Answer: 15 years
Explanation:
A = P(1 + r/n)^(nt)
Where:
A = the final amount
P = the principal (initial deposit)
r = the interest rate (in decimal form)
n = the number of times interest is compounded per year
t = the number of years
In this case, we have:
A = $2469
P = $500
r = 5% = 0.05
n = 12 (compounded monthly)
Substituting these values into the formula, we have:
$2469 = $500(1 + 0.05/12)^(12t)
Dividing both sides of the equation by $500, we get:
4.938 = (1.00417)^(12t)
To solve for t, we need to take the logarithm of both sides of the equation. Let's use the natural logarithm (ln):
ln(4.938) = ln[(1.00417)^(12t)]
Using the property of logarithms, we can bring down the exponent:
ln(4.938) = 12t * ln(1.00417)
Now, we can solve for t by dividing both sides by 12 times ln(1.00417):
t = ln(4.938) / (12 * ln(1.00417))
Using a calculator, we find:
t ≈ 14.58
Rounding to the nearest year, the money was in the bank for approximately 15 years.