Final answer:
To find out how many months it will take for an investment of $10,400 with a 3% compounded monthly interest rate to grow to $20,000, we use the formula for compound interest. Plugging in the values, we find that it takes approximately 6.623 months for the investment to reach $20,000.
Step-by-step explanation:
To find out how many months it will take for your investment to grow to $20,000, we need to use the formula for compound interest:
A = P(1 + r/n)nt
Where:
In this case, P = $10,400, r = 0.03 (3% in decimal form), n = 12 (monthly compounding), and A = $20,000. Let's plug in these values and solve for t:
$20,000 = $10,400(1 + 0.03/12)12t
Divide both sides by $10,400:
1.9231 = (1 + 0.03/12)12t
Take the natural logarithm of both sides:
ln(1.9231) = ln((1 + 0.03/12)12t)
Using the logarithmic identity, ln(ab) = b * ln(a):
ln(1.9231) = 12t * ln(1 + 0.03/12)
Divide both sides by 12 * ln(1 + 0.03/12):
t = ln(1.9231) / (12 * ln(1 + 0.03/12))
Using a calculator, we find that t is approximately 6.623 months.