Final answer:
Leon needs to invest for approximately 8 years at an interest rate of 4% to accumulate $78,008, starting with $57,000.
Step-by-step explanation:
To calculate how many years it will take Leon to accumulate $78,008 at an interest rate of 4% when he has $57,000 to invest, we can use the formula for compound interest:
A = P(1 + r/n)^(nt), where:
A is the future value of the investment/loan, including interest,
P is the principal investment amount (the initial deposit or loan amount),
r is the annual interest rate (decimal),
n is the number of times that interest is compounded per year, and
t is the number of years the money is invested or borrowed for.
We are looking for the variable t, which represents the number of years. First, we need to rearrange the formula to solve for t:
t = ln(A/P) / (n * ln(1 + r/n))
Assuming the interest is compounded annually (n=1), the equation simplifies to:
t = ln(A/P) / ln(1 + r)
Plugging in the values:
t = ln(78,008 / 57,000) / ln(1 + 0.04)
Solving for t, we find:
t = ln(1.36842105) / ln(1.04)
t = 0.312435 / 0.03922071
t ≈ 7.97
Leon will need to invest for approximately 8 years to accumulate enough to pay off a debt of $78,008.