Question 1193568: Please help me with the homework.
1. Ben owes 500 due in eight months. For each of the following cases, what single payment will repay his debt if simple interest is 15%p.a?
1.1 Now
1.2 Six months from now
1.3 in one year
2. A promissory note dated 1 April 2021 for 1500, borrowed at a simple interest discount rate of 16% p a. Due on 1 October 2021 is sold on the 1 July 2021.
2.1. what is the maturity value of the note?
2.2. what is the present value on the date of sale?
Answer by parmen(42)   (Show Source):
You can put this solution on YOUR website! ### 1. Ben's Debt Repayment with Simple Interest at 15% p.a.
The formula for simple interest is:
\[
A = P \times (1 + r \times t)
\]
Where:
- \( A \): Repayment amount,
- \( P = 500 \): Principal,
- \( r = 0.15 \): Annual simple interest rate,
- \( t \): Time in years.
---
#### 1.1. Single Payment **Now**
If Ben repays the debt **now**, no interest has accrued yet:
\[
A = P = 500
\]
---
#### 1.2. Single Payment **Six Months from Now**
From now to six months (\( t = \frac{6}{12} = 0.5 \) years), interest accrues:
\[
A = 500 \times (1 + 0.15 \times 0.5)
\]
\[
A = 500 \times (1 + 0.075)
\]
\[
A = 500 \times 1.075 = 537.50
\]
---
#### 1.3. Single Payment **In One Year**
From now to one year (\( t = \frac{12}{12} = 1 \) year), interest accrues:
\[
A = 500 \times (1 + 0.15 \times 1)
\]
\[
A = 500 \times (1 + 0.15)
\]
\[
A = 500 \times 1.15 = 575
\]
---
### 2. Promissory Note at a Simple Discount Rate of 16% p.a.
#### 2.1. Maturity Value of the Note
The **maturity value** of the note is the amount due at the end of the loan period. For this case, it is explicitly given as **R1500**.
---
#### 2.2. Present Value on the Date of Sale
The note has a simple discount rate of 16% p.a., and the present value is calculated using:
\[
PV = M \times (1 - d \times t)
\]
Where:
- \( M = 1500 \) (maturity value),
- \( d = 0.16 \) (discount rate),
- \( t \): Time remaining to maturity in years. From 1 July 2021 to 1 October 2021 is 3 months or \( t = \frac{3}{12} = 0.25 \) years.
Substitute the values:
\[
PV = 1500 \times (1 - 0.16 \times 0.25)
\]
\[
PV = 1500 \times (1 - 0.04)
\]
\[
PV = 1500 \times 0.96
\]
\[
PV = 1440
\]
---
### Final Summary
#### 1. Ben's Debt Repayment:
1. **Now**: \( R500 \),
2. **Six Months from Now**: \( R537.50 \),
3. **In One Year**: \( R575 \).
#### 2. Promissory Note:
1. **Maturity Value**: \( R1500 \),
2. **Present Value on Sale Date (1 July 2021)**: \( R1440 \).
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