Level 8: Loans, investing and e

Ratio, rates and money · level 8 of 8 · rough age 17 and beyond (a guide for parents; anyone can start anywhere)

In level 7 you valued streams of payments with geometric series. In this level you take loans apart payment by payment, work through a Canadian mortgage, and look at investing with honest averages, bond prices and the long reach of fees. You separate what money says from what it buys, follow compounding to its limit and meet the number \(e\), and end where ratio began, with shape: the paper sizes that halve perfectly and the true and false stories of the golden ratio. All of it is mathematics, not financial advice.

Lesson	You will be able to
Loans and amortization	Derive the loan payment formula, build an amortization table, and prove why early payments are mostly interest
Mortgages	Turn a semi-annually compounded rate into a monthly one, and measure the effect of the down payment, extra payments and renewal
Investing	Compare savings, GICs, bonds, stocks and funds, price a bond, use the geometric mean, and show how fees compound
Real versus nominal rates	Use the exact and approximate links between nominal, real and inflation rates, including after tax
Continuous compounding and the number e	Explain why \((1 + 1/n)^n\) has a limit, use \(A = Pe^{rt}\), and compare it with daily compounding
Ratio in nature and art	Prove why A-series paper uses \(1 : \sqrt2\), derive the golden ratio, and tell real golden-ratio facts from myths

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