There are 70 students in the school band. 40% of them are sixth graders, 20% are seventh graders, and the rest are eighth graders.
The total number of students in the school band is \(70\). We are given that \(40\%\) are sixth graders and \(20\%\) are seventh graders. The remaining students are eighth graders. To find the number of students in each grade, we will multiply the total count by the respective percentage for each grade level. The sum of the percentages for all grades must equal \(100\%\).
First, we calculate the number of sixth graders by finding \(40\%\) of \(70\):
\[S_6 = 0.40 \times 70 = 28.\]
Next, we calculate the number of seventh graders by finding \(20\%\) of \(70\):
\[S_7 = 0.20 \times 70 = 14.\]
Finally, the remaining students are eighth graders. We can find the percentage of eighth graders by subtracting the other two percentages from \(100\%\):
\[P_8 = 100\% - (40\% + 20\%) = 40\%.\]
Then, we calculate the number of eighth graders by finding \(40\%\) of \(70\):
\[S_8 = 0.40 \times 70 = 28.\]
Alternatively, we can find the number of eighth graders by subtracting the sum of sixth and seventh graders from the total:
\[S_8 = 70 - (28 + 14) = 70 - 42 = 28.\]
Verification: \(28 + 14 + 28 = 70\), which confirms our calculations are correct.
This problem demonstrates how to divide a total quantity into parts based on percentages. In real life, this skill is essential for budgeting, managing resources, or organizing groups. For example, if a company has a budget for three departments and allocates specific percentages to each, this method allows for precise calculation of the funds available to each department. Understanding how to work with parts and percentages helps in making informed decisions about resource allocation.