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Homesimple interest
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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. 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 

$0$  $M_{0}$  $0$  $0$ 
$t$  $M_{0}$  $i$  $i$ 
$2t$  $M_{0}$  $i$  $2i$ 
$3t$  $M_{0}$  $i$  $3i$ 
$\vdots$  $\vdots$  $\vdots$  $\vdots$ 
$nt$  $M_{0}$  $i$  $ni$ 
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 If the account is closed and the money withdrawn at the end of $nt$, and the total amount of money received is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
$r(at,bt)=\frac{1}{M_{0}}\frac{i(bt)i(at)}{btat}=\frac{i}{M_{0}t},$ 
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:
$\operatorname{eff.}r(at,bt)=\frac{1}{M(at)}\frac{i(bt)i(at)}{btat}=\frac{1}{% M_{0}+ai}\frac{i}{t},$ 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
$r=\frac{1}{M_{0}}\frac{i(t)i(0)}{t0}$ for $i(t)$, we get $i(t)=M_{0}rt$, or that the accrued interest is a linear function of $t$. It grows directly proportionally with respect to time.
Mathematics Subject Classification
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