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?
To solve this problem, we need to determine the number of years it will take for the loan amount to reach $49,000 or more.
First, let's understand the formula for calculating compound interest:
A = P(1 + r/n)^(nt)
Where:
A = final amount
P = principal amount ($37,000 in this case)
r = annual interest rate (3% expressed as 0.03)
n = number of times interest is compounded per year (annually in this case)
t = time in years
We want to solve for t, so we can rearrange the formula:
t = (log(A/P) / log(1 + r/n)) / n
Plugging in the given values:
P = $37,000
r = 0.03
n = 1 (compounded annually)
A = $49,000
t = (log($49,000/$37,000) / log(1 + 0.03/1)) / 1
Now, we can substitute these values into a calculator to find the solution:
t ≈ 9.263
Since we are asked to write the answer as a whole number, we round up the decimal value to the nearest whole number:
t ≈ 10
So, it will take approximately 10 years for the loan amount to reach $49,000 or more.