Suppose a bank account is opened at time $0$ and $M_0$ is deposited into the account. A simple interest is interest with the following characteristics:

1. it is earned at subsequent time periods $t,2t,\ldots$ where $t$ is the length of the initial time interval (1 for 1 month, 12 for 1 year, etc...)
2. the interest earned at the end of each time period is the same regardless of the time period

The ``total'' interest $i(nt)$ earned (accrued) at the end of time $nt$ is $ni$ If the account is closed and the money withdrawn at the end of $nt$ and the total amount of money received is $$M(nt)=M_0+ni.$$

The interest rate associated with the simple interest as presented above between two time periods, say $at$ and $bt$ is given by $$r(at,bt)=\frac{1}{M_0}\frac{i(bt)-i(at)}{bt-at}=\frac{i}{M_0t},$$ which does not depend on the choice of $a$ and $b$ In other words, the original principal $M_0$ the amount of interest $i$ and the length of the initial time interval $t$ are enough to determine the interest rate.

Remark.

• The expression for the effective interest rate for simple interest is a bit more complicated: $$\operatorname{eff.}r(at,bt)=\frac{1}{M(at)}\frac{i(bt)-i(at)}{bt-at}= \frac{1}{M_0+ai}\frac{i}{t},$$ which decreases with increasing $a$ Imagine as $a$ becomes very large, the increase in interest has practically no impact on the ``accumulated'' principal $M(at)$
• More generally, we say that an interest is simple if its interest rate $r$ is constant with respect to time $t$ Solving $$r=\frac{1}{M_0}\frac{i(t)-i(0)}{t-0}$$ for $i(t)$ we get $i(t)=M_0rt$ or that the accrued interest is a linear function of $t$ It grows directly proportionally with respect to time.