Problem 16 · 1995 AJHSME
Hard
Ratios, Rates & Proportions
unit-rate
Students from three middle schools worked on a summer project. Seven students from Allen school worked for 3 days, four students from Balboa school worked for 5 days, and five students from Carver school worked for 9 days. The total amount paid for the students' work was $774. Assuming each student received the same amount for a day's work, how much did the students from Balboa school earn altogether?
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Answer: C — 180.00 dollars.
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Hint 1 of 2
Pay depends on BOTH how many students and how many days — so the real unit being paid for is one 'student-day' (one student working one day). Count those, not students.
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Hint 2 of 2
Add up all the student-days, find the dollars per student-day, then multiply by Balboa's share.
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Approach: invent the right unit (student-days), then find its rate
- The insight: each student is paid the same per day, so money tracks 'student-days' — one student for one day. Seven students for 3 days is 21 student-days, and so on. Build everything from this unit.
- Student-days: Allen 7 × 3 = 21, Balboa 4 × 5 = 20, Carver 5 × 9 = 45, total = 86.
- The $774 paid for all 86, so each student-day is worth 774 ÷ 86 = $9. Balboa's 20 student-days earned 20 × $9 = $180.
- Why this transfers: when a quantity depends on two factors multiplied together (people × time, machines × hours, workers × days), make the combined unit first — then it's a plain unit-rate problem.
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