Question

A payday loan company charges $4\%$ interest for a two-week period. What would be the annual interest rate from that company?

Answer 1

Finding the annual interest rate:

Step-1: Assumption finding the number of interest periods:

Suppose the principal is $P$.

Given that the payday loan company charges $4\%$ interest for a two-week period.

Now, there are $52$ weeks in a year. so, there are $\frac{52}{2}=26$ interest period in total.

Step-2: Finding the amount:

Formula to be used: We know that for principal $P$ and interest rate $r\%$ for a particular time period, the amount after $n$ such interest periods will be: $A=P{\left(1+\frac{r}{100}\right)}^n$.

Here, $r=4,n=26$.

Therefore,

$$\begin{gathered} A=P{\left(1+\frac{r}{100}\right)}^n \\ =P{\left(1+\frac{4}{100}\right)}^{26} \\ =P\times {\left(1.04\right)}^{26} \\ =2.77P \end{gathered}$$

Step-3: Finding the rate of annual interest

Total annual interest for principal $P$ will be:

$$\begin{gathered} A-P=2.77P-P \\ =1.77P \end{gathered}$$

Formula to be used: We know that if the principal is $P$ and the total interest is $I$, the rate of interest will be: $r=\frac{I}{P}\times 100\%$.

Here, $I=1.77P$. So, the rate of interest will be:

$$\begin{gathered} r=\frac{I}{P}\times 100\% \\ =\frac{1.77P}{P}\times 100\% \\ =177\% \end{gathered}$$

Therefore, the annual interest rate is $177\%$.