Question
Question 2
a) From an initial investment of K23,000.00, what annual interest rate would yield an accrued total of K27,830.00 on simple interest, at the end of 42 months?
b) Calculate the compound interest on an investment of K3,400.00 compounded every 4 months for 3 years at an annual interest rate of 7.5%.
c) A flat screen valued at K7,500.00 is sold at hire purchase for K562.00 deposit plus a regular payment at an interest rate of 12% p.a over 2 years.
a) From an initial investment of K23,000.00, what annual interest rate would yield an accrued total of K27,830.00 on simple interest, at the end of 42 months?
b) Calculate the compound interest on an investment of K3,400.00 compounded every 4 months for 3 years at an annual interest rate of 7.5%.
c) A flat screen valued at K7,500.00 is sold at hire purchase for K562.00 deposit plus a regular payment at an interest rate of 12% p.a over 2 years.
Answers
GPT-4o mini
Let's break down each part of the question step by step.
### a) Simple Interest Calculation
We can use the simple interest formula:
\[
A = P(1 + rt)
\]
Where:
- \( A \) is the total amount accrued (K27,830.00)
- \( P \) is the principal (initial investment, K23,000.00)
- \( r \) is the annual interest rate (in decimal)
- \( t \) is the time in years
First, we convert 42 months into years:
\[
t = \frac{42 \text{ months}}{12} = 3.5 \text{ years}
\]
Rearranging the formula to solve for \( r \):
\[
r = \frac{A - P}{Pt}
\]
Substituting in the values:
\[
r = \frac{27830 - 23000}{23000 \times 3.5}
\]
\[
r = \frac{4820}{80500}
\]
\[
r = 0.0599 \text{ (approximately)}
\]
To get the annual interest rate in percentage:
\[
r \times 100 = 5.99\%
\]
### Answer to a:
The annual interest rate that would yield an accrued total of K27,830 after 42 months is approximately **5.99%**.
### b) Compound Interest Calculation
The compound interest formula is:
\[
A = P \left(1 + \frac{r}{n}\right)^{nt}
\]
Where:
- \( A \) is the total amount after time \( t \)
- \( P \) is the principal (K3,400)
- \( r \) is the annual interest rate (7.5% or 0.075)
- \( n \) is the number of times interest is compounded per year (4 months = 3 times a year)
- \( t \) is the total time in years (3 years)
Using the values:
\[
A = 3400 \left(1 + \frac{0.075}{3}\right)^{3 \times 3}
\]
Calculating \(\frac{0.075}{3} = 0.025\):
\[
A = 3400 \left(1 + 0.025\right)^{9}
\]
\[
A = 3400 \left(1.025\right)^{9}
\]
Calculating \( (1.025)^{9} \):
\[
(1.025)^{9} \approx 1.2460
\]
Now substitute back to find \( A \):
\[
A = 3400 \times 1.2460 \approx 4236.40
\]
Then, calculate the compound interest:
\[
\text{Compound Interest} = A - P = 4236.40 - 3400 = 836.40
\]
### Answer to b:
The compound interest after 3 years on an investment of K3,400.00 compounded every 4 months at an annual interest rate of 7.5% is approximately **K836.40**.
### c) Hire Purchase Calculation
The hire purchase total amount can be calculated using the present value formula for hire purchase:
\[
PV = C + R \cdot \frac{1 - (1 + r)^{-n}}{r}
\]
Where:
- \( C \) is the initial deposit (K562.00)
- \( R \) is the regular payment (unknown)
- \( r \) is the interest rate per period
- \( n \) is the total number of periods (months)
The interest for 12 months at 12% p.a. is:
\[
r = \frac{0.12}{12} = 0.01 \text{ (monthly)}
\]
In 2 years (24 months):
\[
n = 24
\]
We need to find the total amount financed (K7,500) minus the deposit:
\[
PV = K7500 - K562 = K6938
\]
Using the formula:
\[
6938 = R \cdot \frac{1 - (1 + 0.01)^{-24}}{0.01}
\]
Calculating:
\[
\frac{1 - (1 + 0.01)^{-24}}{0.01} \approx 22.6191 \text{ (approximately)}
\]
Thus, solving for \( R \):
\[
R \approx \frac{6938}{22.6191} \approx 307.25
\]
### Answer to c:
The regular payment for the hire purchase is approximately **K307.25**.
### a) Simple Interest Calculation
We can use the simple interest formula:
\[
A = P(1 + rt)
\]
Where:
- \( A \) is the total amount accrued (K27,830.00)
- \( P \) is the principal (initial investment, K23,000.00)
- \( r \) is the annual interest rate (in decimal)
- \( t \) is the time in years
First, we convert 42 months into years:
\[
t = \frac{42 \text{ months}}{12} = 3.5 \text{ years}
\]
Rearranging the formula to solve for \( r \):
\[
r = \frac{A - P}{Pt}
\]
Substituting in the values:
\[
r = \frac{27830 - 23000}{23000 \times 3.5}
\]
\[
r = \frac{4820}{80500}
\]
\[
r = 0.0599 \text{ (approximately)}
\]
To get the annual interest rate in percentage:
\[
r \times 100 = 5.99\%
\]
### Answer to a:
The annual interest rate that would yield an accrued total of K27,830 after 42 months is approximately **5.99%**.
### b) Compound Interest Calculation
The compound interest formula is:
\[
A = P \left(1 + \frac{r}{n}\right)^{nt}
\]
Where:
- \( A \) is the total amount after time \( t \)
- \( P \) is the principal (K3,400)
- \( r \) is the annual interest rate (7.5% or 0.075)
- \( n \) is the number of times interest is compounded per year (4 months = 3 times a year)
- \( t \) is the total time in years (3 years)
Using the values:
\[
A = 3400 \left(1 + \frac{0.075}{3}\right)^{3 \times 3}
\]
Calculating \(\frac{0.075}{3} = 0.025\):
\[
A = 3400 \left(1 + 0.025\right)^{9}
\]
\[
A = 3400 \left(1.025\right)^{9}
\]
Calculating \( (1.025)^{9} \):
\[
(1.025)^{9} \approx 1.2460
\]
Now substitute back to find \( A \):
\[
A = 3400 \times 1.2460 \approx 4236.40
\]
Then, calculate the compound interest:
\[
\text{Compound Interest} = A - P = 4236.40 - 3400 = 836.40
\]
### Answer to b:
The compound interest after 3 years on an investment of K3,400.00 compounded every 4 months at an annual interest rate of 7.5% is approximately **K836.40**.
### c) Hire Purchase Calculation
The hire purchase total amount can be calculated using the present value formula for hire purchase:
\[
PV = C + R \cdot \frac{1 - (1 + r)^{-n}}{r}
\]
Where:
- \( C \) is the initial deposit (K562.00)
- \( R \) is the regular payment (unknown)
- \( r \) is the interest rate per period
- \( n \) is the total number of periods (months)
The interest for 12 months at 12% p.a. is:
\[
r = \frac{0.12}{12} = 0.01 \text{ (monthly)}
\]
In 2 years (24 months):
\[
n = 24
\]
We need to find the total amount financed (K7,500) minus the deposit:
\[
PV = K7500 - K562 = K6938
\]
Using the formula:
\[
6938 = R \cdot \frac{1 - (1 + 0.01)^{-24}}{0.01}
\]
Calculating:
\[
\frac{1 - (1 + 0.01)^{-24}}{0.01} \approx 22.6191 \text{ (approximately)}
\]
Thus, solving for \( R \):
\[
R \approx \frac{6938}{22.6191} \approx 307.25
\]
### Answer to c:
The regular payment for the hire purchase is approximately **K307.25**.