Excess-and-Shortage Method ② Finding the Number of People from Two Conditions
Learn application problems involving seat assignments and group formation, which commonly appear in the excess-and-shortage method.
Excess-and-Shortage Problems with Seats and Groups
Example Problem
Students are seated on benches in a gym. If 6 students sit on each bench, 4 students cannot sit; if 8 students sit on each bench, 2 benches are left unused. How many benches are there?
Solution:Let the number of benches be N.
① Six students each: number of students＝6×N＋4
② Eight students each: number of students＝8×（N－2）＝8N－16 (since 2 benches are left over, N－2 benches are used)
③ Since the number of students is the same,
6N＋4＝8N－16
20＝2N
N＝10 benches
④ Number of students＝6×10＋4＝64 students
Students are seated on benches in a gym. If 6 students sit on each bench, 4 students cannot sit; if 8 students sit on each bench, 2 benches are left unused. How many benches are there?
Solution:Let the number of benches be N.
① Six students each: number of students＝6×N＋4
② Eight students each: number of students＝8×（N－2）＝8N－16 (since 2 benches are left over, N－2 benches are used)
③ Since the number of students is the same,
6N＋4＝8N－16
20＝2N
N＝10 benches
④ Number of students＝6×10＋4＝64 students
How to Interpret "Benches Are Left Over"
"m benches are left over" means that no one is sitting on the last m benches. Set up the equation as number of students＝number of people per bench ×（N－m）.
Key Points
- "○ people cannot sit" means a "shortage," just as when there are not enough items.
- When "m benches are left over," express it as "8×（N－m）," for example.
- Connect the two equations with an equals sign and find the number of benches N.
Practice Problems
- Students are put on buses. If 40 students ride each bus, 2 buses are left over; if 30 ride each bus, 20 students cannot ride. How many buses are there?
- Children are put into tents. If 4 children enter each tent, 3 children cannot enter; if 5 enter each tent, 1 tent is left over. How many children are there?
- If 5 students sit on each bench, 9 cannot sit; if 7 sit on each bench, 1 bench is left over. How many benches are there?
Answers and Explanations
- Answer:There are 8 buses
Explanation: If 40 students ride each bus, 2 buses are left over; if 30 ride each bus, 20 students cannot ride. If there are N buses: 40×（N－2）＝number of students (2 buses are left over, so N－2 buses are used), 30N＋20＝number of students (20 students cannot ride with 30 per bus). 40（N－2）＝30N＋20. 40N－80＝30N＋20. 10N＝100. N＝10 buses. Number of students＝30×10＋20＝320. Check: 40×（10－2）＝40×8＝320✓ - Answer:There are 19 children
Explanation: With 4 children per tent, 3 cannot enter; with 5 per tent, 1 tent is left over. If there are N tents: 4N＋3＝number of children, 5×（N－1）＝number of children (since 1 tent is left over, everyone fits in N－1 tents). 4N＋3＝5N－5. 8＝N. Number of children＝4×8＋3＝32＋3＝35. Check: 5×（8－1）＝5×7＝35✓ - Answer:There are 7 benches
Explanation: With 5 students per bench, 9 cannot sit; with 7 students per bench, 1 bench is left over. If there are N benches: 5N＋9＝number of students, 7×（N－1）＝number of students (since 1 bench is left over, everyone sits on N－1 benches). 5N＋9＝7N－7. 16＝2N. N＝8 benches. Number of students＝5×8＋9＝40＋9＝49. Check: 7×（8－1）＝7×7＝49✓