simple interest
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 $t,2t,\mathrm{\dots}$, 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 following table illustrates the structure of the simple interest.
time period at  principal  interest  interest accrued 

$0$  ${M}_{0}$  $0$  $0$ 
$t$  ${M}_{0}$  $i$  $i$ 
$2t$  ${M}_{0}$  $i$  $2i$ 
$3t$  ${M}_{0}$  $i$  $3i$ 
$\mathrm{\vdots}$  $\mathrm{\vdots}$  $\mathrm{\vdots}$  $\mathrm{\vdots}$ 
$nt$  ${M}_{0}$  $i$  $ni$ 
$The\mathrm{`}\mathrm{`}tota{l}^{\prime \prime}interest$i(nt)$earned(accrued)attheendoftime$nt$is$ni$.Iftheaccountisclosedandthemoneywithdrawnattheendof$nt$,andthetotalamountofmoneyreceivedis$$M(nt)={M}_{0}+ni.$$Theinterestrateassociatedwiththesimpleinterestaspresentedabovebetweentwotimeperiods,say$at$and$bt$,isgivenby$$r(at,bt)=\frac{1}{{M}_{0}}\frac{i(bt)i(at)}{btat}=\frac{i}{{M}_{0}t},$$whichdoesnotdependonthechoiceof$a$and$b$.Inotherwords,theoriginalprincipal$M_0$,theamountofinterest$i$,andthelengthoftheinitialtimeinterval$t$areenoughtodeterminetheinterestrate.\text{\mathbf{R}\mathbf{e}\mathbf{m}\mathbf{a}\mathbf{r}\mathbf{k}}.\text{\u2022 The expression for the effective interest rate for simple interest is a bit more complicated: eff.r(at,bt)= 1 M(at) i(bt)i(at) btat = 1 M 0 +ai i t , which decreases with increasing a . Imagine as a becomes very large, the increase in interest has practically no impact on the \u201caccumulated\u201d principal \u2062 M ( \u2062 a t ) . \u2022 More generally, we say that an interest is simple if its interest rate r is constant with respect to time t . Solving r= 1 M 0 i(t)i(0) t0 for \u2062 i ( t ) , we get = \u2062 i ( t ) \u2062 M 0 r t , or that the accrued interest is a linear function of t . It grows directly proportionally with respect to time.}\begin{array}{cc}\text{Title}\hfill & \text{simple interest}\hfill \\ \text{Canonical name}\hfill & \text{SimpleInterest}\hfill \\ \text{Date of creation}\hfill & \text{20130322 16:40:06}\hfill \\ \text{Last modified on}\hfill & \text{20130322 16:40:06}\hfill \\ \text{Owner}\hfill & \text{CWoo (3771)}\hfill \\ \text{Last modified by}\hfill & \text{CWoo (3771)}\hfill \\ \text{Numerical id}\hfill & \text{7}\hfill \\ \text{Author}\hfill & \text{CWoo (3771)}\hfill \\ \text{Entry type}\hfill & \text{Example}\hfill \\ \text{Classification}\hfill & \text{msc 00A06}\hfill \\ \text{Classification}\hfill & \text{msc 00A69}\hfill \\ \text{Classification}\hfill & \text{msc 91B28}\hfill \\ \text{Related topic}\hfill & \text{CompoundInterest}\hfill \\ \text{Related topic}\hfill & \text{InterestRate}\hfill \end{array}$