The rate at which borrowers take money and the interest on that money or loan is charged at this rate. In other words it is some percentage of the principal that is paid at some fixed intervals on regular basis during some fixed time duration which can be monthly, yearly, weekly or even daily etc. The rate of interest is measured in percentages.

Formulas

Since interest are of two types: simple interest and compound interest, so does formulas of measuring interest rate vary with interest types:

Case 1: Simple Interest:

The formula for finding the simple interest is:

I = $\frac{P * R * T}{100}$

Where I = interest, P = principal which is the amount of loan given, R = rate of interest at which the loan is given and T = time for which loan is given.

From this we can find rate of interest as:

R = $\frac{(100 * I)}{(P * T)}$

The rate deduced from this is then written simply with a percentage sign only to tell the answer.

Case 2: Compound Interest

We know that in compound interest we generally deduce the final amount rather than interest first as amount is easier to be calculated than the interest value.

A = $P (1 + r)^n$

Where A = final amount after the complete tenure, P = principal or the starting amount, r = rate of interest in accordance to the tenure of compounding in decimals and ‘n’ = number of compounding intervals.

Then from this we can deduce our rate of interest as:

$(1 + r)^n$ = $\frac{A}{P}$

$\rightarrow$ $(1 + r)^n$ = $\frac{A}{P}$

$\rightarrow$ 1 + r = $(\frac{A}{P})^{\frac{1}{n}}$

$\rightarrow$ r =  $(\frac{A}{P})^{\frac{1}{n}}$ - 1

Examples

Here are some examples on finding the rate of interest in various cases:

Example 1: A person earned an amount of $460 in a bank account when he deposited $400 for 3 years not compounded. Find the rate of interest applied.

Solution:

We are given that principal, P = 400; amount, A = 460; time, T = 3. We need to find rate of interest, R.

First we need to find interest I = amount - principal = A - P = 460 - 400 = $60.

We know that I = $\frac{P * R * T}{100}$

$\rightarrow$ R = I * $\frac{100}{(P * T)}$

$\rightarrow$ R = 60 * $\frac{100}{(400 * 3)}$

$\rightarrow$ R = 5

Hence the rate of interest applied here is 5% per annum.

Example 2: A woman deposited $900 in a bank to earn $1024 after tenure of 2 years. If the money was compounded annually, find the rate of interest applied.

Solution:

Here principal, P = 900; amount, A = 1024; time, T = 2. Since we are compounding annually, hence n = T = 2. We need to find rate of interest, r. 

We know that A = $P (1 + r)^n$

$\rightarrow$ 1024 = 900 $(1 + r)^2$

$\rightarrow$ $(1 + r)^2$ =$\frac{1024}{900}$

Taking square root both sides and neglecting negative values as rate cannot be negative, we get,

1 + r = $\frac{32}{30}$ = $\frac{16}{15}$

$\rightarrow$ r = $\frac{16}{15}$ - 1 = $\frac{1}{15}$ = 0.0667

Now to get ‘r’ in terms of percentage we will multiply r by 100 and thus the rate of interest applied here is 6.67%.