Problem 20 - Algebra 2 and Trigonometry

Question

Mrs. Gillis gave a test to her two classes of algebra. The mean grade for her class of 20 students was 86 and the mean grade of her class of 15 students was \(79 .\) What is the mean grade when she combines the grades of both classes?

Step-by-Step Solution

Answer

The combined mean grade is 83.

Step 1: Calculate the total sum of grades for each class

For the first class, multiply the mean by the number of students: \(20 \times 86 = 1720\). For the second class, multiply the mean by the number of students: \(15 \times 79 = 1185\).

Step 2: Combine the total sums from both classes

Add the total sums from both classes: \(1720 + 1185 = 2905\).

Step 3: Calculate the total number of students

Add the number of students from both classes: \(20 + 15 = 35\).

Step 4: Calculate the combined mean grade

Divide the combined total sum of grades by the total number of students: \(\frac{2905}{35} = 83\).

Key Concepts

Weighted Average

The concept of a weighted average is crucial when dealing with different group sizes. Here, the grades from two algebra classes are combined. Each class contributes to the overall average based on its size. Calculate each class's total grades first. For a weighted average:

• Multiply each group's mean by its size.
• Add these results together for the total grade sum.
• Divide by the total number of students to find the overall mean.

This way, we account for the fact that the classes have different numbers of students, providing a more accurate combined average grade. In this case, the weighted average ensures that both classes' performances are fairly represented.

Algebra

Algebra often involves manipulating numbers and variables to solve problems. In this exercise, algebraic concepts help compute the combined average grade from two classes. The key steps involve using algebraic expressions:

• Expression for each class's total grade: Class size multiplied by mean.
• Total grade for both classes: Sum of individual class totals.
• Total number of students: Sum of class sizes.

These straightforward algebraic steps showcase how combining information can solve real-world problems, such as determining grades or averages.

Arithmetic Mean

The arithmetic mean refers to the average of a set of numbers. It's a fundamental concept in statistics and mathematics, typically used to summarize data with a single value. For the arithmetic mean, follow these steps:

• Add all numbers in the set together.
• Divide the total by the count of numbers in the set.

In the exercise, the arithmetic mean is crucial for calculating both individual class averages and the combined mean. While each class has its distinct average, they collectively form a larger data set, allowing us to use the same arithmetic process to understand overall performance. This provides insight into how each group and the total contribute to the mean.