Final answer:
The minimum number of times the bank compounds the interest is once per year. The maximum number of times the bank compounds the interest is approximately 14.21 times per year. Assuming an exact 5.00% APR, the account is compounded once per year.
Step-by-step explanation:
To determine the minimum and maximum number of times the bank compounds the interest, we need to find the compounding frequency.
The formula for compound interest is:
A = P(1 + r/n)^(nt)
Where:
- A is the final amount
- P is the principal amount (initial investment)
- r is the annual interest rate (as a decimal)
- n is the number of times the interest is compounded per year
- t is the number of years
In this case, we know A = $105, P = $100, and r = 0.05. Let's calculate the minimum and maximum values:
Minimum:
Let's assume the bank compounds interest annually (n = 1).
$105 = $100(1 + 0.05/1)^(1*t)
$105 = $100(1.05)^t
Dividing both sides by $100, we get:
1.05^t = 1.05
Taking the logarithm of both sides, we get:
t = log(1.05)/log(1.05)
≈ 1
So, the minimum number of times the bank compounds the interest is once per year.
Maximum:
Let's assume the bank compounds interest continuously (n approaches infinity).
The formula for compound interest with continuous compounding is:
A = Pe^(rt)
Substituting the given values, we get:
$105 = $100e^(0.05t)
Dividing both sides by $100, we get:
e^(0.05t) = 1.05
Taking the natural logarithm of both sides, we get:
0.05t = ln(1.05)
Dividing both sides by 0.05, we get:
t = ln(1.05)/0.05
≈ 14.21
So, the maximum number of times the bank compounds the interest is approximately 14.21 times per year.
Assuming an exact 5.00% APR:
If the bank is giving exactly 5.00% APR with no rounding, then we can assume the interest is compounded once per year. Therefore, the account is compounded once per year.