Jay's bank account listed a balance of $3,667.50. He originally opened the account with a $3,070 deposit 2 1/4 years ago. If there were no deposits or withdrawals, what was the simple interest rate (to the nearest hundredth of a percent)?

1 Answer
Somebody N.
Mar 23, 2018

See below.

Explanation:

If you just want the percentage of the total interest after 2.25 years.

#3667.50/3070xx100%=119.46%#

We started with 100%, this was our $3070.

The amount extra is:

#19.56%#

Below is a more realistic answer, since interest is calculated at specified periods. Often monthly, quarterly or yearly.

The amount of interest after 2.25 years is:

We can use the formula for compound interest, with 1 compound per year.

#FV=PV(1+r/n)^(nt)#

Where:

#FV="future value"#

#PV="principal value"#

#r="interest rate as a decimal"#

#n="compounding period"#

#t="time in years"#

Our future value is what we have now. $3667.50

Our principal value is what we started with $3070.00

Compounding period is #1# i.e. once a year.

Time is 2.25 years.

We need to find #bbr#, the interest rate.

Putting in our known values:

#3667.50=3070(1+r/1)^(2.25)#

Divide by 3070:

#3667.50/3070=(1+r)^(2.25)#

Taking logarithms of both sides:

#ln(3667.50/3070)=2.25ln(1+r)#

Divide by 2.25:

#(ln(3667.50/3070))/2.25=ln(1+r)#

Using the laws of logarithms:

#y=ln(b)=>e^y=b#

Using this idea. Raise #bbe# to the power of both sides:

#e^((ln(3667.50/3070))/2.25)=e^(ln(1+r))#

#(3667.50/3070)^(1/2.25)=1+r#

#r=(3667.50/3070)^(1/2.25)-1#

#r~~0.082244085#

This is in decimal form, so multiplying by 100.

#8.22%# percent per year.

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