Problem 53 - Algebra and Trigonometry Real Mathematics, Real People

Question

To get an \(\mathrm{A}\) in a course, you must have an average of at least 90 on four tests of 100 points each. The scores on your first three tests were \(93,91,\) and 84 (a) Write a verbal model for the test average for the course. (b) What is the least you can score on the fourth test to get an \(A\) in the course?

Step-by-Step Solution

Answer

The least you can score on the fourth test to get an \(A\) in the course is 92.

Step 1: Model course average

A course average can be modelled with the equation \(Avg = \frac{t1 + t2 + t3 + t4}{4}\), where \(Avg\) is the course average and \(t1, t2, t3, t4\) are the scores on tests 1 through 4 respectively.

Step 2: Translate scores into model

Substitute the known test scores into the model: \(90 = \frac{93 + 91 + 84 + t4}{4}\). This gives the minimum average needed to get an A in terms of the score on the fourth test.

Step 3: Rearrange and solve

Rearrange the equation to solve for the minimum score on the fourth test: \(t4 = 4 * 90 - 93 - 91 - 84\). Therefore, \(t4 = 92\).

Key Concepts

Average Calculation

Understanding average calculation is essential across various subjects, but in algebra, it plays a unique role. A common scenario involves calculating the mean, or average, of a number of items or scores. To find an average, you sum up all the individual items and then divide by the number of items.

Let's break it down in a classroom context: say you have scores from multiple tests, and you need to find your overall average to determine your grade. If there are four tests each worth 100 points, to find the average score, you add up your scores from all four tests and divide by four. Mathematically, if your scores are represented as \( t1 \) for Test 1, \( t2 \) for Test 2, \( t3 \) for Test 3, and \( t4 \) for Test 4, the average \( Avg \) is calculated like so:
\[ Avg = \frac{t1 + t2 + t3 + t4}{4} \].

By mastering average calculation, you'll not only excel in algebra but also in real-life scenarios, such as calculating your overall performance in a course or budgeting monthly expenses.

Equation Solving

Equation solving is the cornerstone of algebra. It involves finding the values for variables that make a mathematical statement true. In our test scoring example, once the average calculation formula is set up, equation solving comes into play.

You would need to manipulate the formula to find the unknown score for the fourth test. Here's a step-by-step process to follow using the given scores of 93, 91, and 84:

• First, set up the average equation: \( Avg = \frac{t1 + t2 + t3 + t4}{4} \).
• Next, substitute known values into the equation: \( 90 = \frac{93 + 91 + 84 + t4}{4} \).
• Then, multiply both sides of the equation by 4 to eliminate the denominator.
• Now, subtract the known test scores from the result.
• Finally, you get the variable \( t4 \), which is the minimum score needed on the fourth test to achieve the desired average.

This orderly approach not only solves the algebraic problem but also teaches an organized method to handle equations of all types.

Test Scoring in Algebra

Test scoring in algebra often involves applying the concepts of average calculation and equation solving to determine what a student needs to score on future assessments to reach a particular grade. It is a practical application that motivates students to understand and appreciate algebraic concepts.

For instance, after calculating their average from three tests, a student may want to know the score needed on a fourth test to achieve a certain grade, say an \(A\). Using the scores: 93, 91, and 84, and knowing that an \(A\) requires at least an average of 90, the student can set up an equation with the fourth test score \( t4 \) as the unknown. Our previous sections have detailed how to establish and manipulate this equation.

With the conclusion that \( t4 \) must be at least 92, the student can confidently prepare for the fourth test with a clear target score in mind. Understanding how to apply these algebraic methods to test scoring helps students approach their studies strategically and mitigates the stress related to academic performance.