Suppose $5000 is deposited in a bank account that compounds interest four times per year. The bank account contains $9900 after 13 years. What is the annual interest rate for this bank account?
i already set it up but i don't know where to go from there. Please help
To determine the annual interest rate for the bank account, you can use the formula for compound interest:
A = P(1 + r/n)^(nt)
Where:
A is the final amount
P is the principal amount (initial deposit)
r is the annual interest rate (in decimal form)
n is the number of times interest is compounded per year
t is the number of years
In this case, we have the following information:
- P = $5000
- A = $9900
- n = 4 (compounded four times per year)
- t = 13 years
Now, let's substitute the known values into the formula and solve for r.
9900 = 5000(1 + r/4)^(4*13)
Simplifying further:
9900/5000 = (1 + r/4)^52
1.98 = (1 + r/4)^52
To solve for r, we need to isolate r. Take the 52nd root on both sides:
(1 + r/4) = (1.98)^(1/52)
Now, subtract 1 from both sides:
r/4 = (1.98)^(1/52) - 1
To find the value of r, multiply both sides by 4:
r = 4 * [(1.98)^(1/52) - 1]
Using a calculator, you can evaluate this expression to find the annual interest rate for the bank account.
To find the annual interest rate for the bank account, we can use the compound interest formula:
A = P(1 + r/n)^(nt)
Where:
A = the final amount (9900 in this case)
P = the initial principal (5000 in this case)
r = the annual interest rate (what we want to find)
n = the number of times interest is compounded per year (4 in this case)
t = the number of years (13 in this case)
Now, let's plug in the values we have into the formula:
9900 = 5000(1 + r/4)^(4*13)
Simplifying further, we get:
(1 + r/4)^(52) = 9900/5000
Dividing both sides by 5000, we have:
(1 + r/4)^(52) = 1.98
To isolate (1 + r/4), we take the 52nd root of both sides:
1 + r/4 = (1.98)^(1/52)
Finally, subtracting 1 from both sides and multiplying by 4, we can solve for r:
r = 4[(1.98)^(1/52) - 1]
Using a calculator, we find:
r ≈ 4(0.039) ≈ 0.156
Therefore, the annual interest rate for this bank account is approximately 15.6%.