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

Answer:

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

Step-by-step 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:

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

$t=\frac{\ln \left(\frac{119}{100}\right)}{\ln \left(\frac{53}{50}\right)}\approx 2.99$