The School Band Is Comprised Of Middle School Students
The School Band Is Comprised Of Middle School Students And High School
The school band is comprised of middle school students and high school students, but it always has the same maximum capacity. Last year the ratio of the number of middle school students to the number of high school students was 1: 8. However, this year the ratio of the number of middle school students to the number of high school students changed to 2: 7. If there are 18 middle school students in the band this year, how many fewer high school students are in the band this year compared to last year? Explain.
Paper For Above instruction
The problem involves understanding ratios and applying proportional reasoning to find changes in the number of students across two years within a school band with a fixed maximum capacity. It requires analyzing the given ratios for middle school and high school students, understanding the implications of these ratios, and calculating the change in the number of high school students from last year to this year based on the information provided.
To start, we recognize that the school band has a fixed maximum capacity, which remains constant across the years. Last year, the ratio of middle school students to high school students was 1:8, indicating that the number of middle school students last year was one part, and the high school students last year were eight parts. The total capacity last year can thus be represented as 9 parts; the sum of these ratios is 1 + 8 = 9.
This means if the capacity is \( C \) students, then:
\[
C = 9 \times \text{(value of one part)}.
\]
Similarly, this year, the ratio shifted to 2:7, with 2 parts for middle school and 7 parts for high school, totaling 9 parts again. Since the total capacity remains the same, these ratios imply that the total number of students each year is consistent.
Given that in the current year, there are 18 middle school students, and the ratio of middle to high school students is 2:7, the number of middle school students corresponds to 2 parts, so:
\[
2 \text{ parts} = 18 \text{ students}.
\]
Solving for one part:
\[
1 \text{ part} = \frac{18}{2} = 9.
\]
Thus, one part equals 9 students.
Next, the number of high school students this year is represented by 7 parts:
\[
7 \text{ parts} = 7 \times 9 = 63 \text{ students}.
\]
This means that this year, there are 63 high school students in the band.
To determine how many high school students were in the band last year, we need to use the ratio for last year, which was 1:8, meaning the number of middle school students last year was 1 part, and high school students last year was 8 parts. Since the total capacity remains constant, the total last year was also 9 parts:
\[
9 \times \text{(value of one part for last year)} = \text{Total capacity}.
\]
But, notice that the total capacity in students can be calculated based on the current year's total students, which is the same.
For last year, middle school students in terms of parts:
\[
1 \text{ part} = \text{middle school students last year}.
\]
Similarly, the capacity in parts is consistent. Since the school capacity remains constant, and the capacity is 9 parts, the total capacity (or total students) should be the same in both years.
Knowing the total capacity in students from the current year's data:
\[
\text{Total students} = \text{middle school} + \text{high school} = 18 + 63 = 81.
\]
Thus, the school capacity equals 81 students, which aligns with 9 parts:
\[
9 \times \text{(one part)} = 81,
\]
so
\[
\text{one part} = \frac{81}{9} = 9.
\]
Now, the number of high school students last year is 8 parts:
\[
8 \times 9 = 72.
\]
Therefore, in last year, there were 72 high school students. This year, there are 63 high school students, so the number of high school students has decreased by:
\[
72 - 63 = 9.
\]
Conclusion: There are 9 fewer high school students in the band this year compared to last year. This change results from the shift in ratios and the fixed capacity of the band. The analysis highlights how ratios can be used to determine individual quantities within a constrained total, illustrating the practical application of proportional reasoning in real-world contexts.
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