Question

Suzy deposits $500 in a savings account with an interest rate of 6% compounded annually. if suzy does not make any additional deposits or withdrawals, how long will it take for suzy to earn at least $95

Answer 1

It will take about 3 years for Suzy to earn at least $95.

Explanation:

Compound interest is interest computed on the original principal as well as on any accumulated interest.

If you deposit P dollars at rate r, in decimal form, subject to compound interest, then the amount, A, of money in the account after t years is given by

`A=P(1+r)^t`

The amount A is called the account's future value and the principal P is called its present value.

From the information given we know that $500 is the present value, 6% is the rate, and we want to find how long will it take for Suzy to earn at least $95, this means when the future value is $595.

Applying the above formula and solving for t we get that:

`595=500(1+(6)/(100) )^t\\\\500\left(1+(6)/(100)\right)^t=595\\\\\left(1+(6)/(100)\right)^t=(119)/(100)\\\\\mathrm{If\:}f\left(x\right)=g\left(x\right)\mathrm{,\:then\:}\ln \left(f\left(x\right)\right)=\ln \left(g\left(x\right)\right)\\\\\ln \left(\left(1+(6)/(100)\right)^t\right)=\ln \left((119)/(100)\right)\\\\\mathrm{Apply\:log\:rule}:\quad \log _a\left(x^b\right)=b\cdot \log _a\left(x\right)\\\\t\ln \left(1+(6)/(100)\right)=\ln \left((119)/(100)\right)`

`t=(\ln \left((119)/(100)\right))/(\ln \left((53)/(50)\right))\approx 2.99`