Question
8.1 The interest rate on an investment is 𝑥% per annum compounded
monthly. Calculate the value of 𝑥 given that the corresponding
effective interest rate is 9,92%. (3)
8.2 Liyabona bought a computer for R4 700. The value of the computer
depreciated at a rate of 18% p.a. Using the reducing-balance
method, calculate the book value of the computer 4 years after it
was bought. (3)
8.3 Ethan invested R30 000 for 6 years. The investment earned interest
at 12% p.a., compounded monthly for the first two years.
Thereafter, the interest rate changed to 10,8% p.a., compounded
semi-annually for the rest of the period.
Calculate the value of the investment at the end of 6 years. (4)
8.4 Fatima deposited R25 000 into a savings account with an interest
rate of 18% p.a., compounded monthly. Fatima withdrew R8 000
from the account 2 years after depositing the initial amount. She
deposited another R4 000 into this account 3½ years after the
initial deposit.
What amount will Fatima have 5 years after making the initial
deposit in this account? (6)
monthly. Calculate the value of 𝑥 given that the corresponding
effective interest rate is 9,92%. (3)
8.2 Liyabona bought a computer for R4 700. The value of the computer
depreciated at a rate of 18% p.a. Using the reducing-balance
method, calculate the book value of the computer 4 years after it
was bought. (3)
8.3 Ethan invested R30 000 for 6 years. The investment earned interest
at 12% p.a., compounded monthly for the first two years.
Thereafter, the interest rate changed to 10,8% p.a., compounded
semi-annually for the rest of the period.
Calculate the value of the investment at the end of 6 years. (4)
8.4 Fatima deposited R25 000 into a savings account with an interest
rate of 18% p.a., compounded monthly. Fatima withdrew R8 000
from the account 2 years after depositing the initial amount. She
deposited another R4 000 into this account 3½ years after the
initial deposit.
What amount will Fatima have 5 years after making the initial
deposit in this account? (6)
Answers
GPT-4o mini
Let's solve each part of the question step by step.
### 8.1 Effective Interest Rate Calculation
The formula to calculate the effective interest rate from the nominal interest rate \( x \% \) compounded monthly is:
\[
\text{Effective Interest Rate} = \left(1 + \frac{x}{100 \times 12}\right)^{12} - 1 = 0.0992
\]
We need to solve for \( x \):
1. Rearranging gives:
\[
\left(1 + \frac{x}{1200}\right)^{12} = 1.0992
\]
2. Taking the 12th root:
\[
1 + \frac{x}{1200} = (1.0992)^{\frac{1}{12}}
\]
Calculating \( (1.0992)^{\frac{1}{12}} \):
\[
(1.0992)^{\frac{1}{12}} \approx 1.00797
\]
3. Now, solve for \( x \):
\[
\frac{x}{1200} = 1.00797 - 1 \quad \Rightarrow \quad \frac{x}{1200} \approx 0.00797
\]
\[
x \approx 0.00797 \times 1200 \quad \Rightarrow \quad x \approx 9.564
\]
So, rounded to two decimal places, the interest rate \( x \approx 9.56\% \).
### 8.2 Depreciation Calculation Using the Reducing-Balance Method
Using the formula for the book value after \( n \) years with reducing balance depreciation:
\[
\text{Book Value} = P \times (1 - r)^n
\]
Where \( P = 4700 \), \( r = 0.18 \), and \( n = 4 \):
\[
\text{Book Value} = 4700 \times (1 - 0.18)^4
\]
Calculating \( (1 - 0.18)^4 \):
\[
(0.82)^4 \approx 0.4524
\]
Now calculate the book value:
\[
\text{Book Value} = 4700 \times 0.4524 \approx 2126.68
\]
So, the book value of the computer after 4 years is approximately **R2126.68**.
### 8.3 Investment Value Calculation
First we calculate the value of Ethan’s investment for the first 2 years at 12% compounded monthly:
1. The formula for compound interest is:
\[
A = P \left(1 + \frac{r}{n}\right)^{nt}
\]
Where:
- \( P = 30000 \)
- \( r = 0.12 \)
- \( n = 12 \) (monthly)
- \( t = 2 \)
Calculating:
\[
A = 30000 \left(1 + \frac{0.12}{12}\right)^{12 \times 2} = 30000 \left(1 + 0.01\right)^{24} \approx 30000 \times (1.01^{24}) \approx 30000 \times 1.2682418 \approx R38047.25
\]
Next, for the next 4 years at 10.8% compounded semi-annually:
- \( r = 0.108 \)
- \( n = 2 \)
Using the amount from the first 2 years as the principal:
\[
A = 38047.25 \left(1 + \frac{0.108}{2}\right)^{2 \times 4}
\]
Calculating:
\[
A = 38047.25 \left(1 + 0.054\right)^{8} \approx 38047.25 \times (1.054^{8}) \approx 38047.25 \times 1.48024 \approx R56498.29
\]
So, at the end of 6 years, the value of the investment is approximately **R56498.29**.
### 8.4 Final Account Balance Calculation
First, calculate the amount in the savings account after 5 years at 18% compounded monthly:
1. After 2 years:
\[
A_1 = 25000 \left(1 + \frac{0.18}{12}\right)^{12 \times 2} = 25000 \left(1 + 0.015\right)^{24} = 25000 \times (1.015^{24}) \approx 25000 \times 1.404112 \approx R35103
\]
2. After withdrawing R8000:
\[
A_2 = 35103 - 8000 = R27103
\]
3. Compound this for another 3 years (3 to 5 years):
\[
A_2' = 27103 \left(1 + \frac{0.18}{12}\right)^{12 \times 3} = 27103 \left(1.015^{36}\right) \approx 27103 \times 1.72033 \approx R46683.53
\]
4. Now, deposit R4000 after 3.5 years, 0.5 years before the total time (from year 3.5 to year 5):
\[
A_3 = 4000 \left(1 + \frac{0.18}{12}\right)^{12 \times 1.5} = 4000 \left(1.015^{18}\right) \approx 4000 \times 1.30464 \approx R5218.56
\]
5. Add this to the previous total after 5 years:
\[
A_{final} = 46683.53 + 5218.56 \approx R51892.09
\]
So, the total amount Fatima will have after 5 years is approximately **R51892.09**.
### 8.1 Effective Interest Rate Calculation
The formula to calculate the effective interest rate from the nominal interest rate \( x \% \) compounded monthly is:
\[
\text{Effective Interest Rate} = \left(1 + \frac{x}{100 \times 12}\right)^{12} - 1 = 0.0992
\]
We need to solve for \( x \):
1. Rearranging gives:
\[
\left(1 + \frac{x}{1200}\right)^{12} = 1.0992
\]
2. Taking the 12th root:
\[
1 + \frac{x}{1200} = (1.0992)^{\frac{1}{12}}
\]
Calculating \( (1.0992)^{\frac{1}{12}} \):
\[
(1.0992)^{\frac{1}{12}} \approx 1.00797
\]
3. Now, solve for \( x \):
\[
\frac{x}{1200} = 1.00797 - 1 \quad \Rightarrow \quad \frac{x}{1200} \approx 0.00797
\]
\[
x \approx 0.00797 \times 1200 \quad \Rightarrow \quad x \approx 9.564
\]
So, rounded to two decimal places, the interest rate \( x \approx 9.56\% \).
### 8.2 Depreciation Calculation Using the Reducing-Balance Method
Using the formula for the book value after \( n \) years with reducing balance depreciation:
\[
\text{Book Value} = P \times (1 - r)^n
\]
Where \( P = 4700 \), \( r = 0.18 \), and \( n = 4 \):
\[
\text{Book Value} = 4700 \times (1 - 0.18)^4
\]
Calculating \( (1 - 0.18)^4 \):
\[
(0.82)^4 \approx 0.4524
\]
Now calculate the book value:
\[
\text{Book Value} = 4700 \times 0.4524 \approx 2126.68
\]
So, the book value of the computer after 4 years is approximately **R2126.68**.
### 8.3 Investment Value Calculation
First we calculate the value of Ethan’s investment for the first 2 years at 12% compounded monthly:
1. The formula for compound interest is:
\[
A = P \left(1 + \frac{r}{n}\right)^{nt}
\]
Where:
- \( P = 30000 \)
- \( r = 0.12 \)
- \( n = 12 \) (monthly)
- \( t = 2 \)
Calculating:
\[
A = 30000 \left(1 + \frac{0.12}{12}\right)^{12 \times 2} = 30000 \left(1 + 0.01\right)^{24} \approx 30000 \times (1.01^{24}) \approx 30000 \times 1.2682418 \approx R38047.25
\]
Next, for the next 4 years at 10.8% compounded semi-annually:
- \( r = 0.108 \)
- \( n = 2 \)
Using the amount from the first 2 years as the principal:
\[
A = 38047.25 \left(1 + \frac{0.108}{2}\right)^{2 \times 4}
\]
Calculating:
\[
A = 38047.25 \left(1 + 0.054\right)^{8} \approx 38047.25 \times (1.054^{8}) \approx 38047.25 \times 1.48024 \approx R56498.29
\]
So, at the end of 6 years, the value of the investment is approximately **R56498.29**.
### 8.4 Final Account Balance Calculation
First, calculate the amount in the savings account after 5 years at 18% compounded monthly:
1. After 2 years:
\[
A_1 = 25000 \left(1 + \frac{0.18}{12}\right)^{12 \times 2} = 25000 \left(1 + 0.015\right)^{24} = 25000 \times (1.015^{24}) \approx 25000 \times 1.404112 \approx R35103
\]
2. After withdrawing R8000:
\[
A_2 = 35103 - 8000 = R27103
\]
3. Compound this for another 3 years (3 to 5 years):
\[
A_2' = 27103 \left(1 + \frac{0.18}{12}\right)^{12 \times 3} = 27103 \left(1.015^{36}\right) \approx 27103 \times 1.72033 \approx R46683.53
\]
4. Now, deposit R4000 after 3.5 years, 0.5 years before the total time (from year 3.5 to year 5):
\[
A_3 = 4000 \left(1 + \frac{0.18}{12}\right)^{12 \times 1.5} = 4000 \left(1.015^{18}\right) \approx 4000 \times 1.30464 \approx R5218.56
\]
5. Add this to the previous total after 5 years:
\[
A_{final} = 46683.53 + 5218.56 \approx R51892.09
\]
So, the total amount Fatima will have after 5 years is approximately **R51892.09**.