How much more would you earn in the first investment than in the second investment? $48 comma 000 invested for 30 years at 12% compounded annually $48 comma 000 invested for 30 years at 6% compounded monthly LOADING... Click the icon to view some finance formulas. Question content area bottom Part 1 You would earn $ enter your response here more on the first investment than in the second investment. (Round to the nearest dollar as needed.)
To determine how much more you would earn in the first investment than in the second investment, we need to calculate the future value of both investments using the appropriate formulas.
The formula for future value (FV) when compounded annually is:
[ FV = P(1 + r)^n ]
Where:
For the first investment:
Calculating the future value:
[ FV_1 = 48,000(1 + 0.12)^{30} ]
[ FV_1 = 48,000(1.12)^{30} ]
Calculating ( (1.12)^{30} ):
[ (1.12)^{30} \approx 29.959 ]
Now, calculate ( FV_1 ):
[ FV_1 \approx 48,000 \times 29.959 \approx 1,438,992 ]
The formula for future value when compounded monthly is:
[ FV = P(1 + \frac{r}{m})^{mn} ]
Where:
For the second investment:
Calculating the future value:
[ FV_2 = 48,000\left(1 + \frac{0.06}{12}\right)^{12 \times 30} ]
Calculating ( \frac{0.06}{12} = 0.005 ):
[ FV_2 = 48,000(1 + 0.005)^{360} ]
Calculating ( (1.005)^{360} ):
[ (1.005)^{360} \approx 6.022575 ]
Now, calculate ( FV_2 ):
[ FV_2 \approx 48,000 \times 6.022575 \approx 289,082 ]
Now, we can find the difference between the two future values:
[ \text{Difference} = FV_1 - FV_2 ]
[ \text{Difference} \approx 1,438,992 - 289,082 \approx 1,149,910 ]
You would earn approximately $1,149,910 more on the first investment than on the second investment.
Final Answer: $1,149,910 (rounded to the nearest