Payments of $1,800 and $2,400 were made on a $10,000 variable-rate loan 18 and 30 months after the date of the loan. The interest rate was 11.5% compounded semi-annually for the first two years and 10.74% compounded monthy thereafter. What amount was owed on the loan after three years?
This problem has to be solved in steps.
After 18 months (but before the first payment) the amount owed was
10,000*(1 + 0.115/2)^3 = 11,826.09
After making the first $1800 payment, the principal owed is 10,026.09. That increases by a factor 1 + 0.115/2 at 24 months, making the principal 10,602.59.
Then the interest rate goes down. After 30 months, the amount owed is
10,602.59. x (1 + 0.1074/2)
= 11,171.95
Finally, subtract the second payment from the principal and compute the increase during the last six months of the three year period.
A sum of £12,000 is invested for 5 years at an interest rate of 3.5% compounded annually. Calculate the value of the investment after 5 years.
To find the amount owed on the loan after three years, we need to consider the two different interest rates for the different time periods and calculate the remaining principal balance after each payment is made.
Let's break down the problem and calculate the remaining balance step by step:
1. Calculate the interest and remaining balance after the first 18 months:
- For the first two years, the interest rate is 11.5% compounded semi-annually.
- First, convert the annual interest rate to a semi-annual interest rate: 11.5% / 2 = 5.75% per six months.
- Next, calculate the interest on the initial loan amount after 18 months (1.5 years):
- Interest = Loan amount * (1 + Interest rate)^(Number of compounding periods)
- Interest = $10,000 * (1 + 0.0575)^(3), since there are three six-month periods (1.5 years)
- Calculate the remaining balance after the payment of $1,800:
- Remaining balance = Initial loan amount + Interest - Payment
- Remaining balance = $10,000 + Interest - $1,800
2. Calculate the interest and remaining balance after another 12 months (total of 30 months):
- For the subsequent months after the first two years, the interest rate is 10.74% compounded monthly.
- Calculate the interest on the remaining balance from step 1 after 12 months (1 year):
- Interest = Remaining balance from step 1 * (1 + Monthly interest rate)^(Number of compounding periods)
- Interest = Remaining balance from step 1 * (1 + 0.1074)^(12), since there are 12 monthly periods in a year
- Calculate the remaining balance after the payment of $2,400:
- Remaining balance = Remaining balance from step 1 + Interest - Payment
- Remaining balance = Remaining balance from step 1 + Interest - $2,400
3. Calculate the remaining balance after three years (total of 36 months):
- Calculate the interest on the remaining balance from step 2 after 6 months:
- Interest = Remaining balance from step 2 * (1 + Monthly interest rate)^(Number of compounding periods)
- Interest = Remaining balance from step 2 * (1 + 0.1074)^(6), since there are 6 monthly periods in 6 months
- Calculate the remaining balance after the last interest payment:
- Remaining balance = Remaining balance from step 2 + Interest
By following these steps, you can calculate the final remaining balance on the loan after three years.