Question 1106912: Suppose that you want to purchase a used car for $12000. You currently have $8000. If you invest your $8000 at an annual interest rate of 3%, compounded quarterly, how long will it take (in years) before you can purchase the car?
Answer by Theo(13342)   (Show Source):
You can put this solution on YOUR website! the price of the car is 12,000.
you have 8,000.
the annual interest rate is 3% compounded quarterly.
how many years before you can purchase the car?
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.
since you are compounding the interest rate quarterly, your interest rate per quarter is equal to 3% / 4 which is equal to .75% expressed as a percent and .0075 expressed as a rate.
in the formula, you use the rate, not the percent.
n is also expressed in quarters of a year, so if you want to know the number of years, you would have to divide n by 4 as well, once you find it.
your formula becomes 12,000 = 8,000 * (1 + .0075) ^ n.
divide both sides of this equation by 8,000 to get 12,000 / 8,000 = (1 + .0075) ^ n.
take the log of both sies of this equation to get log(12,000/8,000) = log(1.0075^n).
since log(1.0075^n) is equal to n * log(1.0075), your formula becomes log(12,000/8,000) = n * log(1.0075).
divide both sides of this equation by log(1.0075) and you get log(12,000/8,000) / log(1.0075) = n.
solve for n to get n = 54.2644945.
replace n in the original equation with this to get 12,000 = 8,000 * (1.0075)^54.2644945.
the result is 12,000 = 12,000.
recall that n is in quarters, so divide 54.2644945 by 4 to get 13.56612362 years.
an alternate way of analyzing this is to use the formula f = p * (1 + r/c) ^ (n*c).
this formula assumes annual interest rate and number of years and incorporates the number of compounding periods per year into it.
using this formula, you would get 12,000 = 8,000 * (1 + .03/4) ^ (n*4).
you would ue the same procedure to get log(12,000 / 8,000) / log(1 + .03/4) = n*4.
you would solve for n * 4 to get n * 4 = 54.2644945.
you would then solve for n to get n = 13.56612362.
in the formula of f = p * (1 + r/c) ^ (n*c), .....
f = future value as before
p = present value as before
r = annual interest rate (not the annual interest rate percent).
n = number of years
c = number of compounding periods per year.
in the first formula of f = p * (1 + r) ^ n,
r is assumed to be per time period and n is assumed to be number of time periods.
the adjustment from years to time periods is done prior to using the formula.
in the second formula of f = p * (1 + r/c) ^ (n*c),
r is assumed to be per year and n is assumed to be number of years and c is the number of compounding periods per year.
the adjustment from years to time periods is incorporated as part of the formula.
|
|
|
