To determine the number of years it will take for a debt to double at 42% per annum compound interest, we can use the compound interest formula and solve for the time [tex]\( t \)[/tex]:
[tex]\[ A = P(1 + r/n)^{nt} \][/tex]
Here:
- [tex]\( A \)[/tex] is the amount of money accumulated after [tex]\( t \)[/tex] years, including interest.
- [tex]\( P \)[/tex] is the principal amount (the initial amount of debt).
- [tex]\( r \)[/tex] is the annual interest rate (as a decimal).
- [tex]\( n \)[/tex] is the number of times the interest is compounded per year.
- [tex]\( t \)[/tex] is the time the money is invested or borrowed for, in years.
We want to find how long it takes for the debt to double, so [tex]\( A = 2P \)[/tex]. Given that the interest rate is 42% per annum and compounding is annual, [tex]\( r = 0.42 \)[/tex] and [tex]\( n = 1 \)[/tex].
Substitute the known values into the formula:
[tex]\[ 2P = P(1 + 0.42)^t \][/tex]
To isolate [tex]\( t \)[/tex], follow these steps:
1. Divide both sides by [tex]\( P \)[/tex]:
[tex]\[ 2 = (1 + 0.42)^t \][/tex]
2. Simplify inside the parentheses:
[tex]\[ 2 = (1.42)^t \][/tex]
3. To solve for [tex]\( t \)[/tex], take the natural logarithm (ln) of both sides (or common logarithm, the process is similar):
[tex]\[ \ln(2) = \ln((1.42)^t) \][/tex]
4. Use the property of logarithms that allows you to bring the exponent down:
[tex]\[ \ln(2) = t \cdot \ln(1.42) \][/tex]
5. Solve for [tex]\( t \)[/tex]:
[tex]\[ t = \frac{\ln(2)}{\ln(1.42)} \][/tex]
Using a calculator to find the values of the logarithms:
[tex]\[ \ln(2) \approx 0.693147 \][/tex]
[tex]\[ \ln(1.42) \approx 0.350387 \][/tex]
Now divide these values:
[tex]\[ t = \frac{0.693147}{0.350387} \approx 1.9767106726617854 \][/tex]
Therefore, it will take approximately 1.98 years for the debt to double at an annual compound interest rate of 42%.
