1. Lee Holmes deposited $16,700 in a new savings account at 6% interest compounded semiannually. At the beginning of year 4, Lee deposits an additional $41,700 at 6% interest compounded semiannually.
At the end of year 6, what is the balance in Lee’s account?
2.Bill Moore is buying a used Winnebago. His April monthly interest at 10% was $134.
What was Bill’s principal balance at the beginning of April?
3.The Treasury Department auctioned $16 billion in three-month (13-week) bills in denominations of thirty thousand dollars at a discount rate of 4.336%.
What would be the effective rate of interest?
1. To calculate the balance in Lee's account at the end of year 6, we will need to apply the compound interest formula for each deposit separately.
First deposit:
Principal (P) = $16,700
Interest rate (r) = 6% = 0.06
Compounded semiannually means time (t) is in years, so t = 6 years.
The formula to calculate the balance (B) is:
B = P * (1 + r/n)^(n*t)
n represents the number of compounding periods in a year. Since it is compounded semiannually, n = 2.
For the first deposit:
B1 = 16,700 * (1 + 0.06/2)^(2 * 6)
Second deposit:
Principal (P2) = $41,700
t2 = 6 - 4 = 2 years (since the second deposit was made at the beginning of year 4)
For the second deposit:
B2 = P2 * (1 + 0.06/2)^(2 * 2)
Finally, we can calculate the total balance at the end of year 6 by summing up the balances from the two deposits:
Total Balance = B1 + B2
2. To calculate Bill's principal balance at the beginning of April, we can use the formula for simple interest:
Interest for the month (I) = $134
Interest rate (r) = 10% = 0.10
The formula to calculate the principal balance (P) is:
P = I / (r * t)
Here, t represents the time in years. Since the interest is given for one month, t = 1/12 (since there are 12 months in a year).
Principal balance at the beginning of April = $134 / (0.10 * 1/12)
3. To calculate the effective rate of interest for the Treasury Department auctioned bills, we can use the formula for discount rate (d) and effective rate (r):
d = (Discount / Face Value) * (360 / Days)
The Treasury Department auctioned $16 billion in denominations of $30,000, so the number of bills issued is:
Number of bills = $16,000,000,000 / $30,000
Since the bill is for 13 weeks:
Days = 13 * 7 = 91 days
Now we can substitute the values into the discount rate formula:
d = (0.04336 * 100) * (360 / 91)
To find the effective rate of interest (r), we can use the formula:
r = (1 - d) * (365/Days) * 100
To solve these questions, we need to use the formulas for compound interest.
1. The formula for compound interest is: A = P(1 + r/n)^(nt)
Where:
A is the final balance
P is the principal amount (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
In this case, for the first 3 years, we can calculate the balance using:
A = 16700(1 + 0.06/2)^(2*3)
A = 16700(1 + 0.03)^6
After 3 years, Lee makes an additional deposit of $41,700. So the principal amount becomes $16,700 + $41,700 = $58,400.
Now, we can calculate the balance at the end of year 6 using the new principal amount and the same formula:
A = 58400(1 + 0.06/2)^(2*3)
A = 58400(1 + 0.03)^6
2. To find Bill's principal balance at the beginning of April, we can use the formula for simple interest:
I = P * r * t
Where:
I is the interest earned
P is the principal amount
r is the interest rate (in decimal form)
t is the time period
In this case, the interest earned in April is $134, and the interest rate is 10%. Using the formula, we can rearrange it to solve for the principal amount:
P = I / (r * t)
P = 134 / (0.10 * 1)
3. To find the effective rate of interest, we need to use the formula for discount interest:
Discount rate = (D / F) * (360 / t)
Where:
D is the discount (difference between the face value and the purchase price)
F is the face value of the bill
t is the time period in days
In this case, the discount rate is 4.336%. To find the effective rate of interest, we use the formula:
Effective rate = (1 - Discount rate) * (360 / t)
Effective rate = (1 - 0.04336) * (360 / 90)
Effective rate = 0.95664 * 4
Please note that in all these calculations, the interest rates are assumed to be in decimal form. If given as percentages, they need to be divided by 100 before using them in the formulas.