A loan of $37,000 is made at 3%interest, compounded annually. After how many years will the amount due reach $49,000 or more? (Use the calculator provided if necessary.)
Write the answer as a whole number.
****I get 1.03^10 = 49,724.91
So is the answer 10 years at 49,725.00?
Yes, our answers are exactly the same!
To find the number of years it will take for the amount due to reach $49,000 or more, we need to use the formula for compound interest:
A = P(1 + r/n)^(nt)
Where:
A = the amount due ($49,000 in this case)
P = the initial principal ($37,000 in this case)
r = the annual interest rate (3% or 0.03 as a decimal)
n = the number of compounding periods per year (1 for annually compounded interest)
t = the number of years
We want to solve for t, so we can rearrange the formula:
t = (log(A/P)) / (n * log(1 + r/n))
Substituting the given values into the formula:
t = (log(49000/37000)) / (1 * log(1 + 0.03/1))
Using a calculator:
t ≈ (ln(1.324324)) / (ln(1.03))
t ≈ 9.912
Since we need to give the answer as a whole number, we round up to the nearest whole number:
t ≈ 10
So, the loan amount will reach $49,000 or more after approximately 10 years.