doing math

courses › Finance & Money

Compound Growth

level 20 course

Growth on top of growth, and why rates multiply instead of adding.

Pen and paper is fine · no calculator needed why?

Learn first (about 3 minutes)

Opens at level 20.

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Builds on: Money Basics (not open yet) · Percentages in Business (not open yet)

the lesson

The idea, the techniques and a tip for each skill, right here. The Learn page adds worked examples for every skill and untimed practice.

Read the lesson · about 3 minutes

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

An amount grows at one rate for two or three years.

  1. Turn the rate into a factor: 5% is 1.05, 50% is 1.5, 100% is 2.
  2. Multiply the starting amount by the factor once for each year.
  3. Keep the cents after every step.
worked example

Example: You invest $2,000. Assume it grows 10% a year, compounding each year. What is it worth after three years?

  1. Year 1: $2,000 × 1.1 = $2,200.
  2. Year 2: $2,200 × 1.1 = $2,420.
  3. Year 3: $2,420 × 1.1 = $2,662.

Answer: $2,662

Month by month

An account adds interest at the end of each month.

  1. Find this month's interest on the current balance.
  2. Add it to get the new balance.
  3. Next month, find the interest on that new balance, cents included.
worked example

Example: Assume an account adds 1% interest at the end of each month. Starting with $3,000, what is the balance after 2 months?

  1. Month 1: $3,000 + $30 = $3,030.
  2. Month 2: 1% of $3,030 is $30.30.
  3. $3,030 + $30.30 = $3,060.30.

Answer: $3,060.30

Multiply the factors

Two yearly changes, and you want the total change.

  1. Write each change as a factor: up 25% is 1.25, down 20% is 0.8.
  2. Multiply the factors.
  3. Subtract 1 and read it as a percent. A product below 1 gives a minus sign, a decrease.
worked example

Example: Assume revenue grows 10% one year and 20% the next. What is the total percentage change over the two years?

  1. 1.1 × 1.2 = 1.32.
  2. 1.32 − 1 = 0.32, a 32% rise.

Answer: 32%

Tips by skill

  • TipGrow for two years: Multiply by 1 plus the rate, twice. Keep the cents after the first year.
  • TipGrow for three years: Multiply by the growth factor three times: 10% is 1.1, 50% is 1.5, 100% is 2.
  • TipMonthly compounding: Add month 1's interest to the balance before you work out month 2's interest.
  • TipTotal growth over two years: Multiply the two factors, then subtract 1. A fall of 20% is a factor of 0.8.

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.

skills · practice stats

From rounds of this course only: box, review and test-out answers are left out. Once a skill has 40 tries, it compares your first 20 tries with your last 20.

rest ladder

Win 3 of your last 4 rounds and the course rests. A win is 90% right, within 2× the round's par. Pass the review when it comes back and the next rest is longer.

  1. 1 day
  2. 3 days
  3. 7 days
  4. 14 days
  5. 30 days
  6. 60 days
  7. mastered · every 90 days

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