SOLUTION: The formula for an investment worth with interest compounded annually is A=P(1+i)^n, where P represents the initial investment, i is the interest rate, and A is the worth of the in

Question 1203599: The formula for an investment worth with interest compounded annually is A=P(1+i)^n, where P represents the initial investment, i is the interest rate, and A is the worth of the investment after n years.
a) Rearrange the formula for P. What was the initial investment of an investment worth $1000 that compounded 10% interest for 10 years?
b) Rearrange the formula for i. What is the interest rate of an investment whose worth went from $1000 to $1200 in 2 years?
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.
d) Estimate how many years would it take an investment of $2100 at 20% interest to reach a worth of $5225?

Found 2 solutions by MathLover1, Theo:
Answer by MathLover1(20850) (Show Source): You can put this solution on YOUR website!

The formula for an investment worth with interest compounded annually is

where represents the initial investment,is the interest rate, and is the worth of the investment after years.

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

if

%=

b) Rearrange the formula for. What is the interest rate of an investment whose worth went from $ to $ in years?

if

years

or %

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 Rule of is a calculation that estimates the number of years it takes to double your money at a specified rate of return.
for example
$ at % will double to $in: years

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

Note: We don't have to "estimate" but will solve it exactly!!!

to reach a worth of $will be

≈
≈years

Answer by Theo(13342) (Show Source): You can put this solution on YOUR website!
The formula for an investment worth with interest compounded annually is A=P * (1 + i)^n, where P represents the initial investment, i is the interest rate, and A is the worth of the investment after n years.

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

start with A = P * (1 + i) ^ n.
divide both sides of the equation by (1 + i) ^ n to get:
A / ((1 + i) ^ n) = P
switch sides to get:
P = A / ((1 + i) ^ n)
when A = 1000 and i = 10% interest copounded for 10 years, the formula becomes:
P = 1000 / ((1 + .10) ^ 10 = 385.5432894.
that's your solution.
by replacing P with that in the original equation and appling 10% compounded yearly to it for 10 years to get:
A = 385.5432894 * (1 + .10) ^ 10 = 1000.
this confirms that P was calculated correctly.

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

A = P * (1 + i) ^ n becomes:
1200 = 1000 * (1 + i) ^ 2
divide both sides of the 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
subtract 1 from both sides of the equation to get:
1.2 ^ (1/2) - 1 = i
simplify to get:
.095445115
that's your interest rate.
confirm by replacing i in the original equation and solving for A to get:
A = 1000 * (1 + .095445115) ^ 2 = 1200.
this confirms that the value of i is correct.

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 general formula is A = P * (1 + i) ^ n
you know A and you know P and you know i, but you don't know n.
you would use logarithms to find the value of n as follows.
you would divide both sides of the equation by P to get:
A/P = (1 + i) ^ n
you would take the log of both sides of this equation to get:
log(A/P) = log((1+i)^n)
by rule of logs, this becomes:
log(A/P) = n * log(1 + i)
you would divide both sides of the equation by log(1 + i) to get:
log(A/P) / log(1 + i) = n

d) Estimate how many years would it take an investment of $2100 at 20% interest to reach a worth of $5225?
A = P * (1 + i) ^ n becomes:
5225 = 2100 * (1 + .2) ^ n.
simplify to get:
5225 = 2100 * 1.2 ^ n
divie 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 rule of logs, this 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.
replace n in the oiginal equation with that and solve for A to get:
A = 2100 * 1.2 ^ 4.999504552 = 5225, confirming the value of n is good.

note that the log rule that allows this is log(b^a) = a * log(b).

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