Time Value of Money

Concept:Difference between CI and SI for 2 years at the same rate equals $r\%$ of the principal for that period.Explanation:Let principal be $P$ and rate $r\%$ per annum.SI for 2 years: $\frac{P \times r \times 2}{100} = 200$ ⟹ $Pr = 10000$.CI for 2 years: $P\left(1+\frac{r}{100}\right)^2 - P = 209$.Simplify CI: $P\left[\left(1+\frac{r}{100}\right)^2 - 1\right] = P\left[ \frac{2r}{100} + \frac{r^2}{10000} \right] = \frac{2Pr}{100} + \frac{Pr^2}{10000}$.Substitute $Pr = 10000$: CI = $\frac{2 \times 10000}{100} + \frac{10000 \times r^2}{10000} = 200 + r^2$.Given CI = 209, so $200 + r^2 = 209$ ⇒ $r^2 = 9$ ⇒ $r = 3$? Wait, check: the derivation shows $r$ is the rate percent, but we obtained $r=3$? Let's use a known shortcut: CI - SI for 2 years = $\frac{Pr^2}{100^2} = \frac{(10000) r}{100^2}?$ Actually, better: We derived CI = $200 + r$? Let's re-derive carefully. Using $P = \frac{10000}{r}$, then CI = $\frac{10000}{r}\left[\left(1+\frac{r}{100}\right)^2 - 1\right] = \frac{10000}{r}\left[\frac{2r}{100} + \frac{r^2}{10000}\right] = \frac{10000}{r} \cdot \frac{2r}{100} + \frac{10000}{r} \cdot \frac{r^2}{10000} = 200 + r$. Thus $200 + r = 209$ ⇒ $r = 9$. So rate is $9\%$. Answer:Rate percent = $9\%$ per annum.