A rally starts with 120 members. Each hour, 9 members drop out of the group. How many members will remain after 1, 2, 3, … hours? Find a linear expression to represent the number of members at the end of the nth hour.
Answer: After 1, 2, 3, 4, … hours: 111, 102, 93, 84, … members. At the end of the nth hour: 120 − 9n members.
Step-by-step solution
Given: 120 members at the start; 9 members leave every hour
To find: Members after 1, 2, 3, … hours and an expression for the nth hour
To find: Members after 1, 2, 3, … hours and an expression for the nth hour
Idea: The group loses the same 9 members each hour, so after n hours 9n members have left.
- 1 mark
Hours 0 1 2 3 4 5 Members 120 120 − 9 = 111 111 − 9 = 102 102 − 9 = 93 84 75 - After n hours, 9 × n = 9n members have dropped out.1 mark
- Members at the end of the nth hour = 120 − 9n, a linear expression in n (it falls by 9 every hour).1 mark
111, 102, 93, 84, … members remain after 1, 2, 3, 4, … hours; at the end of the nth hour, 120 − 9n members remain.
Check: n = 3: 120 − 27 = 93 ✓.
Answer to write in the exam
After 1 hour: 120 − 9 = 111
After 2 hours: 111 − 9 = 102; after 3 hours: 93; after 4 hours: 84; …
After n hours, members who left = 9n
∴ Members at the end of the nth hour = 120 − 9n
Common mistakes that cost marks
- Writing 120 + 9n. Members drop out, so the number goes down.
- Writing 111n or 120 − 9 − n.
- Subtracting wrongly in the table (for example 111 − 9 = 101).
How this can come in the exam
MCQ (1 mark)
A queue has 150 people and 12 people are served every 10 minutes. The number left after k ten-minute periods is
- 150 − 12k
- 138k
- 150k − 12
- 12k − 150
Show answer
(A) 150 − 12k
12 people leave in each period, so 12k have left after k periods.
Short answer (2 marks)
A rally starts with 200 people and 15 leave every hour. After how many hours will 80 people remain?
Show answer
200 − 15h = 80 (1 mark) ⇒ 15h = 120 ⇒ h = 8 hours (1 mark).Try one yourself
For the rally above (120 members, 9 leave each hour), after how many hours will 57 members remain?
Show answer
120 − 9n = 57 ⇒ 9n = 63 ⇒ n = 7 hours.
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