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?
(1 point)
• 10 years
• 6 years
• 7 years
• 4 years
To solve this problem, we can use the formula for compound interest:
A = P(1 + r/n)^(nt)
where:
A is the final amount
P is the initial principal amount
r is the annual interest rate (as a decimal)
n is the number of times that interest is compounded per year
t is the number of years
In this case, we have:
P = $5,000
r = 3% = 0.03
A = $6,000
We want to solve for t, the number of years. Rearranging the formula, we have:
(1 + r/n)^(nt) = A/P
Substituting the values, we get:
(1 + 0.03/n)^(nt) = 6000/5000
To find the number of years, we will try different values of n (the number of times that interest is compounded per year) and solve for t.
For n = 1 (compounded annually), we have:
(1 + 0.03/1)^(1*t) = 6000/5000
(1 + 0.03)^(t) = 6/5
1.03^t = 1.2
To solve for t, we can take the logarithm of both sides:
t * log(1.03) = log(1.2)
t = log(1.2) / log(1.03)
Using a calculator, we find:
t ≈ 9.07
Therefore, it will take approximately 9.07 years for the account balance to reach at least $6,000 if interest is compounded annually.
The closest option is 10 years.