Interest on interest
Lia and Davi each saved $1,000.00 at 1% a month: every month Lia earns 1% of the $1,000.00 she started with, and Davi earns 1% of whatever his balance is that month. After 5 years, will the difference between them be big or small?
Simulation
Balance in the chosen month
Make your guess first, in task 1. The months move after you lock it in.
Balance month by month
Your guess
At 1% a month, Lia always earns 1% of the starting $1,000.00, and Davi earns 1% of that month's balance.
Tap Next: now the rate goes up.
At 5% a month
Now picture both of them at 5% a month. Lia reaches month 120 with $7,000.00.
Then set Monthly rate to 5% and Months to 120.
Make your guess first. Check unlocks when you tap Lock in my guess or I don't know.
Read What's going on, right below, and then tap Next.
When the money doubles
How many months does it take for Davi's money to double, that is, to reach $2,000.00? Find out with the simulation: change the rate and look for the first month when Davi's balance goes over $2,000.00.
You found all three. Now multiply the rate by the months:
| Monthly rate | Months | Rate × months |
|---|---|---|
| 1% | 70 | 70 |
| 2% | 36 | 72 |
| 3% | 24 | 72 |
This is the rule of 72: at compound interest, 72 ÷ the monthly rate gives you roughly how many months the money takes to double. It's an estimate, and it works well for small rates. Compare with Lia: at 1% a month, she only reaches $2,000.00 in month 100.
One more
Use the rule of 72 to predict: at 4% a month, how many months does it take for Davi's money to double? Then check it in the simulation.
Month-by-month table
| Month | Lia | Lia earned | Davi | Davi earned |
|---|
What's going on
Lia gets simple interest: each month's interest is always calculated on the starting amount. At 1% a month, she earns $10.00 every month, from the first to the last. Davi gets compound interest: each month's interest is calculated on that month's balance. In the 1st month he also earns $10.00. In the 2nd, he earns 1% of $1,010.00, which is $10.10.
Those extra 10 cents are interest on interest: interest Davi earns on top of interest he had already earned. Every month his balance is bigger, so his 1% is bigger too. At first that adds up to a few cents. By month 120, he's earning $32.68 a month, and Lia is still at $10.00. That's why Davi's curve keeps getting steeper while Lia's line keeps the same slope.
Here are the two rules written as formulas. is the amount saved at the start, is the monthly rate as a decimal (1% is 0.01), is the number of months, and is the balance at the end. In finance, is called the principal and the future value.
Lia:
Davi:
The power is what the simulation did month by month: multiplying the balance by 1.01, 120 times, gives , 230% more than at the start. Adding 1% 120 times would give only Lia's 120%. Savings accounts, certificates of deposit, and credit card debt all work with compound interest. The rates used here are only examples to calculate with.
For teachers
Objective
By the end, students can explain why an amount at compound interest pulls away faster and faster from the same amount at simple interest, using the idea that each month's interest is calculated on the balance, which already includes earlier interest. The formula shows up as a summary of what they saw; the focus is the mechanism, month by month.
Prerequisites
Finding a percentage of an amount and successive increases (EF09MA05 in Brazil's national curriculum, BNCC, a 9th-grade skill). Reading a line graph. Powers with whole-number exponents help with reading the formula, but students don't need them to use the activity.
How to use in class
About 25 minutes. Project task 1 from the panel and ask each student to write down an amount, before any calculation; enter the class median as the guess, because the months only move once it's locked in (3 min). Drag the "Months" slider to month 12 and stop there (once the slider is selected, the arrow keys move one month at a time): ask what the $6.83 difference will be at month 60 if it keeps growing by the same amount (5 × $6.83, about $34). Then take the same slider to month 60. In task 2, the class votes on one of the four ranges for 5% a month at month 120 before changing the rate (5 min). Have someone read aloud the sentence "This month, Davi earned..." for three different months and explain why the number changes (5 min). Then read the explanation with the class and let students, in pairs, do the task 3 challenge on a phone or computer (10 min). Wrap up with the question about the table's last column and the rule of 72.
Common mistakes
Adding percentages: the student says that 1% a month for 60 months gives 60% for Davi too. This shows up as a guess of zero difference in task 1 ($1,600.00 for both). Judging by the start: in the first months the difference is a few cents, and they conclude compound interest makes no difference; this shows up as guesses under $50 in task 1 and as the "less than $10,000" range in task 2. Calculating Davi's interest always on $1,000.00: this shows up when they expect "Davi earned" to be $10.00 every month. In the challenge, answering 100 months at 1% means using Lia's rule for Davi.
Discussion questions
- Why is there almost no difference between Lia and Davi in the first months?
- If the rate doubles, from 1% to 2%, does the time it takes the money to double get cut in half? Why is the answer "almost"?
- Credit card debt also grows at compound interest. What does the simulation tell you about putting off paying a debt like that?
- Why doesn't the rule of 72 work for Lia?
Based on Contemporary Mathematics, OpenStax, sections 6.3 and 6.4, under the CC BY-NC-SA 4.0 license. This activity is under the same license. Access the book for free at openstax.org.