Question 1161707
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Assume contributions at the end of each month, and interest compounding monthly, the formula for the future value is<br>
{{{A = P(((1+r/n)^(nt)-1)/(r/n))}}}<br>
A = future value
P = periodic contribution
r = (annual) interest rate
n = # of contributions per year = $ of compounding periods per year
t = # of years<br>
In this problem,
A = 20000
P = 200
r = .05
n = 12
t = to be determined<br>
{{{20000 = 200(((1+.05/12)^(12t)-1)/(.05/12))}}}<br>
The unknown is in an exponent, so you will need to use logarithms or a utility like a graphing calculator.<br>
{{{100 = (((1.0041667)^(12t)-1)/(.0041667))}}}<br>
{{{0.41667 = (1.0041667)^(12t)-1)}}}<br>
{{{1.41667 = (1.0041667)^(12t)}}}<br>
{{{ln((1.41667)) = 12t*ln((1.0041667))}}}<br>
{{{t = ln((1.41667))/(12*ln((1.0041667)))}}}<br>
{{{t = 6.98}}} to 2 decimal places<br>
ANSWER: 6.98 years.<br>
Since contributions are made monthly, her savings will be a bit over $20,000 after 7 years.<br>