Problem 54 - Algebra 1: Concepts and Skills

Question

You have 35 hits in 140 times at bat. Your batting average is \(\frac{35}{140}=0.250 .\) How many consecutive hits must you get to increase your batting average to \(0.300 ?\) Use the following verbal model to answer the question. Desired Batting average \(=\frac{\text { Past hits }+\text { Future hits }}{\text { Past times at bat }+\text { Future times at bat }}\)

Step-by-Step Solution

Answer

You must get 10 consecutive hits to increase your batting average to 0.300.

Step 1: Identify the Knowns and Unknowns

The past hits is 35, past times at bat is 140, and the desired batting average is 0.300. The future hits is unknown but will be equal to the future times at bat as each successful attempt increases the hits and times at bat by 1. Therefore, it can be represented as \(x\).

Step 2: Formulate the Equation

Using these variables, the model as per the exercise becomes: \(0.300 = \frac{35 + x}{140 + x}\)

Step 3: Solve the Equation

Rearranging the equation to solve for \(x\) we have: \(0.300*(140 + x) = 35 + x\). Simplify it to: \(42 + 0.3x = 35 + x\). Further simplifying gives: \(7 = 0.7x\). Then solve for \(x\): \(x = \frac{7}{0.7} = 10\).

Key Concepts

Algebraic Equations

Algebraic equations are at the heart of solving many mathematical problems, particularly when it comes to finding unknown values.
An algebraic equation is a statement of equality between two expressions that contain numbers, variables (representing unknown values), and arithmetic operations like addition, subtraction, multiplication, and division. For example, in the batting average problem, we used the equation \(0.300 = \frac{35 + x}{140 + x}\) to represent the relationship between the number of hits, times at bat, and the desired batting average.
This equation includes numbers that we know from the problem (35 and 140), the variable \(x\) representing an unknown quantity we wish to solve for (future hits and future times at bat), and the desired batting average. Solving such equations involves manipulating the expressions to isolate the variable, a process that often includes combining like terms, distributing multiplication over addition or subtraction, and inversely applying arithmetic operations.

Verbal Models in Algebra

Verbal models are a way to translate word problems into algebraic equations. They allow us to describe a scenario or a logic puzzle using mathematical language.

Building a Verbal Model

To create a verbal model, you first need to define what the words mean in terms of mathematical operations and variables. In our batting average problem, we were given a description that could be translated this way: the desired batting average equals the sum of past and future hits divided by the sum of past and future times at bat. This is represented algebraically as \(\text{desired batting average} = \frac{\text{past hits} + \text{future hits}}{\text{past at bats} + \text{future at bats}}\).

Applying the Model

After we express the relationship verbally, we can turn it into an equation that can be solved. The exercise improvement advice often emphasizes interpreting the text correctly to build accurate verbal models. This step minimizes errors and increases understanding of the algebraic process. Building a strong verbal model is crucial as it serves as a guide to form the algebraic expression that you will manipulate to find the unknown.

Solving for Unknowns

In algebra, solving for unknowns is essentially a detective work where you decipher clues to find the value of a variable.

Isolating the Variable

The main goal when solving equations is to isolate the variable on one side of the equation. This requires performing operations that 'undo' other operations. For example, if an equation has a term \( +5 \), you'll use \( -5 \), its opposite, to cancel that out on both sides of the equation.
Using the problem at hand, once we have an equation like \(0.300*(140 + x) = 35 + x\), we perform several steps - distribute the 0.300, combine like terms, and eventually isolate \(x\) to find its value.

Checking Your Solution

Once you find the value of the unknown, it's important to verify it by plugging it back into the original equation. In our case, when we solve \(x = \frac{7}{0.7}\), we get \(x = 10\). We check this by replacing \(x\) with 10 in the original equation to ensure that the left and right sides are equal, reaffirming the correctness of our solution.