---
title: "Car loan interest rate and lump sum at year 3"
subject: "Engineering Economics"
chapter: 2
key: "semester_6/engineering_economics:car-loan-installments-lump-sum-year3"
source: https://www.bctengineer.com/q/semester_6/engineering_economics/car-loan-installments-lump-sum-year3
site: BCT Engineering
---

# Car loan interest rate and lump sum at year 3

Engineering Economics · Chapter 2: Interest and Time Value of Money

- Asked in: 2067 Mangsir, 4+4 marks
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_6/engineering_economics/car-loan-installments-lump-sum-year3
- Whole chapter: https://www.bctengineer.com/md/semester_6/engineering_economics/chapter-2

## Question

Mr. Basnet purchases a car which costs Rs. 20,00,000. He pays 40% as down payment. Remaining amount will be paid on installment basis and he wishes to pay Rs. 25,000 per month for next five years. What annual interest rate will he be paying? At the end of 3rd year, what lump sum amount should he pay to clear all his dues?

## Short answer

Loan amount Rs 12,00,000. Solving 12,00,000 = 25,000(P/A,i,60) gives monthly i ≈ 0.7629%, so nominal annual rate ≈ 9.15%. At end of year 3, the lump sum to clear dues is the present worth of remaining 24 payments: Rs 5,46,381.

## Answer

#### Loan amount
- Down payment = 40% × 20,00,000 = Rs 8,00,000
- Loan P = 20,00,000 − 8,00,000 = Rs 12,00,000
- Monthly installment A = Rs 25,000, n = 5 × 12 = 60 months

```text
 +12,00,000                             (loan received)
 0    1     2     3   ...   59    60    (months)
      ↓     ↓     ↓         ↓     ↓
     25,000 each month (end of month)
```

#### (a) Annual interest rate
Equation: 12,00,000 = 25,000 (P/A, i, 60), so (P/A, i, 60) = 48.

Trial and error (monthly i):

| i per month | (P/A, i, 60) | Compare with 48 |
|---|---|---|
| 0.70% | 48.8559 | too high |
| 0.80% | 47.5042 | too low |
| 0.7629% | 48.0000 | = 48 |

Interpolating between 0.70% and 0.80%: i = 0.70 + 0.10 × (48.8559 − 48)/(48.8559 − 47.5042) ≈ 0.763% per month.

- Nominal annual rate = 12 × 0.7629% = **9.15% per year**
- Effective annual rate = (1.007629)^12 − 1 = 9.55%

#### (b) Lump sum at the end of year 3
After 36 installments are paid, 24 installments remain. The amount that clears the dues is the present worth of the remaining 24 payments at month 36:

- (P/A, 0.7629%, 24) = 21.8553
- Lump sum = 25,000 × 21.8553 = Rs 546,381

This is the outstanding balance just after the 36th installment is paid. If he has not yet paid the 36th installment, he pays 25,000 + 546,381 = Rs 571,381.

**Answer: interest ≈ 9.15% per year (nominal); lump sum at end of year 3 ≈ Rs 546,381.**

## Check yourself

The answer's key points as multiple-choice questions, with the correct option and why.

1. Loan amount?
   A. Rs 8,00,000
   B. Rs 20,00,000
   C. Rs 15,00,000
   D. Rs 12,00,000 (60% of 20,00,000)
   Answer: D. 40% is paid as down payment. (Key point: Loan)

2. Equation for interest rate?
   A. (P/A, i, 60) = 12,00,000/25,000 = 48
   B. (F/A, i, 60) = 48
   C. (P/A, i, 5) = 48
   D. (A/P, i, 60) = 48
   Answer: A. Solved by trial and interpolation. (Key point: Equation)

3. Annual nominal rate?
   A. 0.76%
   B. About 9.15% (12 x 0.7629%)
   C. 9.55%
   D. 7.629%
   Answer: B. Monthly rate times 12. (Key point: Nominal rate)

4. Lump sum at end of year 3?
   A. 25,000 x 36
   B. Rs 12,00,000
   C. PW of remaining 24 payments: 25,000 (P/A,0.7629%,24) = Rs 546,381
   D. 25,000 (F/A,0.7629%,36)
   Answer: C. Outstanding balance equals PW of remaining installments. (Key point: Lump sum)

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