Question 1114867: how long it takes a $3100 investment to earn $500 interest if it is invested at 9% compounded semiannually?
please can you show me step by step
Answer by Theo(13342)   (Show Source):
You can put this solution on YOUR website! the formula to use is f = p * (1 + r) ^ n
f is the future value.
p is the present value
r is the interest rate per time period.
n is the number of time periods.
your investment is 3100 and you want it to grow to 3100 + 500 = 3600.
your interest rate is 9% per year compounded semi-annually.
divide 9% by 2 to get 4.5% per semi-annual period.
your number of time periods will be in half years, or semi-annual periods.
in your problem, the formula becomes 3600 = 3100 * (1 + .045) ^ n
the formula uses rate, not percent rate, therefore you need to divide 4.5% by 100 to get a rate of .045 per half year.
divide both sides of that equation by 3100 to get 3600 / 3100 = (1 + .045) ^ n
take the log of both sides of that equation to get log(3600 / 3100) = log((1 + .045) ^ n)
since log(1 + .045) ^ n) is equal to n * log(1 + .045), your equation becomes;
log(3600 / 3100) = n * log(1 + .045)
divide both sides of that equation by log(1 + .045) and solve for n to get:
n = log(3600 / 3100) / log(1 + .045) = 3.397144814.
that's the number of semi-annual periods you will requirie to earn 500 dollars interest on your investment of 3100.
3100 * (1 + .045) ^ 3.97144814 = 3600.
3600 minus 3100 = 500 interest.
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