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When interest is calculated for every period only on the principal then the total interest gained on all the period is called simple interest.
The principal plus simple interest gained is called amount.
i.e. Amount = Principal + Simple Interest
A = P + S.I
Before derive the formula for finding simple interest and the amount, we consider an example first. Suppose a person borrowed Rs.100 with 10% rate of simple interest for 3 years.
Then simple interest for 1st year = 
Amount at the end of 1st year = 100 + 10 = 110
Amount at the end of 2nd year = 110 + 10 = 120
Amount at the end of 3rd year = 120 + 10 = 130
i.e. every time the interest of Rs.10 calculated on the principal of Rs.100 is added to get the amount, after 1st, 2nd and 3rd year.
Total simple interest gained in 3 years = 3 × 10 = 30
It can be obtained as
S.I = Amount – Principal = 130 – 100 = Rs.30
Now, using above concept we can drive the formula of simple interest and the amount.
Suppose,
Principal = P
Rate of Interest = r
Number of periods = n
Simple Interest of 1st period = Pr (same for all periods) = S.I
Amount at the end of 1st period = Principal + Interest
= P + Pr = A1
Amount at the end of 2nd period = A1 + S.I = (P + Pr) + Pr
= P + 2Pr = A2
Amount at the end of 3rd period = A2 + S.I = (P + 2Pr) + Pr
= P + 3Pr = A3
And so on
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Amount at the end of nth period = P + nPr = P + Prn
Amount = Principal + Interest for n-Periods
Amount = P + Prn 
Simple Interest = Prn 
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