Percentages (II) · Ages 13–14
Growth and depreciation
Growth: times (1 + r%)n. Depreciation: times (1 − r%)n.
How to do it
Method
Growth: times (1 + r%)^n. Depreciation: times (1 − r%)^n
200000 × 0.93 = 145800
Another way
Year by year
200000 → 180000 → 162000 → 145800
In full
Growing by the same percentage every year, the value after n years = the start × (1 + r%)n; losing the same percentage every year (depreciating), the value after n years = the start × (1 − r%)n. A $200000 car loses 10% of its value a year: one way, the formula, after 3 years it is worth 200000 × 0.93 = $145800; another, year by year, 200000 → 180000 → 162000 → 145800. That is 1 − 0.729 = 27.1% lost in all, not 3 × 10% = 30%: each year's 10% is of that year's lower value. Back from the value after n years to the start: divide by (1 ± r%)n. Asked in which year it first passes (or falls below) an amount, multiply year by year until it does.
Worked examples
Have a go first, then open each one for the answer and the working.
Example 1
A city of 100000 people grows 5% a year. To the nearest whole number, how many people will it have after 3 years?
Show the answer and working
Answer: 115763 people
100000 × (1 + 5%)3 = 100000 × 1.053, about 115763 people.
Example 2
A machine loses 10% of its value each year. After 6 years, its value is its first value times which of these?
- A. 0.16
- B. 1 − 0.1 × 6
- C. 0.96
- D. 0.9 × 6
Choose the right answer
Show the answer and working
Answer: C. 0.96
Each year leaves 1 − 10% = 0.9 of it; over 6 years, 0.96.
Example 3
A new car worth $280,000 loses 20% of its value in its first year and 12% a year after that. To the nearest dollar, what is it worth after 3 years?
Show the answer and working
Answer: 173,466 dollars
The first year leaves 1 − 20%, each of the next 2 years 1 − 12%: 280000 × (1 − 20%) × (1 − 12%)2, about 173466.
Practice questions to try
Try these 5; the answers are at the end. For a whole worksheet, see VIP worksheets below.
1.A number grows 10% a year, so in 3 years it has grown 30% in all.
True or false?
Answer:
2.A machine loses 20% of its value each year. After 4 years, its value is its first value times which of these?
- A. 0.84
- B. 0.24
- C. 0.8 × 4
- D. 1 − 0.2 × 4
Choose the right answer
Answer:
3.A website has 5000 members, and the number grows 10% a year. At the end of which year does it first pass 15000?
- A. The end of year 13
- B. The end of year 20
- C. The end of year 12
- D. The end of year 11
Choose the right answer
Answer:
4.A machine loses 20% of its value a year and is worth $112,640 after 3 years. What was it worth new?
Answer:
5.A new car worth $310,000 loses 25% of its value in its first year and 10% a year after that. To the nearest dollar, what is it worth after 3 years?
Answer:
All the answers
- 1. False
- 2. A. 0.84
- 3. C. The end of year 12
- 4. 220,000 dollars
- 5. 188,325 dollars
VIP practice notes and papers
Aimed at tests and exams: questions on this topic, new every time, to print or save as a PDF.
- Practice notes: The key point, worked examples step by step, then questions of the same kind to follow, hints fading
- Practice paper: Set out in parts like a school paper, marked, with answers and full working
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