To solve this problem, we want to compare the time it takes for both Maya's and Jaxon's investments to triple. To do this, we'll use the compound interest formula:
\[ A = P(1 + \frac{r}{n})^{nt} \]
Where:
- \( A \) is the final amount of money after interest.
- \( P \) is the principal amount (the initial amount of money).
- \( r \) is the annual interest rate (in decimal form).
- \( n \) is the number of times the interest is compounded per year.
- \( t \) is the time the money is invested, in years.
We need to rearrange the formula to solve for the time (\( t \)) it takes for the investment to triple. We want \( A = 3P \). The rearranged formula for \( t \) is:
\[ t = \frac{\log(A/P)}{n \cdot \log(1 + r/n)} \]
Let's calculate this for both Maya's and Jaxon's investments.
For Maya's investment:
Principal \( P = \$19,000 \)
Annual interest rate \( r = 8\% = 0.08 \) (converted to a decimal)
Compounded annually, so \( n = 1 \).
We want the investment to triple, so we set \( A = 3P = 3 \times \$19,000 = \$57,000 \). Using the formula for \( t \):
\[ t_{\text{Maya}} = \frac{\log(3)}{\log(1 + 0.08)}\]
For Jaxon's investment:
Principal \( P = \$19,000 \)
Annual interest rate \( r = 83\% = 0.83 \)
Compounded monthly, so \( n = 12 \).
Again, we want the investment to triple, so \( A = 3P \). Using the formula for \( t \):
\[ t_{\text{Jaxon}} = \frac{\log(3)}{12 \cdot \log(1 + \frac{0.83}{12})}\]
To find the difference in time, we subtract the two values.
Let's calculate these values.
For Maya:
\[ t_{\text{Maya}} = \frac{\log(3)}{\log(1 + 0.08)}\approx \frac{1.098612}{0.076961} \approx \frac{1.098612}{0.076961} \approx 14.27 \text{ years} \]
For Jaxon:
\[ t_{\text{Jaxon}} = \frac{\log(3)}{12 \cdot \log(1 + \frac{0.83}{12})} \approx \frac{1.098612}{12 \cdot 0.052456} \approx \frac{1.098612}{0.629472} \approx 1.74 \text{ years} \]
Now we find the difference:
\[ \text{Time difference} = t_{\text{Maya}} - t_{\text{Jaxon}} \approx 14.27 - 1.74 \approx 12.53 \text{ years} \]
Therefore, to the nearest hundredth of a year, it would take approximately 12.53 years longer for Maya's money to triple than for Jaxon's money to triple.
