SOLUTION: A P38,000 loan bears interest at 10% compounded semi-annually and is to be repaid in semi-annual payments of P2,000 each. a. How many semi-annual payments must be the debtor make

Algebra.Com
Question 1193768: A P38,000 loan bears interest at 10% compounded semi-annually and is to be repaid in semi-annual payments of P2,000 each.
a. How many semi-annual payments must be the debtor make?
b. What smaller final payment should he make six months after the last payment of P2,000 is made?
Found 2 solutions by proyaop, ikleyn:
Answer by proyaop(69)   (Show Source): You can put this solution on YOUR website!
**a. Determine the Number of Semi-Annual Payments**
* **Find the effective semi-annual interest rate:**
* Semi-annual interest rate = (1 + Annual Interest Rate / Number of compounding periods per year)^(Number of compounding periods per year) - 1
* Semi-annual interest rate = (1 + 0.10 / 2)^(2) - 1
* Semi-annual interest rate = 0.0525 or 5.25%
* **Use a financial calculator or spreadsheet software (like Excel or Google Sheets) to determine the number of payments.**
* **In Excel, you can use the NPER function:**
* `=NPER(rate, pmt, pv, [fv], [type])`
* rate: Semi-annual interest rate (0.0525)
* pmt: Payment amount (-2000)
* pv: Present value (38000)
* fv: Future value (0, as the loan will be fully paid off)
* type: 0 for payments at the end of each period (default)
* This will give you the number of semi-annual payments required to repay the loan.
**b. Calculate the Smaller Final Payment**
1. **Calculate the remaining balance after the last full payment of P2,000:**
* This involves calculating the remaining balance after the number of full payments determined in part (a). You can use the financial calculator or spreadsheet functions like `FV` (future value) to calculate the remaining balance after these payments.
2. **Calculate the interest accrued on the remaining balance:**
* Multiply the remaining balance by the semi-annual interest rate.
3. **Calculate the final payment:**
* Final payment = Remaining balance + Interest accrued on the remaining balance
**Note:**
* This approach provides a general framework for solving this type of loan repayment problem.
* The specific calculations will require the use of financial tools or software.
Let me know if you'd like to explore the calculations using a specific financial calculator or spreadsheet software.

Answer by ikleyn(52776)   (Show Source): You can put this solution on YOUR website!
.
A P38,000 loan bears interest at 10% compounded semi-annually and is to be repaid
in semi-annual payments of P2,000 each.
a. How many semi-annual payments must be the debtor make?
b. What smaller final payment should he make six months after the last payment
of P2,000 is made?
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ 

Use the standard formula for the semi-annual payment for a loan

    P = ,


where L is the loan amount; r =  is the effective semi-annual compounding interest rate;
n is the number of payments; P is the semi-annual payment.


In this problem  P = $2000;  r =  = 0.05.


Substitute these values into the formula and get for semi-annual payment

    2000 = .


In this equation, n is the unknown: we should find n from this equation.


Simplify step by step

     = ,

    0.052631579 = ,

     = ,

    1.05263158 = ,

     = ,

    0.95 = ,

     = 1 - 0.95,

     = 0.05,

     = 0.05,

    1.05^n = 1/0.05,

    1.05^n = 20,

    n*log(1.05) = log(20),

    n =  = 61.4.


So, 61 full semi-annual payments should be made of 2,000 each,
and then the last,62-th payment, should be made of the lesser amount.


ANSWER.  61 full semi-annual payments should be made of 2,000 each,
         and then the last, 62-th payment, should be made of the lesser amount.

         The total number of semi-annual payments is 62.

Solved.



RELATED QUESTIONS

A ₱38,000 loan bears interest at 10% compounded semi-annually and is to be repaid in... (answered by proyaop,ikleyn)
A ₱38,000 loan bears interest at 10% compounded semi-annually and is to be repaid in (answered by yurtman,ikleyn)
What is the future value of a semi-annual payments of 8,000 pesos for 12 years with... (answered by Theo)
Gladys borrows 400 000 pesos at an interest rate of 4% per year compounded... (answered by Theo)
Gladys borrows ₱ 400,000.00 at an interest rate of 4% per year compounded... (answered by Theo)
To pay a ₱370,000 loan, Aling Vergie promises to pay ₱42,000 at the end of every 6... (answered by Theo)
A loan at 14% interest compounded semi-annually must be repaid with one single payment of (answered by greenestamps)
The Ram Company borrowed $20 000 at 10% compounded semi-annually and made payments toward (answered by Theo)
Find the present value and the future value of semi-annual payments of 105,000 pesos with (answered by Theo)