intricate problems involving multiple unknowns.