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.
Sixth ? 215
seventh 0.5 190
eight 1/3 240
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 determine how many sixth-grade students must attend the dance given the goals for attendance and fundraising, let’s break down the information provided.
1. **Calculate the Attendance of Seventh and Eighth Graders:**
The seventh-grade class has 190 students and the eighth-grade class has 240 students. According to the problem, they are expected to meet their attendance goals.
2. **Finding the Total Attendance from Seventh and Eighth Graders:**
- If all 190 seventh-graders attend, they will contribute 190 students.
- If all 240 eighth-graders attend, they will contribute 240 students.
Therefore, the total attendance from the seventh and eighth grades combined is:
\[
190 + 240 = 430 \text{ students}
\]
3. **Setting the Goal for Total Attendance:**
The total attendance goal for the dance is 261 students.
4. **Determine the Required Attendance from the Sixth Grade:**
Since the seventh and eighth grades will have exceeded the total attendance goal of 261 students as calculated (430 students), it seems there is either a mistake in understanding the numbers or the question.
However, let's assume the goal is still to reach a total of 261 including all grades:
\[
\text{Total Attendance Required} = 261 - (\text{Seventh Grade}) - (\text{Eighth Grade}) = 261 - 430
\]
This results in a negative number, which indicates a misunderstanding or conflict here since the seventh and eighth grades alone have already exceeded the goal.
5. **Checking the Fundraising Calculation:**
The seventh and eighth grades (with their attendance) will raise money. The problem states they expect to raise $787.50.
Each student raises $4.50, so the total amount raised from both seventh and eighth grades can be calculated as:
\[
\text{Total Raised} = \text{Number of students} \times 4.50
\]
If 190 students from seventh and 240 students from eighth align to that data, they would contribute:
\[
\text{Total from 7th and 8th} = (190 + 240) \times 4.5 = 430 \times 4.5 = 1935
\]
This would suggest that if those students attend, they would significantly exceed the goal of $1174.50.
### The Inequality and Solution
You proposed:
\[
4.50x \ge 787.50
\]
Where x is the number of sixth-grade students needing to attend.
- Dividing both sides by 4.50 gives:
\[
x \ge 175
\]
This is correct if you are assuming the fundraising should exactly meet the total after accounting for seventh and eighth grade incomes.
However, it appears from our calculations that the seventh and eighth grades alone already have sufficient numbers to exceed the entire needed amount of 261 students.
### Conclusion
- **To answer the question:**
Amalia did **not** correctly determine the number of sixth-grade students needed to attend the dance since the attendance goal of 261 is already surpassed with the 430 students coming from seventh and eighth grades.
### Correct findings could conclude:
It may be necessary to revise the goals, as the sheer attendance from seventh and eighth students alone fills the quota without the sixth-grade students needing to add to it. There’s no need for sixth-grade students since the goal is already satisfied.
1. **Calculate the Attendance of Seventh and Eighth Graders:**
The seventh-grade class has 190 students and the eighth-grade class has 240 students. According to the problem, they are expected to meet their attendance goals.
2. **Finding the Total Attendance from Seventh and Eighth Graders:**
- If all 190 seventh-graders attend, they will contribute 190 students.
- If all 240 eighth-graders attend, they will contribute 240 students.
Therefore, the total attendance from the seventh and eighth grades combined is:
\[
190 + 240 = 430 \text{ students}
\]
3. **Setting the Goal for Total Attendance:**
The total attendance goal for the dance is 261 students.
4. **Determine the Required Attendance from the Sixth Grade:**
Since the seventh and eighth grades will have exceeded the total attendance goal of 261 students as calculated (430 students), it seems there is either a mistake in understanding the numbers or the question.
However, let's assume the goal is still to reach a total of 261 including all grades:
\[
\text{Total Attendance Required} = 261 - (\text{Seventh Grade}) - (\text{Eighth Grade}) = 261 - 430
\]
This results in a negative number, which indicates a misunderstanding or conflict here since the seventh and eighth grades alone have already exceeded the goal.
5. **Checking the Fundraising Calculation:**
The seventh and eighth grades (with their attendance) will raise money. The problem states they expect to raise $787.50.
Each student raises $4.50, so the total amount raised from both seventh and eighth grades can be calculated as:
\[
\text{Total Raised} = \text{Number of students} \times 4.50
\]
If 190 students from seventh and 240 students from eighth align to that data, they would contribute:
\[
\text{Total from 7th and 8th} = (190 + 240) \times 4.5 = 430 \times 4.5 = 1935
\]
This would suggest that if those students attend, they would significantly exceed the goal of $1174.50.
### The Inequality and Solution
You proposed:
\[
4.50x \ge 787.50
\]
Where x is the number of sixth-grade students needing to attend.
- Dividing both sides by 4.50 gives:
\[
x \ge 175
\]
This is correct if you are assuming the fundraising should exactly meet the total after accounting for seventh and eighth grade incomes.
However, it appears from our calculations that the seventh and eighth grades alone already have sufficient numbers to exceed the entire needed amount of 261 students.
### Conclusion
- **To answer the question:**
Amalia did **not** correctly determine the number of sixth-grade students needed to attend the dance since the attendance goal of 261 is already surpassed with the 430 students coming from seventh and eighth grades.
### Correct findings could conclude:
It may be necessary to revise the goals, as the sheer attendance from seventh and eighth students alone fills the quota without the sixth-grade students needing to add to it. There’s no need for sixth-grade students since the goal is already satisfied.