Question
Constructed Response Question
King Middle School has three grade levels. The associated student body, a club on campus, has decided that they would like to host an after-school dance to raise money. The table shows the number of students in each of these grade levels.

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.
Answers
GPT-4o mini
To find out how many sixth-grade students need to attend the dance for the associated student body to meet their goal of 261 students total, we need to first break down the attendance goals based on the provided information.
1. **Understanding the Problem**:
- Total attendance goal: 261 students.
- The seventh-grade and eighth-grade students' attendance will meet their goals, and together they will raise $787.50.
- Each student contributes $4.50, so we can determine how many seventh and eighth graders there are in total from the amount they raised.
2. **Calculating Attendance from Seventh and Eighth Grades**:
- The total amount raised by these students is $787.50.
- To find out how many students that represents, we divide the total amount raised by the amount each student raises:
\[
\text{Number of students from 7th and 8th grade} = \frac{787.50}{4.50} = 175 \text{ students}.
\]
3. **Calculating Attendance Needed from Sixth Grade**:
- We know that the total attendance goal is 261 students.
- Since the seventh and eighth grades account for 175 students, we can find out how many sixth-graders (let's denote this as \(x\)) need to attend to meet the goal:
\[
x + 175 = 261.
\]
- Solving for \(x\):
\[
x = 261 - 175 = 86.
\]
4. **Determining the Percentage of Sixth-Grade Students who Need to Attend**:
- Now, let’s find out what percentage of sixth-grade students this number (86) represents. Assume there are \(N\) sixth-grade students enrolled.
- The percentage of sixth-grade students who need to attend is given by:
\[
\text{Percentage} = \left(\frac{86}{N}\right) \times 100.
\]
5. **Evaluating Amalia's Inequality**:
- Amalia set up the inequality \(4.50x \geq 787.50\) to find the number of sixth-grade students needed to attend based solely on the money raised. This inequality results in \(x \geq 175\).
- This is incorrect because \(x\) represents only the number of sixth-grade students and not the total number attending to achieve the overall attendance goal of 261 students.
- The correct number of sixth-grade students attending to reach the total of 261 is 86.
Thus, **the correct inequality** to find the required number of sixth-grade students attending would be:
\[
x + 175 \geq 261.
\]
This tells us that \(x\) should be equal to 86, meaning 86 sixth-graders need to attend if the seventh and eighth grades achieve their expected attendance.
1. **Understanding the Problem**:
- Total attendance goal: 261 students.
- The seventh-grade and eighth-grade students' attendance will meet their goals, and together they will raise $787.50.
- Each student contributes $4.50, so we can determine how many seventh and eighth graders there are in total from the amount they raised.
2. **Calculating Attendance from Seventh and Eighth Grades**:
- The total amount raised by these students is $787.50.
- To find out how many students that represents, we divide the total amount raised by the amount each student raises:
\[
\text{Number of students from 7th and 8th grade} = \frac{787.50}{4.50} = 175 \text{ students}.
\]
3. **Calculating Attendance Needed from Sixth Grade**:
- We know that the total attendance goal is 261 students.
- Since the seventh and eighth grades account for 175 students, we can find out how many sixth-graders (let's denote this as \(x\)) need to attend to meet the goal:
\[
x + 175 = 261.
\]
- Solving for \(x\):
\[
x = 261 - 175 = 86.
\]
4. **Determining the Percentage of Sixth-Grade Students who Need to Attend**:
- Now, let’s find out what percentage of sixth-grade students this number (86) represents. Assume there are \(N\) sixth-grade students enrolled.
- The percentage of sixth-grade students who need to attend is given by:
\[
\text{Percentage} = \left(\frac{86}{N}\right) \times 100.
\]
5. **Evaluating Amalia's Inequality**:
- Amalia set up the inequality \(4.50x \geq 787.50\) to find the number of sixth-grade students needed to attend based solely on the money raised. This inequality results in \(x \geq 175\).
- This is incorrect because \(x\) represents only the number of sixth-grade students and not the total number attending to achieve the overall attendance goal of 261 students.
- The correct number of sixth-grade students attending to reach the total of 261 is 86.
Thus, **the correct inequality** to find the required number of sixth-grade students attending would be:
\[
x + 175 \geq 261.
\]
This tells us that \(x\) should be equal to 86, meaning 86 sixth-graders need to attend if the seventh and eighth grades achieve their expected attendance.