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The question asks for the amount of money saved by Suraj when paying in two equal yearly installments instead of a single payment after 2 years.

### Given Information:
- Suraj borrows a sum of Rs. \( P \).
- He pays it back in two equal yearly installments of Rs. \( 1,41,120 \).
- The rate of interest is \( 5\% \) compounded annually.
- We need to find the amount (in Rs.) saved by Suraj by paying in installments instead of a single payment after 2 years.

### Approach:

1. **Installment Method**:
   The amount paid annually is Rs. \( 1,41,120 \) for two years. We will calculate the present value of these installments discounted at 5% interest.

   - For the first installment, Rs. \( 1,41,120 \) is paid at the end of 1 year, so its present value is discounted for 1 year:
     \[
     PV_1 = \frac{1,41,120}{(1 + 0.05)} = \frac{1,41,120}{1.05}
     \]

   - For the second installment, Rs. \( 1,41,120 \) is paid at the end of 2 years, so its present value is discounted for 2 years:
     \[
     PV_2 = \frac{1,41,120}{(1.05)^2}
     \]

   The total present value of the installments is the sum of \( PV_1 \) and \( PV_2 \).

2. **Single Payment Method**:
   If Suraj makes a single payment after 2 years, the total amount he would have to pay (with compound interest) is the principal \( P \) compounded for 2 years:
   \[
   A = P(1 + 0.05)^2 = P \times 1.1025
   \]

3. **Find the Amount Saved**:
   The amount saved by paying in installments is the difference between the total amount paid in installments and the equivalent single payment amount.

I'll calculate the exact values now.### Calculation Summary:

1. **Installment Method**:
   - Present value of the first installment (after 1 year): Rs. \( 1,34,400 \)
   - Present value of the second installment (after 2 years): Rs. \( 1,28,000 \)
   - Total present value of both installments: Rs. \( 2,62,400 \)

2. **Equivalent Single Payment**:
   - The equivalent principal \( P \) for a single payment compounded for 2 years: Rs. \( 2,38,004.54 \)

3. **Amount Saved**:
   - By paying in installments, Suraj effectively saved Rs. \( 19,840 \) compared to making a single payment at the end of 2 years.

Would you like further details or have any questions?

Here are 5 related questions for deeper understanding:
1. How would the savings change if the interest rate were higher, say 10%?
2. What if the loan had to be repaid in three installments instead of two?
3. How does the concept of compound interest affect the calculation of savings?
4. Can we apply the same formula for monthly or quarterly installments?
5. What is the difference between simple and compound interest in such loan scenarios?

**Tip**: For long-term loans, paying in installments usually leads to less interest accrued, compared to a lump sum payment later.

A sum of Rs. P was borrowed from a bank by Suraj and paid back in two equal yearly installments, each of Rs. 1,41,120. If the rate of interest is 5% compounded annually, then what is the amount (in Rs.) saved by Suraj by paying in installments instead of a single payment after 2 years?

Learn how to calculate the amount saved by paying loan installments instead of a single payment after 2 years with a 5% compound interest rate. Includes step-by-step solution for present value of installments and compound interest. Suitable for Grades 9-12.

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