Problem 63 - General Chemistry: Principles and Modern Applications

Question

In a class of 76 students, the results of a particular examination were \(7 \mathrm{A}^{\prime} \mathrm{s}, 22 \mathrm{B}^{\prime} \mathrm{s}, 37 \mathrm{C}^{\prime} \mathrm{s}, 8 \mathrm{D}^{\prime} \mathrm{s}, 2 \mathrm{F}^{\prime} \mathrm{s}\). What was the percent distribution of grades, that is, \(\%\) A's, \(\%\) B's, and so on?

Step-by-Step Solution

Answer

The percentage of grades are: A's = 9.21%, B's = 28.95%, C's = 48.68%, D's = 10.53%, F's = 2.63%.

Step 1: Analyze the data given

Note that the grades distribution is as follows: A's = 7, B's = 22, C's = 37, D's = 8, F's = 2. Total number of students is 76.

Step 2: Calculate the Percentage of A's

To calculate the percentage of students who received A's, divide the number of A's by the total number of students and multiply by 100. The formula is: \(\%A's = (\frac{7}{76}) * 100\)

Step 3: Calculate the Percentage of B's

Using the same method, the percentage of students who received B's is: \(\%B's = (\frac{22}{76}) * 100\)

Step 4: Calculate the Percentage of C's

The percentage of students who received C's is: \(\%C's = (\frac{37}{76}) * 100\)

Step 5: Calculate the Percentage of D's

The percentage of students who received D's is: \(\%D's = (\frac{8}{76}) * 100\)

Step 6: Calculate the Percentage of F's

Lastly, the percentage of students who received F's is: \(\%F's = (\frac{2}{76}) * 100\)

Key Concepts

Percentage Calculation

Calculating percentages is a fundamental skill in mathematics that is widely used in various real-life situations. Percentage simply refers to how much out of a hundred. For example, when you say 50%, it means 50 out of 100. Here's how you calculate the percentage of each grade category in a class:

To begin, take the number of students who received a particular grade, divide it by the total number of students in the class, and then multiply the result by 100 to convert it to a percentage.

Let's take the calculation of A's in the class as an example. You have 7 students who got A's out of a total of 76 students. Therefore, the percentage of students who received A's is \(\%A = \frac{7}{76} \times 100\). Breaking it down:

• 7 divided by 76 gives you the fraction of students with A's.
• Multiply this fraction by 100 to convert it into a percentage.

This method can be applied to any situation where you need to determine the percentage from a given total, making it an essential tool in both academics and everyday life.

Grade Distribution Analysis

Grade distribution analysis involves examining how grades are spread out across a class to provide valuable insights into student performance patterns. This process helps educators understand the strengths and weaknesses in their teaching methods and how students are absorbing the material. In this example, we've analyzed grades in a class of 76 students to determine the percentage distribution of each grade.

Here's what you typically analyze:

• The number of students achieving each grade.
• The percentage of the total class that this number represents.

Armed with this data, educators can make informed decisions about curriculum adjustments or additional support services.

For example, if a large percentage of students are receiving C's, it might indicate that the material is at an appropriate level but could benefit from additional depth or explanation. Similarly, a high percentage of A's and B's might suggest that the students are well-prepared and understanding the course material effectively. Either way, grade distribution analysis is crucial for ongoing educational improvements.

Mathematics Problem-Solving

Mathematical problem-solving is a process that not only involves performing calculations but also using logic and reasoning to interpret and make decisions based on those calculations. In the case of analyzing exam results, you have to:

• Clearly define the problem, which in this instance, is to calculate the percentage distribution of grades.
• Gather necessary data, such as the numbers of students per grade and the total number of students.
• Apply mathematical concepts, like percentages, to find solutions.

A strong problem-solving approach involves breaking down a problem into simpler, manageable parts, performing calculations, and then analyzing the outcomes for meaningful insights.

For instance, from the percentages calculated in the grade distribution, educators can infer if the majority of the class grasped the material well or if further instructional interventions are needed. Mathematical problem-solving, therefore, not only provides answers but also aids in strategic planning and decision-making processes.