# Mary said there is no difference between an 8% annual interest rate compounded quaterly and annual interest rate. Explain to mary why she is wrong.with examples

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If an 8% annual interest rate is compounded quarterly, the rate per quarter is 8/4 = 2%.

So an amount **$100**, in a year would become $100*( 1 + .02)^4

=> $100* 1.02^4

=> $100* 1.0824

=> $108.24

If the compounding is done annually the amount would increase by 8% or become $100*(1 + .08)

=> $108.

So, there is a difference of 24 cents in the two cases.

**Even if we take the notional annual interest rate to be the same, as the period of compounding decreases (3 months instead of 12 months), the amount compounded increases after a year.**

To find the difference between (i) 8% annual interest compounded quarterly and (ii)8 % interest annually.

Let Mary invests **$10000** with 8% interest quarterly.

So at at each quarter she gets a quarterly compound interest of (8/4)% = 2% or 0.02 per dollar.

The amount A after the 4 quarters is given by A = Principle * (1+r/4)^4, where r = 8% by the formula for compound interest with interest.

So at the end of the year $10000 becomes with interest $10000(1+0.2)^4 = $10824.32.

I Mary invests the principle P = $10000 in annual interest 8%, then the amount A with interest at the end of the year = A = P(1+r)^1 = $10000(1+8%)^1 = $10000(1+ 8/100) = $100001.08 = $10800.

**The difference Mary should know is that**** she earns a higher amount ($10824.32-10800) = $24.32 if she invests in a quarterly compounding system. **