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Question 304916: Venture capital. Henry invested $12,000 in a new
restaurant. When the restaurant was sold two years
later, he received $27,000. Find his average annual
return by solving the equation 12,000(1 + r)^2 = 27,000.
Answer by Theo(13342)   (Show Source):
You can put this solution on YOUR website! Average annual return is given by the equation:
F = P * (1+r)^n
F = the future value of the money received
P = the present value of the money invested
r = the annual interest rate
n = the number of years.
In your problem:
F = $27,000
P = $12,000
n = 2
Your equation becomes:
27000 = 12000 * (1+r)^2
Divide both sides of this equation by 12000 to get:
27000/12000 = (1+r)^2
Simplify to get:
2.25 = (1+r)^2
Take the log of both sides of this equation to get:
log(2.25) = log((1+r)^2)
Since, in general, log(x^y) = y*log(x), then your equation becomes:
log(2.25) = 2 * log(1+r)
Divide both sides of this equation by 2 to get:
log(2.25)/2 = log(1+r)
Use the LOG function of your calculator to get:
.352182518 / 2 = log(1+r)
Simplify to get:
.176091259 = log(1+r)
Find the number whose log is equal to .176091259 using your calculator to get:
1+r = 1.5
Subtract 1 from both sides of this equation to get:
r = .5
The annual interest rate is equal to .5 which is 50% per year.
Substitute in your original equation of 2.25 = (1+r)^2 to get:
2.25 = (1.5)^2
Use your calculator to find the result of (1.5)^2 to get:
2.25 = 2.25 confirming that your answer is good.
Your answer is that the annual interest rate is equal to 50% per year compounded annually.
In most hewlett packard calculators, you find the log of a number by entering the number and then hitting the LOG key.
In most hewlett packard calculators, you find the number whose log is by entering the log and then hitting the 2nd key and then hitting the LOG key.
Example:
I want the log of 100.
I enter 100 and then hit the LOG key to get 2.
I want the number whose log is equal to 2.
I enter 2 and then hit the 2nd key and then hit the LOG key to get 100.
If you have a different type of calculator, then follow the instructions given with that calculator.
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