Question

At Valley Middle School, the sixth grade has 160 students, the seventh grade has 320 students, and the eighth grade has 240 students. The student congress is traditionally made up of eight representatives from each grade. a. For each grade, find the percent of students in the congress. Give your answers to the nearest tenth of a percent. b. Tom is in the seventh grade class. He says that, since his class has more students, it should have more representatives. He suggests that each grade be represented by the same percent of its students. Devise a plan for setting up the student congress this way. Tell how many representatives each grade should elect, and the percent of each grade that is represented. c. Which plan do you think is fairer: the original plan or the plan you devised in Part b? Defend your answer.

Short answer

The revised plan is fairer because it gives representation based on the same percent of students from each grade.

Step-by-step solution

Calculate Percent of Students in Congress for Each Grade

First, find the percent of students in the congress for each grade. To calculate this, divide the number of representatives by the total number of students in each grade, then multiply by 100.For sixth grade: \[\text{Percent} = \left( \frac{8}{160} \right) \times 100 = 5\text{%} \]For seventh grade: \[\text{Percent} = \left( \frac{8}{320} \right) \times 100 = 2.5\text{%} \]For eighth grade: \[\text{Percent} = \left( \frac{8}{240} \right) \times 100 = 3.3\text{%} \]

Determine Representatives Based on Percentages

To make the representation based on the same percent of students in each grade, choose a suitable percent. Assume a 3% representation for simplicity.For sixth grade (3% of 160): \[\text{Representatives} = 160 \times 0.03 = 4.8 \approx 5 \]For seventh grade (3% of 320): \[\text{Representatives} = 320 \times 0.03 = 9.6 \approx 10 \]For eighth grade (3% of 240): \[\text{Representatives} = 240 \times 0.03 = 7.2 \approx 7 \]

Compare Fairness of the Plans

Reflect on both plans: The original plan assigns 8 representatives to each grade irrespective of size. The revised plan adjusts the number to 5, 10, and 7. Considering fairness, the revised plan accounts for the different sizes of each grade, ensuring equal representation by percent.

Conclusion on the Fairer Plan

Decide which plan is fairer. The revised plan is fairer as it represents each grade based on the same percent of students, accommodating the difference in grade sizes.

Key concepts

percent calculation

When working with percentages, you are typically comparing a part to a whole. In this exercise, we want to find out what percent of each grade is represented in the student congress at Valley Middle School. To do this, we use the formula: To calculate percent, we use the formula: \(\text{{Percent}} = \frac{{\text{{Part}}}}{{\text{{Whole}}}} \times 100\). For instance, if you have 8 representatives from a class of 160 students, the percent calculation would be: \[ \text{{Percent}} = \frac{{8}}{{160}} \times 100 = 5\text{{%}} \]. This means 5% of the sixth grade are representatives in the student congress.

Repeat this calculation for each grade:

- Seventh grade: \[ \text{{Percent}} = \frac{{8}}{{320}} \times 100 = 2.5\text{{%}} \]
- Eighth grade: \[ \text{{Percent}} = \frac{{8}}{{240}} \times 100 = 3.3\text{{%}} \]

Now you see the variations among grades. This is the first indicator that equality in numbers does not always mean fairness in representation.

representative allocation

Representative allocation deals with distributing representatives among grades based on certain criteria, in this case, their student population. The goal is to be fairer by ensuring each grade is represented proportionally. To achieve this, you must determine what percentage of each grade should be represented and allocate representatives based on these percentages.

Suppose we decide each grade should have 3% of its students as representatives:

- For sixth grade, calculate 3% of 160 students: \[ 160 \times 0.03 = 4.8 \text{approximately 5 representatives} \]
- For seventh grade, calculate 3% of 320 students: \[ 320 \times 0.03 = 9.6 \text{approximately 10 representatives} \]
- For eighth grade, calculate 3% of 240 students: \[ 240 \times 0.03 = 7.2 \text{approximately 7 representatives} \]

This method ensures representation is equal by percentage across different student populations instead of an arbitrary fixed number. While the numbers are no longer uniform, this proportional allocation is fairer as it considers the actual size of each grade.

comparative fairness

Comparative fairness involves evaluating different approaches to determine which is fairer. In our scenario, we need to compare the original method where each grade gets 8 representatives regardless of size, with the new method where representation is based on a consistent percentage.

Under the original plan:

- Sixth grade gets 8 representatives (5%)
- Seventh grade gets 8 representatives (2.5%)
- Eighth grade gets 8 representatives (3.3%)

This method does not account for the smaller or larger class sizes, leading to uneven representation.

In the revised plan:

- Sixth grade gets 5 representatives (3%)
- Seventh grade gets 10 representatives (3%)
- Eighth grade gets 7 representatives (3%)

This approach ensures that each grade is represented equally by percentage, despite the differences in class size.

Considering fairness, the revised plan is preferable. By focusing on representing each grade proportionally to its size, it recognizes the diversity in student numbers and offers a more balanced approach.