4. The Average Of G Numbers Is 50. If One Number, 40, Is Replaced By 70, What Is The New Average? 5. A Student's Scores Are 85, 90, And 80 On Three Tests. The First Two Tests Are Each Worth 25% Of The Final Grade, And The Third Test Is Worth 50%. What Is The Student's Weighted Average Score?

In the realm of mathematics, understanding averages and weighted scores is paramount. These concepts serve as foundational pillars for various applications, ranging from academic grading to statistical analysis. In this comprehensive guide, we will delve into the intricacies of calculating means and navigating weighted scores, equipping you with the knowledge and skills to conquer these mathematical challenges.

The mean, often referred to as the average, is a fundamental measure of central tendency. It represents the sum of a set of numbers divided by the total count of numbers in the set. In essence, the mean provides a single value that encapsulates the typical or central value within a dataset. Understanding how changes to a dataset affect the mean is crucial for data analysis and problem-solving.

1.1. The Impact of Replacement on the Mean

Let's consider a scenario where we have a set of G numbers with a mean of 50. This means that the sum of these G numbers is 50 * G. Now, imagine that one of these numbers, say 40, is replaced by 70. This replacement alters the sum of the numbers, and consequently, the mean. To calculate the new mean, we need to determine the change in the sum and then divide by the total number of values (G).

The initial sum of the numbers is 50 * G. When 40 is replaced by 70, the sum increases by 70 - 40 = 30. Therefore, the new sum becomes 50 * G + 30. To find the new mean, we divide this new sum by the total number of values, which is still G. The new mean is (50 * G + 30) / G, which can be simplified to 50 + 30/G. This formula highlights how the change in the sum (30) and the number of values (G) influence the new mean. For example, if G is 10, the new mean would be 50 + 30/10 = 53. This concept is vital for understanding how modifications within a dataset affect its average value.

1.2. Real-World Applications of Mean Calculation

The concept of the mean extends far beyond textbook problems. In everyday life, we encounter averages in various contexts. For instance, calculating the average score on a series of tests helps students gauge their academic performance. Similarly, determining the average fuel efficiency of a car provides insights into its operational costs. In business, calculating the average sales per month helps track revenue trends and forecast future performance. Understanding how the mean is affected by changes in the data is crucial in these applications. For example, a salesperson who closes a particularly large deal in one month will see their average sales increase significantly, while a month with unusually low sales will decrease the average. This knowledge enables informed decision-making and strategic planning in diverse fields.

Weighted scores are used when different items or scores have different levels of importance. Unlike a simple average, which treats all values equally, a weighted average assigns different weights to each value, reflecting their relative significance. This concept is particularly relevant in academic grading, where different assessments may contribute differently to the final grade. For example, a final exam might carry more weight than a class quiz, reflecting its comprehensive nature.

2.1. Calculating Weighted Averages

To calculate a weighted average, you multiply each value by its corresponding weight, sum the products, and then divide by the sum of the weights. This process ensures that values with higher weights contribute more to the final average. Let's consider a student who scored 85, 90, and 80 on three tests. The first two tests each count for 25% of the grade, and the third test counts for 50%. To calculate the weighted average, we perform the following steps:

1. Multiply each score by its weight: (85 * 0.25), (90 * 0.25), and (80 * 0.50).
2. Sum the products: (85 * 0.25) + (90 * 0.25) + (80 * 0.50) = 21.25 + 22.5 + 40 = 83.75.
3. Divide by the sum of the weights: 83.75 / (0.25 + 0.25 + 0.50) = 83.75 / 1 = 83.75.

Therefore, the student's weighted average score is 83.75. This calculation demonstrates how the weights affect the final grade. The third test, with its 50% weight, significantly influences the average, while the first two tests have a lesser impact due to their lower weights. Understanding weighted averages is crucial for students to strategically focus their efforts on assessments that contribute the most to their final grade.

2.2. Applications of Weighted Scores in Various Fields

The application of weighted scores extends beyond academic grading. In finance, portfolio managers use weighted averages to calculate the return on investment for a portfolio of assets, where the weights represent the proportion of the portfolio invested in each asset. This allows for a more accurate reflection of the overall portfolio performance, considering the varying contributions of different investments. In market research, weighted averages are used to analyze survey data, where responses from different demographic groups may be given different weights to ensure the results accurately represent the population. Furthermore, weighted scoring systems are used in project management to prioritize tasks and allocate resources effectively. By assigning weights to different criteria, such as cost, time, and risk, project managers can make informed decisions and optimize project outcomes. This versatile tool is crucial in any situation where contributions have unequal importance, enabling fair and accurate aggregation of data.

To solidify your understanding of means and weighted scores, let's tackle a couple of practice problems.

3.1. Problem 1: The Fluctuating Test Scores

Imagine a student who has taken four tests. The student's scores on the first three tests are 75, 80, and 85. The student wants to achieve an average score of 82 across all four tests. What score does the student need to achieve on the fourth test to reach this goal?

To solve this problem, we first calculate the total score needed for an average of 82 across four tests. This is done by multiplying the desired average (82) by the number of tests (4), giving us 328. Next, we sum the scores of the first three tests: 75 + 80 + 85 = 240. To find the score needed on the fourth test, we subtract the sum of the first three scores from the total score needed: 328 - 240 = 88. Therefore, the student needs to score 88 on the fourth test to achieve an average of 82. This problem reinforces the concept of how individual scores influence the overall mean and highlights the importance of planning and goal-setting in academic performance.

3.2. Problem 2: The Weighted Grade Calculation

A student's grade in a course is determined by the following weights: homework (20%), quizzes (30%), and a final exam (50%). The student has an average of 90 on homework, 85 on quizzes, and scored 92 on the final exam. What is the student's overall grade in the course?

To calculate the student's overall grade, we use the weighted average formula. First, we multiply each grade by its corresponding weight: (90 * 0.20), (85 * 0.30), and (92 * 0.50). These products represent the weighted contributions of each component to the final grade. Next, we sum the products: (90 * 0.20) + (85 * 0.30) + (92 * 0.50) = 18 + 25.5 + 46 = 89.5. Therefore, the student's overall grade in the course is 89.5. This problem illustrates the practical application of weighted averages in academic settings, where different assessments contribute differently to the final grade. It highlights how a strong performance on a heavily weighted assessment, such as the final exam, can significantly impact the overall grade.

In this comprehensive guide, we have explored the concepts of means and weighted scores, equipping you with the knowledge and skills to navigate these mathematical challenges. Understanding the impact of replacements on the mean and the nuances of weighted averages empowers you to analyze data effectively and make informed decisions in various contexts. By mastering these concepts, you unlock a deeper understanding of averages and their applications in the real world, paving the way for success in mathematics and beyond.