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King School has an enrollment of \(\mathrm{900}\) students. The school day consists of \(\mathrm{6}\) class periods during which each class is taught by one teacher. There are \(\mathrm{30}\) students per class. Each teacher teaches a class during \(\mathrm{5}\) of the \(\mathrm{6}\) class periods and has one class period free. No students have a free class period. How many teachers does the school have?
Let's start by organizing what we know about King School in everyday terms:
Think of this school like a big scheduling puzzle. We have:
Process Skill: TRANSLATE - Converting the word problem into clear, manageable pieces of information
Now let's figure out how many individual classes the school needs to run.
Think about it this way: if we have 900 students and each class holds 30 students, then during any single period, we need:
\(900 \text{ students} ÷ 30 \text{ students per class} = 30 \text{ classes}\)
Since there are 6 periods in a day, and we need 30 classes during each period:
Total class sessions per day = \(30 \text{ classes} × 6 \text{ periods} = 180 \text{ class sessions}\)
This means 180 individual classes must be taught throughout the entire school day.
Now here's the key insight: each teacher only works 5 periods out of 6.
Let's say we have T teachers total. Since each teacher teaches 5 periods:
Total teaching periods available = \(T \text{ teachers} × 5 \text{ periods each} = 5T \text{ teaching periods}\)
Think of it like this: if you have 10 teachers and each works 5 periods, you have 50 teacher-periods of work available.
Here's where we connect our two calculations:
The school needs 180 class sessions to be taught (from step 2)
The teachers can provide 5T teaching periods (from step 3)
For the school to function, these must be equal:
\(5T = 180\)
Solving for T:
\(T = 180 ÷ 5 = 36\)
Process Skill: INFER - Recognizing that teacher availability must exactly match teaching demand
The school has 36 teachers.
Let's verify: \(36 \text{ teachers} × 5 \text{ periods each} = 180 \text{ teaching periods}\), which matches exactly the 180 class sessions needed.
The answer is C. 36
1. Misunderstanding the constraint that teachers work 5 out of 6 periods
Students often overlook this crucial constraint and assume teachers work all 6 periods. This leads them to think they need fewer teachers since each teacher would handle more classes. They might calculate: \(180 \text{ total class sessions} ÷ 6 \text{ periods per teacher} = 30 \text{ teachers}\), leading to answer choice B.
2. Confusing students and classes when calculating capacity
Students may incorrectly think about "30 students per class" and get confused about whether they should divide 900 by 30 or multiply. Some students calculate total classes needed incorrectly, thinking they need 900 classes total rather than understanding that with 30 students per class, they need \(900 ÷ 30 = 30\) classes running simultaneously during each period.
3. Not recognizing this as a capacity-matching problem
Students may not realize they need to match "teaching demand" (180 class sessions needed) with "teaching supply" (teachers × periods worked). Instead, they might try other approaches like dividing total students by teachers directly, leading to incorrect setups.
1. Arithmetic errors in basic calculations
Students make simple calculation mistakes such as: \(900 ÷ 30 = 25\) instead of 30, or \(30 × 6 = 150\) instead of 180, or \(180 ÷ 5 = 30\) instead of 36. These arithmetic errors lead directly to wrong answer choices.
2. Using 6 instead of 5 when calculating teacher capacity
Even if students understand the constraint conceptually, they may accidentally use 6 periods per teacher in their calculation: \(6T = 180\), giving \(T = 30\). This gives answer choice B instead of the correct answer C.
No likely faltering points