Question 1175340
the formula you show is:


A = P * (1 + i) ^ n


in this formula:


A is the future value
P is the present value
i is the interest rate per time period
n is the number of time periods


your problems will be taken in turn below:


a) Solve the formula for P. What was the initial investment of an investment worth $1000 that compounded 10% interest for 10 years?


in this problem, A is equal to 1000 and i is equal to .10 per year and n is equal to 10 years.


the interest rate is not shown as percent.
the interest rate percent divided by 100 is equal to the interest rate.


the formula becomes 1000 = P * (1 + .10) ^ 10


solve for P to get P = 1000 / (1 + .10) ^ 10 = 385.5432894.


confirm by solving for A using that value for P.


you get A = 385.5432894 * (1 + .10) ^ 10 = 1000.


the value of P is confirmed to be good.


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b) Solve the formula for i. What is the interest rate of an investment whose worth went from $1000 to $1200 in 2 years?


the same basic form of the equation is used.


A = P * (1 + i) ^ n becomes:


1200 = 1000 * (1 + i) ^ 2


divide both sides of this equation by 1000 to get:


1.2 = (1 + i) ^ 2


take the square root of both sides of the equation to get:


1.2 ^ (1/2) = 1 + i


solve for 1 + i to get:


1.095445115 = 1 + i


subtract 1 from both sides of the equation to get:


.095445115 = i


that's your answer.


confirm by replacing i in the original equation and solve for A to get:


A = 1000 * (1 + .095445115) ^ 2 = 1200.


this confirms the rate is good.


multiply the rate by 100 and you get 9.5445115%.


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c) Explain a method with which you could estimate how many years it would take for an investment to reach a certain worth at a certain interest rate.


the formula is, once again, A = P * (1 + i) ^ n


you want to solve for  n.


divide both sides of the equation by P to get:


A/P = (1 + i) ^ n


take the log of both sides of the equation to get:


log(A/P) = log((1+i)^n)


by the law of logs that says log(x^a) = a * log(x), your equation becomes:


log(A/P) = n * log(1 + i)


divide both sides of the equation by log(1 + i) to get:


log(A/P) / log(1 + i) = n


that's the procedure you would take to find n.


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d) Estimate how many years would it take an investment of $2100 at 20% interest to reach a worth of $5225?


you would use the method described in (c) to find the value of n.


the basic formula is, once again, A = P * (1 + i) ^ n


the formula becomes 5225 = 2100 * (1 + .2) ^ n


divide both sides of the equation by 2100 to get:


5225/2100 = (1 + .2) ^ n


take the log of both sides of the equation to get:


log(5225/2100) = log((1 + .2)^n)


by the law of logs that says log(x^a) = a * log(x), your equation becomes:


log(5225/2100) = n * log(1.2)


divide both sides of the equation by log(1.2) to get:


log(5225/2100) / log(1.2) = n


solve for n to get:


n = 4.999504552.


confirm by replacing n with 4.999504552 in the equation and solve for A to get:


A = 2100 * (1 + .2) ^ 4.999504552 = 5225.


this confirms the value of n is good.