SOLUTION: Three years ago Mr Smith borrowed 4000 from Sarah for a duration of five years at 12% interest rate per annum and compounded monthly. One year ago He borrowed 8000 at 16% interest

Question 1186531: Three years ago Mr Smith borrowed 4000 from Sarah for a duration of five years at 12% interest rate per annum and compounded monthly. One year ago He borrowed 8000 at 16% interest per annum and compounded quarterly,also for five years. He agrees to pay his debt in two equal instalments,one now and another one five years later. If Sarah charges 20% interest rate per annum compounded half yearly, how much will each payment be?.
Please help with the homework.thanks in advance
Answer by CPhill(1987) (Show Source): You can put this solution on YOUR website!
Here's how to break down this compound interest problem:
**Loan 1:**
* Principal: $4000
* Interest Rate: 12% per annum (0.12/year)
* Compounding: Monthly (12 times/year)
* Loan Duration: 5 years
Since the loan was taken three years ago, it has been accruing interest for three years already. We need to calculate the accumulated amount after three years.
1. **Monthly interest rate:** 0.12 / 12 = 0.01
2. **Number of compounding periods:** 3 years * 12 months/year = 36 months
3. **Accumulated amount:** $4000 * (1 + 0.01)^36 = $4000 * 1.43076878 ≈ $5723.08
**Loan 2:**
* Principal: $8000
* Interest Rate: 16% per annum (0.16/year)
* Compounding: Quarterly (4 times/year)
* Loan Duration: 5 years
Since the loan was taken one year ago, it has been accruing interest for one year.
1. **Quarterly interest rate:** 0.16 / 4 = 0.04
2. **Number of compounding periods:** 1 year * 4 quarters/year = 4 quarters
3. **Accumulated amount:** $8000 * (1 + 0.04)^4 = $8000 * 1.16985856 ≈ $9358.87
**Total Debt:**
Mr. Smith's total debt now is $5723.08 + $9358.87 = $15081.95
**Repayment:**
Mr. Smith will make two equal payments: one now and one five years from now. Sarah charges 20% interest compounded semi-annually.
Let 'x' be the amount of each payment. The present value of these two payments must equal the total debt.
1. **Semi-annual interest rate:** 0.20 / 2 = 0.10
2. **Number of compounding periods for the future payment:** 5 years * 2 periods/year = 10 periods
The present value equation is:
$15081.95 = x + \frac{x}{(1 + 0.10)^{10}}$
$15081.95 = x + \frac{x}{2.59374246}$
$15081.95 = x + 0.38554329x$
$15081.95 = 1.38554329x$
$x = \frac{15081.95}{1.38554329}$
$x ≈ 10885.56$
**Answer:**
Each payment will be approximately $10885.56.

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