# What is the formula for the difference between CI and SI?

if principal, time and rate are given how do i find the difference between Compound interest and Simple Interest?

P=12,000
n=1 and a  1/2 yrs.
R=10% per year


formulae that i know:

CI - SI for 2 years = P(R/100)^2
CI-SI for 3 years = P(R/100)^2 (R/100 + 3)


but none of these will work for 1 and a half years, so what formula do i use? or how do i use these formulae in this context?

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Formula for simple interest:

$$P_{SI} = P \left(1 + \frac{nR}{100}\right)$$

Formula for compound interest:

$$P_{CI} = P \left( 1+\frac{R}{100} \right)^n$$

Therefore their difference is

$$P_{CI} - P_{SI} = P \left( \left(1+\frac{R}{100}\right)^n - \left(1+\frac{nR}{100}\right)\right)$$

If you substitute $n=2$ and $n=3$ into this formula, and expand out the brackets, you will get the formula you quoted in your question.

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alright, thanks –  Nirvan Jul 26 '12 at 16:14

The general compound interest formula says that after $n$ terms at a rate of $R$ percent per term is that the final principal is $P(1+\frac R{100})^n$. Often $R$ is quoted as an annual rate, but if you compound monthly you need to use $\frac R{12}$ per month and $n$ is the number of months. If you have partial terms, you need to specify what happens for a partial term. Maybe you get nothing for the last half year, maybe you get half the interest, or whatever. For simple interest, the final principal is $P(1+\frac {nR}{100})$, so the difference is just the difference of these: $P(1+\frac R{100})^n-P(1+\frac {nR}{100})$

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so are the formulae i mentioned correct? –  Nirvan Jul 26 '12 at 16:08
@Nirvan: What Ross is trying to say is that you didn't give enough information. Is the rate an anuual rate, or is it the rate per period? How long is a period for the purpose of compounding? Without those information it is hard to say whether you have the correct formulae or not. –  Willie Wong Jul 26 '12 at 16:12
The 2 year one is correct, which you can see by expanding the square (n=2) and the only term which survives is the R^2 term. The other is also correct if by $r$ you meant $R$, as it comes from $(1+a)^3-(1-3a)=(1-3a+3a^2+a^3)-(1-3a)=3a^2-a^3)$. My equation works for any $n$. You can see the effect is proportional to $(nR)^2$, so if that is very small you can ignore the difference. –  Ross Millikan Jul 26 '12 at 16:15
@ Willie Wong okay i specified the period of rate, is there anything else? –  Nirvan Jul 26 '12 at 16:17
@Nirvan: So when you have a half year, how much interest do you get? Your formulas, as you say, are specific to 2 or 3 periods. Chris and I have given you equivalent formulations for any number of periods. –  Ross Millikan Jul 26 '12 at 16:19