Final answer:
To find out how many years it would take for a compound interest loan to grow by 42% at an annual rate of 3.5%, we use the compound interest formula, solve for the time variable, and then calculate the result which tells us the number of years needed.
Step-by-step explanation:
To calculate how many years it would take for a compound interest loan to grow by 42% at an annual compound interest rate of 3.5%, we can use the formula for compound interest growth. This formula is A = P(1 + r/n)^{nt}, where:
- A is the amount of money accumulated after n years, including interest.
- P is the principal amount (the initial amount of money).
- r is the annual interest rate (decimal).
- n is the number of times that interest is compounded per year.
- t is the time the money is invested for, in years.
Since the problem states that the debt grows by 42%, the final amount A is equal to 1.42 times the principal P. Assuming the interest is compounded once per year (n=1), the formula simplifies to A = P(1 + r)^t.
To find t, we would set the formula up as follows: 1.42P = P(1 + 0.035)^t. Then, we need to solve this equation for t:
- Divide both sides by P to get 1.42 = (1 + 0.035)^t.
- Take the natural logarithm of both sides: ln(1.42) = ln((1.035)^t).
- Use the power rule of logarithms to bring down the exponent: ln(1.42) = t * ln(1.035).
- Divide by ln(1.035) to solve for t: t = ln(1.42) / ln(1.035).
Calculating this will give us the number of years needed for the debt to increase by 42%.