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?
P = Po(1+r)^n.
r = Quarterly % rate.
n = 4 comp/yr + 13yrs = 52 = The # of
compounding periods.
9900 = 5000(1=r)^52.
(1+r)^52 = 9900 / 5000 = 1.98
Take Log of both sides:
52*Log(1+r) = Log1.98 = 0.296665
Log(1+r) = 0.00571.
1+r = 10^0.00571. = 1.013223.
r = 1.013223 - 1 = 0.013223 = Quarterly % rate.
APR = 4 * 0.013223 = 0.0529 = 5.29 %.
= Annual % rate.
thanks!
To find the annual interest rate for this bank account, we can use the formula for compound interest:
A = P(1 + r/n)^(n*t)
Where:
A = the final amount (in this case, $9900)
P = the principal amount (in this case, $5000)
r = the annual interest rate (what we are looking to find)
n = the number of times interest is compounded per year (four times in this case)
t = the number of years (13 years in this case)
Substituting the given values into the formula:
9900 = 5000(1 + r/4)^(4*13)
Divide both sides by 5000:
9900/5000 = (1 + r/4)^(4*13)
1.98 = (1 + r/4)^52
Next, take the 52nd root of both sides:
(1.98)^(1/52) = 1 + r/4
1.030247 = 1 + r/4
Subtract 1 from both sides:
0.030247 = r/4
Multiply both sides by 4:
r = 0.121
Therefore, the annual interest rate for this bank account is 12.1%.
To find the annual interest rate for the bank account, we can use the formula for compound interest:
\[A = P \left(1 + \frac{r}{n}\right)^{nt}\]
Where:
- A is the amount of money accumulated after the specified time period
- P is the principal amount (the initial deposit)
- r is the annual interest rate (in decimal form)
- n is the number of times the interest is compounded per year
- t is the number of years the money is invested for
In this case, we are given:
- P = $5000
- A = $9900
- n = 4, as interest is compounded four times per year
- t = 13 years
Now, we can rearrange the formula to solve for r:
\[r = n \left(\left(\frac{A}{P}\right)^{\frac{1}{nt}} - 1\right)\]
Substituting the given values:
\[r = 4 \left(\left(\frac{9900}{5000}\right)^{\frac{1}{4 \times 13}} - 1\right)\]
Simplifying the expression:
\[r = 4 \left(\left(\frac{99}{50}\right)^{\frac{1}{52}} - 1\right)\]
Using a calculator to evaluate this expression, we find:
\[r \approx 0.0862\]
Therefore, the annual interest rate for this bank account is approximately 8.62%.