We are given three financial problems to solve: (8) A man invests Tk. 10,000 and withdraws Tk. 1,500 at the end of each year for 7 years. The money is invested at 4% per annum. We need to find the amount left after 7 years. (9) Mr. Karim can purchase a machine by paying Tk. 40,000 in cash or by 8 equal yearly installments paid at the beginning of each year. The interest rate is 12%. We need to find the amount of each installment. (10) You borrowed Tk. 42,000 at 12% to be repaid over 8 years with equal installment payments at the end of each year. We must prepare an amortization schedule for a lender providing an 8% return. This seems to be a typo: should be for a lender requiring a 12% return, which is the initial interest rate. I will create the schedule using the correct 12% interest rate and loan amount of 42,000.

1. Problem Description

We are given three financial problems to solve:

(8) A man invests Tk. 10,000 and withdraws Tk. 1,500 at the end of each year for 7 years. The money is invested at 4% per annum. We need to find the amount left after 7 years.

(9) Mr. Karim can purchase a machine by paying Tk. 40,000 in cash or by 8 equal yearly installments paid at the beginning of each year. The interest rate is 12%. We need to find the amount of each installment.

(10) You borrowed Tk. 42,000 at 12% to be repaid over 8 years with equal installment payments at the end of each year. We must prepare an amortization schedule for a lender providing an 8% return. This seems to be a typo: should be for a lender requiring a 12% return, which is the initial interest rate. I will create the schedule using the correct 12% interest rate and loan amount of 42,

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2. Solution Steps

Problem 8:

We need to calculate the future value of the initial investment and subtract the future value of the series of withdrawals.

Let $P = 10000$ be the initial investment, $W = 1500$ be the annual withdrawal, $r = 0.04$ be the interest rate, and $n = 7$ be the number of years.

The future value of the initial investment after 7 years is:

$FV = P(1+r)^n = 10000(1+0.04)^7 = 10000(1.04)^7 \approx 10000(1.3159) = 13159.32$

The future value of the series of withdrawals is:

$FV_W = W \frac{(1+r)^n - 1}{r} = 1500 \frac{(1.04)^7 - 1}{0.04} = 1500 \frac{1.3159 - 1}{0.04} = 1500 \frac{0.3159}{0.04} = 1500(7.8929) \approx 11839.35$

The amount left after 7 years is:

$Amount = FV - FV_W = 13159.32 - 11839.35 = 1319.97$

Problem 9:

This is an annuity due problem.

Let $P = 40000$ be the present value of the machine, $n = 8$ be the number of installments, and $r = 0.12$ be the interest rate. Let $A$ be the amount of each installment.

The present value of an annuity due is given by:

$P = A \frac{1 - (1+r)^{-n}}{r} (1+r)$

$40000 = A \frac{1 - (1.12)^{-8}}{0.12} (1.12)$

$40000 = A \frac{1 - 0.40388}{0.12} (1.12)$

$40000 = A \frac{0.59612}{0.12} (1.12)$

$40000 = A (4.96766) (1.12)$

$40000 = A (5.56378)$

$A = \frac{40000}{5.56378} \approx 7188.34$

Problem 10:

We are given a loan of Tk. 42,000 at 12% to be repaid over 8 years. We need to create an amortization schedule.

Let $P = 42000$ be the loan amount, $r = 0.12$ be the interest rate, and $n = 8$ be the number of years.

First, we calculate the annual payment A using the annuity formula:

$P = A \frac{1 - (1+r)^{-n}}{r}$

$42000 = A \frac{1 - (1.12)^{-8}}{0.12}$

$42000 = A \frac{1 - 0.40388}{0.12}$

$42000 = A \frac{0.59612}{0.12}$

$42000 = A (4.96766)$

$A = \frac{42000}{4.96766} \approx 8454.47$

Now, we construct the amortization schedule.

| Year | Beginning Balance | Payment | Interest | Principal | Ending Balance |

|------|-------------------|---------|----------|-----------|----------------|

| 0 | 42000.00 | | | | 42000.00 |

| 1 | 42000.00 | 8454.47 | 5040.00 | 3414.47 | 38585.53 |

| 2 | 38585.53 | 8454.47 | 4630.26 | 3824.21 | 34761.32 |

| 3 | 34761.32 | 8454.47 | 4171.36 | 4283.11 | 30478.21 |

| 4 | 30478.21 | 8454.47 | 3657.39 | 4797.08 | 25681.13 |

| 5 | 25681.13 | 8454.47 | 3081.74 | 5372.73 | 20308.40 |

| 6 | 20308.40 | 8454.47 | 2437.01 | 6017.46 | 14290.94 |

| 7 | 14290.94 | 8454.47 | 1714.91 | 6739.56 | 7551.38 |

| 8 | 7551.38 | 8454.47 | 906.17 | 7548.30 | 3.08 |

3. Final Answer

Problem 8: Tk. 1319.97

Problem 9: Tk. 7188.34

Problem 10: See the amortization schedule above.