Question

1. Suppose you pay back $680 on a $650 loan you had for 75 days. What was your simple annual interest rate?

Answer 1

well, let's assume a year has 365 days, so 75 days is simply 75/365 of a year.  We also know that your loan was $650 and you paid back $680 or namely 30 bucks extra, so the interest on those $650 is really 30 bucks.

$~~~~~~ \textit{Simple Interest Earned} \\\\ I = Prt\qquad \begin{cases} I=\textit{interest earned}\dotfill &\ 30\\ P=\textit{original amount deposited}\dotfill & \ 650\\ r=rate\to r\%\to \frac{r}{100}\\ t=years\to \frac{75}{365}\dotfill &\frac{15}{73} \end{cases} \\\\\\ 30 = (650)(\frac{r}{100})(\frac{15}{73})\implies 30=\cfrac{9750r}{7300}\implies 30=\cfrac{195r}{146}\implies 4380=195r \\\\\\ \cfrac{4380}{195}=r\implies \stackrel{\%}{22.46}\approx r$