Final answer:
The interest rate needed to grow $1,000 to $1,500 when compounded quarterly for 4 years is not provided in the multiple-choice options. After using the compound interest formula, the calculated rate should be around 10.7%, which suggests there might be an error in the question or the provided options.
Step-by-step explanation:
To determine the interest rate needed to grow $1,000 to $1,500 when compounded quarterly for 4 years, we can use the compound interest formula:
A = P(1 + \frac{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 or borrowed for, in years.
Plugging in the values we get:
$1,500 = $1,000(1 + \frac{r}{4})^{4\times4}
We need to solve for r, which will require the use of algebra and possibly a calculator for precision.
After simplifying, we would find that r = 0.107 or 10.7%, which is not an option given in the question. Hence, we would go through our calculations once again to ensure accuracy.
In this scenario, there might have been either a miscalculation or the options provided by the question could be incorrect. Assuming the calculations have been done correctly and the options are incorrect, it's important to communicate this discrepancy to the student.
It's worth noting that compound interest can have significant effects over time. Starting to save early and allowing investments to benefit from compound interest can lead to substantial growth of savings, as illustrated by the examples provided in the reference information.