Problem 23 · 1985 AJHSME
Stretch
Ratios, Rates & Proportions
double-counting
King Middle School has 1200 students. Each pupil takes 5 classes a day. Each teacher teaches 4 classes. Each class has 30 students and 1 teacher. How many teachers are there at King Middle School?
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Answer: E — 50.
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Hint 1 of 2
You can't jump straight from students to teachers — but both connect to the same middle thing: CLASSES. Find the number of classes first, then convert classes to teachers.
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Hint 2 of 2
Count one quantity two different ways. The total 'student-seats' (a student sitting in a class) equals 1200 × 5 counted from the students' side, and also equals (number of classes) × 30 counted from the classes' side. Setting those equal unlocks the class count.
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Approach: count student-class slots two ways
- Count total enrollments (one student in one class) two ways. From students: 1200 × 5 = 6000 seats. From classes: each class has 30 students, so (# classes) × 30 = 6000, giving 6000 ⁄ 30 = 200 classes.
- Each teacher covers 4 classes, so teachers = 200 ⁄ 4 = 50.
- Why this transfers: 'counting the same thing two ways' is a powerhouse — pick a quantity both groups touch (here, class enrollments), tally it from each side, and set them equal. It chains students → classes → teachers without ever needing a direct student-to-teacher link.
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