The formula for finding the annual installment, when A is the amount taken on loan, where r% is the rate of interest, n is the number of installments, is:
Understanding the Annual Loan Installment Formula
The question asks for the specific formula used to determine the amount of each annual installment payment required to repay a loan. This formula takes into account the initial loan amount, the annual interest rate, and the total number of installments over which the loan will be repaid.
Based on the options provided, the formula for finding the annual installment amount is given as:
\(\text{Installment}= \frac{A}{{1 + {{\left( {\frac{{100}}{r}} \right)}^n}}} \times \frac{r}{{100}}\)
This formula connects the principal loan amount with the interest rate and the repayment period to calculate the fixed annual payment.
Key Components of the Annual Installment Calculation
Let's define the terms used in this annual installment formula:
- \(A\): This variable represents the principal amount of the loan taken. It is the original sum borrowed before any interest is added or payments are made.
- \(r\): This variable denotes the annual interest rate. It is typically expressed as a percentage (e.g., if the rate is 5%, \(r\) is 5). This is the cost per year of borrowing the money.
- \(n\): This variable signifies the total number of annual installments planned for the repayment of the loan. This is the term of the loan in years if payments are made annually.
- Installment: This is the resulting amount calculated by the formula, representing the fixed sum paid annually to cover both principal and interest until the loan is fully repaid.
Applying the Annual Installment Formula Step-by-Step
To calculate the annual installment using the given formula, you would follow these mathematical steps:
- Take the interest rate \(r\) and calculate the ratio \(\frac{{100}}{r}\).
- Raise this ratio \(\left(\frac{{100}}{r}\right)\) to the power of \(n\) (the number of installments). This gives you \({\left( {\frac{{100}}{r}} \right)}^n\).
- Add 1 to the result from step 2: \(1 + {{\left( {\frac{{100}}{r}} \right)}^n}\).
- Divide the loan amount \(A\) by the result from step 3: \(\frac{A}{{1 + {{\left( {\frac{{100}}{r}} \right)}^n}}}\). This calculates a part of the formula involving the loan amount and the time/rate factors.
- Calculate the ratio \(\frac{r}{{100}}\), which is the interest rate expressed as a decimal.
- Multiply the result from step 4 by the result from step 5. This final multiplication yields the annual installment amount.
Executing these steps with the specific values for the loan amount \(A\), interest rate \(r\), and number of installments \(n\) will provide the required annual payment amount according to this particular formula.
Revision Table: Key Variables for Annual Installment
| Variable | Represents | Notes |
|---|---|---|
| \(A\) | Loan Principal Amount | The initial sum borrowed. |
| \(r\) | Annual Interest Rate | Used as a percentage value (e.g., 7 for 7%). |
| \(n\) | Number of Installments | Total number of annual payments. |
| Installment | Annual Payment Amount | The fixed amount paid each year. |
Additional Information on Loan Repayments
Understanding loan installment calculations involves several core concepts in finance:
- Principal vs. Interest: Each loan payment typically consists of two parts: a portion that repays the original principal amount and a portion that pays the interest accumulated since the last payment. Early payments are often more interest-heavy, while later payments reduce the principal more significantly.
- Compounding: Interest is usually compounded, meaning that in subsequent periods, interest is calculated on the principal amount plus any accumulated interest from previous periods. Annual compounding means interest is calculated and added to the principal once a year.
- Amortization Schedule: For loans with equal installments, an amortization schedule details how each payment is allocated between principal and interest, showing the remaining loan balance after each payment. This table helps visualize the repayment process over the loan term.
While different formulas exist for calculating loan payments based on compounding frequency (annual, monthly, etc.) and payment structure, the formula provided is presented for calculating the annual installment under the specific conditions of loan amount \(A\), annual rate \(r\), and \(n\) annual installments.
Important Questions from Installments
A loan of Rs. 1,50,000 is availed with compound interest rate of 10% per annum for two years compounded annually. It is to be paid in equal yearly installments, and the installment is to be paid at the end of each year. The value of the equal yearly installment is : (Rounded off to two places of decimal)
A sum of Rs. P was borrowed and paid back in two equal yearly instalments, each of Rs. 35,280. If the rate of interest was 5% per annum and interest is compounding annually, then the value of P is ________.
A sum of Rs. 16400 is borrowed to be paid back in 2 years by equal payments allowing 5% compound interest. Find the annual payment.
A sum of Rs. 1100 was taken as a loan. This is to be paid in two equal installments. If the rate of interest is 20% per annum, compounded annually, find the amount payable in each installment.
A loan of Rs 15000 is to be repaid in 4 equal annual installments. If the compound interest rate is 10% per annum, what is the approximate amount of each installment?
