To determine which rate yields a larger amount after 4 years, we need to compare two different interest compounding methods: 12% compounded monthly and 11.94% compounded continuously.
Step 1: Calculate the amount with 12% compounded monthly
For interest compounded monthly, we use the formula for compound interest:
[tex]\[
A = P \left(1 + \frac{r}{n}\right)^{nt}
\][/tex]
Where:
- [tex]\(P\)[/tex] is the principal amount ([tex]$12,000)
- \(r\) is the annual nominal interest rate (12% or 0.12 as a decimal)
- \(n\) is the number of times interest is compounded per year (12 for monthly compounding)
- \(t\) is the time the money is invested for in years (4 years)
Plugging in these values:
\[
A = 12000 \left(1 + \frac{0.12}{12}\right)^{12 \times 4}
\]
Calculating inside the parentheses first:
\[
1 + \frac{0.12}{12} = 1 + 0.01 = 1.01
\]
Now raise this value to the power of \(12 \times 4 = 48\):
\[
1.01^{48}
\]
The amount after 4 years with 12% compounded monthly is:
\[
A = 12000 \times 1.01^{48} \approx 19346.71
\]
Step 2: Calculate the amount with 11.94% compounded continuously
For interest compounded continuously, we use the formula:
\[
A = Pe^{rt}
\]
Where:
- \(P\) is the principal amount ($[/tex]12,000)
- [tex]\(e\)[/tex] is the base of the natural logarithm (approximately 2.71828)
- [tex]\(r\)[/tex] is the annual nominal interest rate (11.94% or 0.1194 as a decimal)
- [tex]\(t\)[/tex] is the time the money is invested for in years (4 years)
Plugging in these values:
[tex]\[
A = 12000 \times e^{0.1194 \times 4}
\][/tex]
First, calculate the exponent:
[tex]\[
0.1194 \times 4 = 0.4776
\][/tex]
Then, raise [tex]\(e\)[/tex] to this exponent:
[tex]\[
e^{0.4776}
\][/tex]
The amount after 4 years with 11.94% compounded continuously is:
[tex]\[
A = 12000 \times e^{0.4776} \approx 19346.41
\][/tex]
Comparison and Conclusion
- 12% compounded monthly yields approximately [tex]$19,346.71.
- 11.94% compounded continuously yields approximately $[/tex]19,346.41.
When comparing the two amounts:
[tex]\[
19346.71 > 19346.41
\][/tex]
Therefore, the 12% compounded monthly rate yields a larger amount after 4 years.
Answer: The 12% compounded monthly option yields the larger amount.
