Suppose you deposit $5,000 in a savings account that earns 3% annual interest. If you make no other withdraws or deposits, how many years will it take the account balance to reach at least $6,000?
a. 6 years
b. 7 years
c.10 years
d. 4 years
To solve this problem, we need to use the formula for compound interest:
A = P(1 + r/n)^(nt)
Where:
A = the future value of the investment/loan, including interest
P = the principal investment amount (initial deposit)
r = annual interest rate (as a decimal)
n = number of times that interest is compounded per year
t = number of years
In this case, we have:
P = $5,000
r = 3% = 0.03
A = $6,000
Substituting the values into the formula, we get:
$6,000 = $5,000(1 + 0.03/n)^(n*t)
As we make no other withdraws or deposits, we can assume the interest is compounded annually, which means n = 1.
$6,000 = $5,000(1 + 0.03/1)^(1*t)
Simplifying the equation, we have:
1.2 = (1.03)^t
To solve for t, we can use logarithms:
log(1.2) = log(1.03)^t
0.079181246 = t*log(1.03)
t = 0.079181246 / log(1.03)
Using a calculator, we find:
t ≈ 6.89
Rounding up to the nearest whole number, we get:
t ≈ 7
Therefore, it will take 7 years for the account balance to reach at least $6,000.
The correct answer is b. 7 years.