SOLUTION: May has $10,000 to invest for five years. How much additional interest will she earn if the investment provides a 5% annual return, when compared to a 4.5% annual return. How

Question 887043: May has $10,000 to invest for five years.
How much additional interest will she earn if the investment provides a 5% annual return, when compared to a 4.5% annual return.
How long will it take her $10,000 to double in value if it earns 5% annually? What annual rate has been earned if $1,000 grows into $4,000 in 20 years?

Answer by Theo(13342) (Show Source): You can put this solution on YOUR website!
f = p * (1+i)^n
that's the formula for future value of a present amount.
f is the future value
p is the present amount
i is the interest rate per time period
n is the number of time periods.

10,000 invested for 5 years at 5% interest would be solved as follows:
p = 10,000
i = .05
n = 5
formula becomes:
f = 10,000 * (1.05)^5 = 12762.81563

10,000 invested for 5 years at 4.5% interest would be solved as follows:
p = 10,000
i = .045
n = 5
formula becomes:
f = 10,000 * (1.045)^5 = 12461.81938

the interest at 5% is equal to 12762.81563 - 10000 = 2762.81564
the interest at 4.5% is equal to 12461.81938 - 10000 = 2461.81938
the difference is equal to 2762.81564 - 2461.81938 = 300.9962485 which can be rounded to 301.

to find out how long it takes for the money to double at 5%, the formula is the same except now you are solving for n.
the formula is f = p * (1+i)^n
the formula becomes:
20000 = 10000 * (1.05)^n
divide both sides of this equation by 10000 to get:
2 = 1 * (1.05)^n which is the same as 2 = 1.05^n
since the variable is in the exponent, you need to take the log of both sides of this equation to solve.
you get:
log(2) = log(1.05^n) which becomes log(2) = n*log(1.05)
divide both sides of this equation by log(1.05) to get log(2) / log(1.05) = n
solve for n to get n = 14.20669908
that's your solution.
confirm by replacing n in the original equation to get f = 10000 * (1.05)^14.20669908 which results in f = 20000.
your solution is confirmed.

your equation remains the same for the last part of you problem.
f = p * (1+i)^n
you know f and you know p and you know n.
you have to solve for i.
the equation becomes:
4000 = 1000 * (1+i)^20
divide both sides of the equation by 1000 to get 4000/1000 = (1+i)^20 which becomes:
4 = (1+i)^20
take the 20th root of both sides of this equation to get:
4^(1/20) = 1+i
subtract 1 from both sides of this equation to get:
4^(1/20) - 1 = i
use your calculator to solve for i to get i = .0717734625
replace i with that in the original equation to get:
4000 = 1000 * (1.0717734625)^20 which becomes 4000 = 4000
this confirms the solution is correct.

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