How Mortgages
& Interest Work
A guide to understanding the math behind borrowing & saving
What is Interest?
🍕 The Pizza Analogy
You lend your friend $100 to buy pizza.��
Your friend says: "I'll pay you back $106 in a year."��
That extra $6 is the interest — the price of borrowing money.
The Key Idea
After 1 year, you owe:
(1 + r) × P
P
Principal — how much you borrow
r
Rate — the annual interest rate (e.g. 0.06 = 6%)
Compound Interest: The Snowball Effect
⛄ Like a Snowball Rolling Downhill
Year 1: Interest is charged on $10,000
→ You now owe $10,600��
Year 2: Interest is charged on $10,600
→ You now owe $11,236��
Year 3: Interest is charged on $11,236
→ You now owe $11,910��
Each year, the interest is bigger
because the balance is bigger!
The Formula
Amount = P × (1 + r)ⁿ
n = number of years
📐 Quick Example
P = $10,000 r = 6% = 0.06 n = 10 years��
$10,000 × (1.06)¹⁰ = $17,908
Borrowing $10,000 at 6% — Year by Year
After 20 years�
$32,071
Original loan
was only $10k!
The Numbers: Watching Debt Grow
$10,000 borrowed at 6% annual interest
Year | What You Owe | Interest That Year | Total Interest Paid |
1 | $10,600 | $600 | $600 |
2 | $11,236 | $636 | $1,236 |
3 | $11,910 | $674 | $1,910 |
5 | $13,382 | $757 | $3,382 |
10 | $17,908 | $1,014 | $7,908 |
15 | $23,966 | $1,357 | $13,966 |
20 | $32,071 | $1,815 | $22,071 |
💡 Notice: The interest each year keeps getting larger — because you're paying interest on interest!
The Bright Side: Savings! 🎉
Same Formula, Different Feeling!
When you save money in a bank, compound interest works FOR you.��
The bank pays YOU interest on your savings — and then pays interest on that interest too!
Same Formula
Amount = P × (1 + r)ⁿ
n = number of years
But where does the interest money come from?
The Bright Side: Savings! 🎉
Same Formula, Different Feeling!
When you save money in a bank, compound interest works FOR you.��
The bank pays YOU interest on your savings — and then pays interest on that interest too!
Same Formula
Amount = P × (1 + r)ⁿ
n = number of years
The bank uses your savings to give loans — to people buying homes, students paying for college, businesses getting started. ��When those loans get paid back with interest, the bank passes some of that back to you.
The Bright Side: Savings! 🎉
After 10 years:
$17,908
After 20 years:
$32,071
After 30 years:
$57,435
What If Interest Is Added Monthly?
Monthly Compounding
Instead of adding interest once a year, the bank adds a little bit every month.��Monthly rate = r ÷ 12�(6% ÷ 12 = 0.5% per month)��
But interest compounds 12 times per year instead of once — so it adds up to slightly more!
Annual
P × (1 + r)ⁿ
Monthly
P × (1 + r/12)12n
📐 After 20 years ($10,000 at 6%)
Annual: $32,071
Monthly: $33,102
Difference: $1,031 extra!
💡 Rule of thumb: More frequent compounding → more total interest. Daily compounding adds even more, but the extra gets smaller and smaller. (In the limit → Euler's number e!)
Annual vs Monthly Compounding
The gold bar is always slightly taller — that's the monthly compounding advantage!
Key Takeaways
1
Interest is the price of borrowing money (or the reward for saving it).
2
Compound interest means interest on interest — it grows exponentially, not linearly.
3
The same formula P×(1+r)ⁿ works for both debt and savings — it's whether it helps or hurts you that changes.
4
Monthly compounding grows faster than annual, because interest gets added (and earns more interest) more often.