Suppose you deposit $5,000 in a savings account that earns 3% annual interest. If you make no other withdrawals or deposits, how many years will it take the account balance to reach at least $6,000?
To find out how many years it will take for the account balance to reach at least $6,000, we can use the formula for compound interest:
A = P(1 + r/n)^(nt)
Where:
A = the amount of money accumulated after n years, including interest.
P = the principal amount ($5,000 in this case).
r = the annual interest rate (3% or 0.03).
n = the number of times that interest is compounded per year (assuming it is compounded annually).
t = the number of years the money is invested for.
We want to find the number of years (t) it will take for the account balance to reach at least $6,000, so we set A = $6,000 and solve for t:
$6,000 = $5,000(1 + 0.03/1)^(1t)
$6,000 = $5,000(1.03)^t
$6,000/$5,000 = (1.03)^t
1.2 = 1.03^t
To solve for t, we take the natural logarithm of both sides:
ln(1.2) = ln(1.03^t)
ln(1.2) = t ln(1.03)
t = ln(1.2) / ln(1.03)
t ≈ 9.89
It will take approximately 9.89 years for the account balance to reach at least $6,000. This means it will take 10 years for the account balance to actually surpass $6,000.