# Find the effective annual interest rate of the 7 % annual interest rate compounded quarterly. Round your answer to the nearest 0.01%

## Expert Answers Hello!

There are 4 quarters in a year, so compounding occurs 4 times, and the quarterly rate is 1/4 of the (stated) annual rate.

After these 4 quarters, an initial amount of money `x` will be

`x*(1+(7%)/4)^4,`

the profit will be

`x*(1+(7%)/4)^4-x=x*[(1+(7%)/4)^4-1],`

and therefore the effective annual interest rate `e` is

`e=(1+(7%)/4)^4-1.`

Remember that 1% of something is 1/100 of that something, we obtain

`e=[(1+(0.07)/4)^4-1]*100% = [(1+0.0175)^4-1]*100%.`

This is approximately 7.1859%, or 7.19% when rounded to the nearest 0.01% as required.

The answer: the effective annual interest rate is approximately 7.19%.

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