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Saving money

Compound interest with ↑, ↓, and ⇓

For students, teachers, and the simply curious. No prior knowledge of logarithms needed — if you can do 3 + 5, you can follow this to the end.

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Saving money

Imagine you put $100 in a savings account. The bank pays 3% interest per year. What happens?

Building the formula step by step

After one year, the bank adds 3% to your money. That means you keep what you had (100%) and get 3% extra, so you now have 103% of your money:

100 + 3% of 100 = 103
But there is a shortcut: adding 3% is the same as multiplying by 1.03.
100 × 1.03 = 103
Where does the 1.03 come from? It is 1 + 3/100 = 1 + 0.03 = 1.03.

After the second year, you get 3% interest again — but this time on $103, the amount that already includes last year's interest:

103 × 1.03 = 106.09

But 103 was itself 100 × 1.03. So we can write the whole chain:

100 × 1.03 × 1.03 = 100 × 1.03 ↑ 2 = 106.09
Repeated multiplication is a power. Two years means ↑ 2.

See the pattern? After three years: 100 × 1.03 ↑ 3. After ten years: 100 × 1.03 ↑ 10. In general:

After n years: m × r ↑ n
m = starting amount, r = 1 + interest rate (1.03 for 3%), n = number of years.

One formula. Three letters. Let's see what we can do with it.

0510152025$100$150$200≈ 23 yearsyears →

$100 at 3% interest. The money doubles after about 23 years.

Question 1: How much money do I need to put in the bank now, to have $1000 after 10 years at 3% interest?

Our formula says: after n years, your money is m × r ↑ n. We know the interest rate (r = 1.03), the time (n = 10), and we know we want to end up with 1000. So:

m × 1.03 ↑ 10 = 1000
We want m, but it's trapped in a multiplication.

The m is multiplied by 1.03 ↑ 10. To get m alone, we apply the inverse of multiplication — which is division:

m = 1000 ÷ 1.03 ↑ 10
1000 ÷ 1.03 ↑ 10(1000 divided by 1.03 up 10)

About $744. If you put $744 in the bank today at 3% interest, in 10 years it will have grown to $1000.

Question 2: What interest rate do I need to double my money in 10 years?

Doubling means going from m to 2 × m. Our formula says m × r ↑ 10 = 2 × m. The m appears on both sides, so it cancels out. We are left with:

r ↑ 10 = 2
We want r (the left side of ↑).

Compare with the rule card: this has the shape a ↑ b = c, and r is in the position of a — the left side. Left side unknown → use ↓:

r = 2 ↓ 10
2 ↓ 10(2 down 10)

About 1.072 — meaning an interest rate of about 7.2%. In school notation, this would be written as ¹⁰√2. Same answer, but you can't see where the root came from or why.

Question 3: How many years does it take for my money to double at 3% interest?

Again: m × 1.03 ↑ y = 2 × m. The m's cancel:

1.03 ↑ y = 2
We want y (the right side of ↑).

Compare with the rule card: a ↑ b = c, and y is in the position of b — the right side. Right side unknown → use ⇓:

y = 2 ⇓ 1.03
2 ⇓ 1.03(2 double‑down 1.03)

About 23.4 years. In school, this would be log(2) / log(1.03) — correct, but it doesn't show you why. The ⇓ tells you: it's the inverse that recovers the right side of ↑.

Three questions. In the old system, each one feels like a different trick: "move the exponent down", "use a root sign", "use log and divide." In the new notation, every answer is the same move: check the rule card, pick the right inverse (↓ or ⇓), apply it. Done.

Try it yourself — click m, r, or n to choose what you want to find, and slide the others to see the answer change live:

Goal ($)$1,000
= $744
3.0%
10 yr
m = 1,000 ÷ 1.03 ↑ 10 = 744division undoes multiplication
05$1,00010Click m, r, or n to change what you're solving for

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2 ↑ 3 = 82 = 8 ↓ 3left unknown? use down3 = 8 ⇓ 2right unknown? use double-down