Answer:
14 yeards
Step-by-step explanation:
We can solve this problem using the compound interest formula and working a bit with trial and error or estimation.
Formula:
The formula for compound interest is:
A = P * (1 + r/n)^(n*t)
where:
A - Ending balance (amount you want to reach, $10,000 in this case)
P - Principal amount (initial investment, $6,000)
r - Annual interest rate (as a decimal, 0.037 for 3.7%)
n - Number of times interest is compounded per year (quarterly, so n = 4)
t - Number of years (unknown, what we need to find)
Solving by Estimation:
Estimate the time: We know the initial amount is $6,000, and we need to reach $10,000. With a 3.7% interest compounded quarterly, it likely takes more than a few years but less than decades to achieve this growth. Let's start with an estimate of 10 years (t = 10).
Calculate the ending balance with the estimated time:
Plug the values into the formula (assuming you have a calculator that can handle exponents):
A = 6000 * (1 + 0.037 / 4)^(4 * 10)
Note: Even though the interest rate is given annually (3.7%), we use it divided by the number of compounding periods (n) because the interest is compounded quarterly (4 times a year).
Compare the result with the target balance:
If the calculated ending balance (A) is less than $10,000, the estimated time (t) is too low. We need to try a higher value for t.
If the calculated ending balance (A) is more than $10,000, the estimated time (t) is too high. We need to try a lower value for t.
Iterate and Refine:
In this case, with t = 10, the calculated ending balance will likely be less than $10,000. So, we need to increase the estimated time. You can continue by trying t = 12, 14, and so on, until the calculated A gets closer to $10,000.
Alternatively:
Some financial calculators or spreadsheet tools have built-in functions for compound interest calculations. You can use these tools to enter the initial amount, interest rate, compounding frequency, and target balance to find the time it takes to reach the goal.
Final Answer:
Using trial and error or a financial calculator, you will find that it approximately takes 14 years for the investment to reach $10,000 with a starting amount of $6,000, an interest rate of 3.7% compounded quarterly.
Note: This is an approximate answer due to rounding during calculations.