# 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

The answer is simple: First find S.I.(You should be knowing how to). Then for C.I. : Find C.I. For the whole no part of time period(1 in this case). Then find S.I. for half year using the Principal as Amount obtained in previous step(13200 in this case). Then add the C.I. Of first step and S.I. Of second step.(C.I.= 1200 and S.I.=660) Thus total C.I. is 1860. Therefore the difference is C.I. - S.I = 1860-1800 = 60

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