Problem 44 - Algebra 1

Question

A baseball player's batting average is found by dividing the number of hits \(h\) by the official times at bat \(b .\) During the 1998 baseball season, Alex Rodriguez of the Seattle Mariners had 686 official times at bat and made 213 hits. Use a verbal model to find his batting average. Round your answer to the nearest thousandth.

Step-by-Step Solution

Answer

Alex Rodriguez's batting average for the 1998 baseball season was approximately 0.311 when rounded off to the nearest thousandth.

Step 1: Understanding the Problem Statement

Note down the data provided in the problem: the Number of Hits \(h = 213\) and Official Times at Bat \(b = 686\). Batting average is calculated by dividing the number of hits \(h\) by the official times at bat \(b\).

Step 2: Formulating the Mathematical Equation

As stated earlier, the batting average is calculated using the formula: Batting Average = \(h / b\). Substituting given values, it becomes Batting Average = \(213 / 686\).

Step 3: Calculation and Rounding Off

Using a calculator, perform the division operation \(213 / 686\) to get 0.310497. But since the problem mentioned answering to the nearest thousandth, we need to round off the calculated result to three decimal digits. Hence, 0.3105 is the batting average.

Key Concepts

Verbal Model

A verbal model is a way of transforming a word problem into something we can calculate. It's about translating descriptions into expressions or equations. When dealing with a batting average, a verbal model helps us to understand the relationship between different quantities involved, like hits and times at bat. In this exercise, we begin by identifying what we need to find: the batting average. We know it's calculated by looking at two numbers: - **Number of Hits (h):** This is the total times a player successfully hits the ball. - **Official Times at Bat (b):** This is the total times a player has batted, not counting walks or sacrifices. So, the verbal model here is: "The batting average is the number of hits divided by the official times at bat." By turning descriptive words into a mathematical relationship, we move one step closer to finding a numerical solution.

Mathematical Equation

Once we have a verbal model, the next step is to write a mathematical equation. This involves using mathematical symbols and operations to represent the verbal model. For a batting average, we use the equation:\[\text{Batting Average} = \frac{h}{b}\]In our example, Alex Rodriguez has:- **213 Hits** - **686 Times at Bat**Substituting these values into the equation gives us:\[\text{Batting Average} = \frac{213}{686}\]The equation is a compact representation of the verbal model. It's just a more precise way to communicate the same relationship. By solving this equation, we can find the actual batting average value.

Rounding Numbers

After finding a mathematical solution, rounding numbers is often needed to provide a more understandable answer. Rounding is the process of adjusting a number to ensure ease of communication and to meet specified precision standards.For the batting average, after calculating \( \frac{213}{686} \), we get a long decimal: 0.310497. The problem asks us to present this to the nearest thousandth. Here's how to round the number:

• Locate the thousandth place, which is three digits to the right of the decimal point.
• Look at the digit immediately to the right of the thousandth place (in this case, the digit 9).If this digit is 5 or more, round the thousandth place up.
• Since 9 is greater than 5, we increase the digit in the thousandth place (0) by one, making it 5.

Thus, 0.310497 becomes 0.3105, providing a clearer, rounded batting average.