i.	A person borrows Rs. 10,000 at 8% per annum simple interest and repays the loan in two equal instalments, one at the end of 3 years and the other at the end of 5 years. Inspect the amount of each instalment?

To find the amount of each instalment, let's solve this problem using the concept of simple interest and the equation of value. The borrower repays the loan in two equal instalments: one at the end of 3 years and the other at the end of 5 years. 

### Step 1: Understand the Terms
- Principal amount (P) = Rs. 10,000
- Rate of interest (R) = 8% per annum
- The two instalments are paid at the end of 3 years and 5 years.

### Step 2: Equation of Value at the Time of Loan (Start Time)
Since the loan is being paid in two equal instalments, we need to find the value of the instalments so that the present value of these instalments is equal to the initial loan amount, considering the time value of money at 8% per annum.

Let the instalment amount be Rs. \(X\).

#### Present Value Calculation
- First instalment of Rs. \(X\) is paid at the end of 3 years. Its present value at the time of loan (start time) is:
  \[
  \text{Present Value of First Instalment} = \frac{X}{(1 + 0.08)^3}
  \]
- Second instalment of Rs. \(X\) is paid at the end of 5 years. Its present value at the time of loan (start time) is:
  \[
  \text{Present Value of Second Instalment} = \frac{X}{(1 + 0.08)^5}
  \]

### Step 3: Set Up the Equation of Value
The sum of the present values of both instalments should be equal to the initial loan amount (Rs. 10,000):
\[
\frac{X}{(1.08)^3} + \frac{X}{(1.08)^5} = 10,000
\]

### Step 4: Solve for X
Now, let's solve this equation to find the value of \(X\).

\[
X \left(\frac{1}{(1.08)^3} + \frac{1}{(1.08)^5}\right) = 10,000
\]

\[
X \left(\frac{1}{1.25971} + \frac{1}{1.46933}\right) = 10,000
\]

\[
X \left(0.7938 + 0.6806\right) = 10,000
\]

\[
X \times 1.4744 = 10,000
\]

\[
X = \frac{10,000}{1.4744}
\]

Let's calculate the value of \(X\).The amount of each instalment is approximately Rs. 6,782.35.

This means the borrower will repay the loan in two equal instalments of Rs. 6,782.35 each, one at the end of 3 years and the other at the end of 5 years.

Would you like more details or have any questions?

Here are some related questions you might find useful:
1. How is compound interest different from simple interest in loan repayments?
2. What would be the total interest paid if the entire loan amount was paid at the end of 5 years?
3. How would the instalments change if the interest rate was 10% instead of 8%?
4. What is the effect of changing the instalment periods (e.g., 4 years and 6 years) on the instalment amounts?
5. How can you calculate the instalments if the interest was compounded annually?

**Tip:** When dealing with loan repayments in instalments, always discount future payments to their present value using the given interest rate to ensure accurate calculations.

A person borrows Rs. 10,000 at 8% per annum simple interest and repays the loan in two equal instalments, one at the end of 3 years and the other at the end of 5 years. Inspect the amount of each instalment?

Learn how to calculate the equal instalments required to repay a Rs. 10,000 loan at 8% simple interest with payments at the end of 3 and 5 years. This problem involves key financial concepts like the time value of money, simple interest, and the equation of value. Ideal for students in Grades 9-12.

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