SOLUTION: how long it takes a $1200 investment to earn $200 interest if it is invested at 9% interest compounded quarterly?

Algebra ->  Customizable Word Problem Solvers  -> Finance -> SOLUTION: how long it takes a $1200 investment to earn $200 interest if it is invested at 9% interest compounded quarterly?      Log On

Ad: Over 600 Algebra Word Problems at edhelper.com


   



Question 543333: how long it takes a $1200 investment to earn $200 interest if it is invested at 9% interest compounded quarterly?
Answer by jpg7n16(66) About Me  (Show Source):
You can put this solution on YOUR website!
You're looking for the compounding formula:
A=p%2A%281%2B%28i%2Fm%29%29%5E%28m%2An%29
where
"A" = amount you will ultimately have in the account (A=1200+200=1400)
"P" = amount you started with, aka principal (p=1200)
"i" = interest rate per year (i=9%)
"m" = number of compounding periods per year (Quarterly means m=4)
"n" = number of years (n=??? - this is what you're solving for)
Plug into the formula and solve for N.
.
A=p%2A%281%2B%28i%2Fm%29%29%5E%28m%2An%29
1400=1200%2A%281%2B%28.09%2F4%29%29%5E%284%2An%29
1400=1200%2A%281.0225%29%5E%284%2An%29
1400%2F1200=%281.0225%29%5E%284%2An%29
7%2F6=%281.0225%29%5E%284%2An%29
.
Here's where most people give up. They don't remember how to bring the variable in the exponent down to work with. The solution -> natural log function!
ln%28n%5Ex%29=x%2Aln%28n%29
Example: Take 5%5Ex
ln%285%5Ex%29=x%2Aln%285%29
and ln%285%29=1.609
So x%2Aln%285%29=1.609x
It allows you to bring the exponent down so it's workable.
.
So what we need to do now is apply the ln function to both sides of the equation. And remember ln(some number) is a constant. You can calculate it or divide by it.
.
ln%287%2F6%29=ln%28%281.0225%29%5E%284%2An%29%29
ln%287%2F6%29=%284%2An%29%2Aln%281.0225%29
ln%287%2F6%29%2Fln%281.0225%29=4%2An
n=%281%2F4%29%28ln%287%2F6%29%2Fln%281.0225%29%29
Time to bust out the calculator
n=%281%2F4%29%28ln%287%2F6%29%2Fln%281.0225%29%29=%281%2F4%29%286.9279%29=1.732
So it takes approx 1.732 years (20.78 months) to earn $200 of interest.