Compound Growth
Growth on top of growth, and why rates multiply instead of adding.
Pen and paper is fine · no calculator needed why?
Opens at level 20. You're level 1. You can read and practice here now.
the idea
Compounding means each period's growth is added to the amount, so the next period grows the bigger amount: growth on growth, interest on interest.
The tool is the growth factor, 1 plus the rate as a decimal. Growing 10% is multiplying by 1.1; falling 20% is multiplying by 0.8. Two years of growth is two multiplications, never the two rates added. So 10% a year for two years is 1.1 × 1.1 = 1.21, a 21% rise, not 20%.
Match the rate to its period: a monthly rate compounds once a month.
techniques
Multiply by the factor each year
- Turn the rate into a factor: 5% is 1.05, 50% is 1.5, 100% is 2.
- Multiply the starting amount by the factor once for each year.
- Keep the cents after every step.
worked example
You invest $2,000. Assume it grows 10% a year, compounding each year. What is it worth after three years?
- Year 1: $2,000 × 1.1 = $2,200.
- Year 2: $2,200 × 1.1 = $2,420.
- Year 3: $2,420 × 1.1 = $2,662.
Answer: $2,662
Month by month
- Find this month's interest on the current balance.
- Add it to get the new balance.
- Next month, find the interest on that new balance, cents included.
worked example
Assume an account adds 1% interest at the end of each month. Starting with $3,000, what is the balance after 2 months?
- Month 1: $3,000 + $30 = $3,030.
- Month 2: 1% of $3,030 is $30.30.
- $3,030 + $30.30 = $3,060.30.
Answer: $3,060.30
Multiply the factors
- Write each change as a factor: up 25% is 1.25, down 20% is 0.8.
- Multiply the factors.
- Subtract 1 and read it as a percent. A product below 1 gives a minus sign, a decrease.
worked example
Assume revenue grows 10% one year and 20% the next. What is the total percentage change over the two years?
- 1.1 × 1.2 = 1.32.
- 1.32 − 1 = 0.32, a 32% rise.
Answer: 32%
watch out for
- Adding the rates. $2,000 at 5% a year for two years grows to $2,205, not $2,200: year 2 also earns 5% on year 1's $100.
- Giving only the growth when the question asks what the amount is worth at the end.
- Adding a rise and a fall. Up 25% and then down 20% is no change at all, not a 5% rise.
practice
Grow for two years
worked example
You invest $9,700. Assume it grows 5% a year, compounding each year. What is it worth after two years?
Answer: $10,694.25
- Year 1: $9,700 × 1.05 = $10,185.
- Year 2: $10,185 × 1.05 = $10,694.25.
Grow for three years
worked example
You invest $20,000. Assume it grows 10% a year, compounding each year. What is it worth after three years?
Answer: $26,620.00
- Multiply by 1.1 three times: $20,000 → $22,000 → $24,200 → $26,620.
Monthly compounding
worked example
Assume an account adds 0.5% interest at the end of each month. Starting with $10,000, what is the balance after 2 months?
Answer: $10,100.25
- Month 1: $10,000 + $50 = $10,050.
- Month 2: $10,050 + $50.25 (0.5% of $10,050) = $10,100.25.
Total growth over two years
worked example
Assume revenue grows 5% one year and 25% the next. What is the total percentage change over the two years? (Use a minus sign for a decrease.)
Answer: 31.25%
- Multiply the growth factors: 1.05 × 1.25 = 1.3125.
- So the total change is 31.25%.