Question
If the associated student body decides to host this dance, their goal is to have a total of 261 students attend. If the seventh-grade class and the eighth-grade class exactly meet their goals for attendance, what percent of sixth-grade students would need to attend for the club to meet its goals? Explain or show your reasoning. You may use a combination of diagrams, drawings, expressions/equations, and words.
If the dance occurs, 261 students will need to attend to reach the goal of raising at least $1174.50. Each student will raise $4.50 for attending. Together, the seventh-grade and eighth-grade students expect to raise $787.50.
The president of the associated student body, Amalia, wrote and solved the following inequality to determine x, the number of sixth-grade students that will need to attend the dance.
4.50x ≥ 787.50
x ≥ 175
Do you think that Amalia correctly determined the number of sixth-grade students who need to attend?
If your answer is yes, provide a justification. If your answer is no, provide a justification by writing and solving the correct inequality.
If the dance occurs, 261 students will need to attend to reach the goal of raising at least $1174.50. Each student will raise $4.50 for attending. Together, the seventh-grade and eighth-grade students expect to raise $787.50.
The president of the associated student body, Amalia, wrote and solved the following inequality to determine x, the number of sixth-grade students that will need to attend the dance.
4.50x ≥ 787.50
x ≥ 175
Do you think that Amalia correctly determined the number of sixth-grade students who need to attend?
If your answer is yes, provide a justification. If your answer is no, provide a justification by writing and solving the correct inequality.
Answers
GPT-4o mini
To determine if Amalia correctly calculated the number of sixth-grade students that need to attend the dance, let's first analyze the situation step-by-step.
1. **Total Attendance Goal**: The goal is for a total of **261** students to attend the dance.
2. **Seventh-Grade and Eighth-Grade Attendance**: The problem states that the seventh-grade class and the eighth-grade class will meet their goals for attendance, but it does not specify those goals. Therefore, we need to denote their combined attendance with a variable:
- Let \( y \) be the number of seventh and eighth graders combined that attend the dance.
3. **Sixth-Grade Attendance Requirement**: To find how many sixth-grade students (\( x \)) need to attend, we can express the total attendance goal as the sum of all three grades:
\[
x + y = 261
\]
4. **Rearranging to find the number of sixth-grade students**:
\[
x = 261 - y
\]
5. **Funding Goal**: It's stated that together the seventh-grade and eighth-grade students will raise $787.50. The total amount that needs to be raised is $1174.50, and each attending student raises $4.50. This means:
\[
4.50x + 787.50 = 1174.50
\]
6. **Calculating the required sixth graders to attend**:
- We can rearrange the above equation to find \( x \):
\[
4.50x = 1174.50 - 787.50
\]
\[
4.50x = 387
\]
\[
x = \frac{387}{4.50} = 86
\]
7. **Conclusion**: According to this calculation, **86 sixth-grade students would be required to attend** in order to meet the total funding goal.
Now let's address Amalia's inequality:
Amalia wrote the inequality as:
\[
4.50x \ge 787.50
\]
When solved, this yields:
\[
x \ge 175
\]
This solution is incorrect because it does not account for the total funding goal. Instead, we found that the correct number of sixth-grade students that need to attend is \( 86 \).
### Summary
- **Amalia's conclusion** that \( x \ge 175 \) is incorrect.
- The correct inequality based on the funding needs is:
\[
4.50x + 787.50 = 1174.50
\]
resulting in \( x = 86 \), which represents the actual number of sixth graders needed to meet the fundraising goal.
1. **Total Attendance Goal**: The goal is for a total of **261** students to attend the dance.
2. **Seventh-Grade and Eighth-Grade Attendance**: The problem states that the seventh-grade class and the eighth-grade class will meet their goals for attendance, but it does not specify those goals. Therefore, we need to denote their combined attendance with a variable:
- Let \( y \) be the number of seventh and eighth graders combined that attend the dance.
3. **Sixth-Grade Attendance Requirement**: To find how many sixth-grade students (\( x \)) need to attend, we can express the total attendance goal as the sum of all three grades:
\[
x + y = 261
\]
4. **Rearranging to find the number of sixth-grade students**:
\[
x = 261 - y
\]
5. **Funding Goal**: It's stated that together the seventh-grade and eighth-grade students will raise $787.50. The total amount that needs to be raised is $1174.50, and each attending student raises $4.50. This means:
\[
4.50x + 787.50 = 1174.50
\]
6. **Calculating the required sixth graders to attend**:
- We can rearrange the above equation to find \( x \):
\[
4.50x = 1174.50 - 787.50
\]
\[
4.50x = 387
\]
\[
x = \frac{387}{4.50} = 86
\]
7. **Conclusion**: According to this calculation, **86 sixth-grade students would be required to attend** in order to meet the total funding goal.
Now let's address Amalia's inequality:
Amalia wrote the inequality as:
\[
4.50x \ge 787.50
\]
When solved, this yields:
\[
x \ge 175
\]
This solution is incorrect because it does not account for the total funding goal. Instead, we found that the correct number of sixth-grade students that need to attend is \( 86 \).
### Summary
- **Amalia's conclusion** that \( x \ge 175 \) is incorrect.
- The correct inequality based on the funding needs is:
\[
4.50x + 787.50 = 1174.50
\]
resulting in \( x = 86 \), which represents the actual number of sixth graders needed to meet the fundraising goal.