Problem 74 - Algebra for College Students

Question

Scott scored 92 and 96 on his first two tests in "Methods in Teaching Mathematics." What score must he make on his third test to keep an average of 90 or greater?

Step-by-Step Solution

Answer

Scott must score at least 82 on his third test.

Step 1: Understand the problem

Scott wants an average score of at least 90 across three tests. We need to find the score of his third test that will ensure this average.

Step 2: Set up the equation for average

The average of three test scores is given by \[\text{Average} = \frac{\text{Score 1} + \text{Score 2} + \text{Score 3}}{3} \] Given: Score 1 = 92 Score 2 = 96 Score 3 = x

Step 3: Insert the known values into the equation

Substitute the known scores and the desired average into the equation: \[90 = \frac{92 + 96 + x}{3} \]

Step 4: Solve the equation for the unknown score (x)

Multiply both sides of the equation by 3 to clear the fraction: \[270 = 92 + 96 + x \] Combine the known scores: \[270 = 188 + x\] Subtract 188 from both sides to solve for x: \[x = 270 - 188\] \[x = 82\]

Step 5: Verify the result

Check the solution by calculating the average with the found value of the third score. \[\text{Average} = \frac{92 + 96 + 82}{3} = \frac{270}{3} = 90 \] This confirms that an 82 on the third test will ensure an average of 90.

Key Concepts

algebra

Algebra is a branch of mathematics dealing with symbols and the rules for manipulating those symbols. In our exercise, algebra comes into play when setting up the equation to find the unknown test score. Algebra often involves combining like terms, using variables to represent unknowns, and solving equations. Understanding algebra is essential for solving various types of mathematical problems because it forms the basis for more advanced topics. For example, in this exercise, we represented the unknown test score with the variable \(x\), simplifying the problem-solving process.

equation solving

Equation solving involves finding the value(s) that satisfy a given mathematical statement, often called an equation. In the context of our exercise, we set up an equation to find out what score Scott needs on his third test to keep an average of 90. The steps were:

• Understand the problem and determine what we need to find.
• Write down the equation based on the given information.
• Substitute the known values into the equation.
• Solve for the unknown variable by isolating it on one side of the equation.

Equation solving is a crucial skill because it enables us to find unknowns in various contexts, like determining a necessary test score to achieve a desired average.

mathematical problem-solving

Mathematical problem-solving is the process of identifying, analyzing, and solving problems using mathematical concepts and techniques. For the given exercise, problem-solving included several important steps:
1. Understanding what the problem is asking. Scott needs an average of at least 90 from three tests.
2. Setting up the right equation to represent the scenario.
3. Writing down the given data and the unknown we need to find.
4. Solving the equation using proper algebraic methods.
5. Verifying the result to ensure its correctness.
This step-by-step approach ensures that the solution is accurate and comprehensible, making mathematical problem-solving a systematic and logical process.