Vlad purchased some furniture for his apartment. The total cost was $ 2943.37. He paid $ 850 down and financed the rest for 18 months. At the end of the finance period .Vlad owed 4 2147.28 . What annual interest rate, compounded monthly , was he being charged ? Round your answer to two decimal places
To calculate the annual interest rate, we can use the formula for compound interest:
A = P(1 + r/n)^(n*t)
Where:
A = final amount
P = principal amount
r = annual interest rate (as a decimal)
n = number of times interest is compounded per year
t = number of years
In this case, the final amount (A) is $4,214.28, the principal amount (P) is $2,093.37 (the remaining amount financed), the interest is compounded monthly (n = 12), and the time (t) is 18 months (or 1.5 years).
Plug in these values into the formula and solve for r:
4,214.28 = 2,943.37 * (1 + r/12)^(12*1.5)
To isolate r, divide both sides by 2,943.37:
4,214.28 / 2,943.37 = (1 + r/12)^(18)
Now, take the natural logarithm of both sides to remove the exponent:
ln(4,214.28 / 2,943.37) = ln(1 + r/12)^(18)
The left side of the equation represents the natural logarithm of the ratio of the final amount to the principal amount. Use a calculator to find this value.
Next, divide the result by 18:
(ln(4,214.28 / 2,943.37)) / 18 = ln(1 + r/12)
Now, multiply both sides by 12:
12 * (ln(4,214.28 / 2,943.37)) / 18 = r/12
Simplify the left side:
(12/18) * (ln(4,214.28 / 2,943.37)) = r/12
To find r, multiply both sides by 12:
12 * (12/18) * (ln(4,214.28 / 2,943.37)) = r
Finally, round the result to two decimal places to get the annual interest rate.