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What annual compound interest rate is required for the debt of a compound interest loan to grow by 42% in 14 years?
Round your answer to the nearest tenth of a percent.
Solution
Let D = (A+I) = Amount loaned + Interest = total debt, t = time (14 years) and r = interest rate.
D=1.42 which is 42% greater than the original loaned amount A
1.42A = A(1+r)t
Divide both sides by A to get 1.42 = (1+r)t ==> 1.42 = (1+r)14
Take the 14th root of both sides to get 1.421/14 = 1+r ==> 1.420.07142871 = 1+r
Using a scientific calculator yields 1.02536229 = 1+r, so r = 0.02536 ~ 2.5%.
Test this answer by selecting A = $2222 (original loan).
A+I = A(1+0.02536)14
A+I = 2222(1.02536)14
A+I = 2222*1.41994 = $3155.00 to nearest dollar
Interest I = 3155 – 2222 = $933 which adds ~ 42% debt to the original loan.