Compound Interest, Reverse Percentage and Compound Proportional Change
Senior 3 · Mathematics
Compound Interest, Reverse Percentage and Compound Proportional Change
Lesson 4 of 4
25 min
Mugisha buys a moto for 1,800,000 Frw to work as a moto-taxi driver. Each year it is worth 20% less than the year before (an illustrative rate). After one year it has lost 360,000 Frw; after the second year it loses 20% of 1,440,000 Frw, which is only 288,000 Frw. The value falls by the same percentage each year, not by the same amount. This is compound interest running backwards.
Definition
Compound proportional change: a quantity that changes by the same percentage in every period. Starting from , after periods
where is the rate per period as a decimal.
Depreciation is the loss of value of an asset (a moto, a phone, a machine) over time; when it is a fixed percentage per year, it is compound decay.
Example: the moto's value after years is Frw.
Worked example 1: the value of the moto
Find the value of the moto after 3 years, and the total loss of value.
Solution Frw. The total depreciation is Frw, which is 48.8% of the price, not .
| Year | 0 | 1 | 2 | 3 | 4 | 5 |
|---|---|---|---|---|---|---|
| Value (Frw) | 1,800,000 | 1,440,000 | 1,152,000 | 921,600 | 737,280 | 589,824 |
Worked example 2: Rwanda's population
The Fifth Rwanda Population and Housing Census counted 13,246,394 people in August 2022, and the National Institute of Statistics of Rwanda (NISR) reports an average annual growth rate of 2.3% since the 2012 census, which counted 10,515,973.
(a) Check the 2.3%: what would 2.3% a year for 10 years give, starting from the 2012 count? (b) If the rate stayed at 2.3% a year, estimate the population 10 years after 2022.
Solution (a) , within 0.4% of the 2022 count, so 2.3% a year fits the two censuses well.
(b) , about 16.6 million. This is only a model: birth and death rates change, and NISR makes its own projections with several scenarios.
Definition
Continued proportion: the numbers , , , are in continued proportion when each one is the same multiple of the one before:
Example: the moto's values 1,800,000; 1,440,000; 1,152,000 are in continued proportion with multiplier 0.8. Check: .
Important
Successive changes: multiply the multipliers. Several percentage changes in a row have one overall multiplier, the product of the separate multipliers. Percentages do not simply add.
Worked example 3: a chain of changes
The price of a bag of fertiliser rises 5% in January, rises 8% in April, then falls 3% in July (illustrative). Find the overall percentage change.
Solution Overall multiplier: . The price rose by about 10.0% overall, not exactly: here the two answers happen to be close, but they are rarely equal.
Worked example 4: back to the original value
A machine loses 20% of its value each year. After 2 years it is worth 2,560,000 Frw. What was it worth when new?
Solution , so Frw.
Check: ✓.
Warning
Percentages over several periods do not add. 20% depreciation for 3 years is a loss of 48.8%, not 60%, because each year's 20% is taken from a smaller value.
A decrease does not undo an equal increase. After and then you are 1% below where you started. To undo you must divide by 1.1, a decrease of about 9.1%.
Tip
One multiplier per step, then one product. Write each change as a multiplier (up 5% is 1.05, down 3% is 0.97), multiply them all, and read the overall change from the product: 1.09998 means , and 0.512 means .
ExamQA: Compound Interest and Depreciation. Watch how the same multiplier method handles growth (a multiplier above 1) and depreciation (a multiplier below 1), raised to the number of years.
Practice
The last three are harder.
Further reading for this unit
Sources and credits
The bar chart, worked examples, practice items and quiz questions were written by Ganzaa. The moto, machine and fertiliser figures are illustrative. This lesson adapts CC BY-SA material and is shared under CC BY-SA 4.0.
