Let's break down each part of the question and solve it step-by-step:
### Part (a)
P \left(1+\frac{ R }{400}\right)^{4 T} refers to the amount received when the principal P is compounded quarterly at an annual interest rate [tex]\(R\%\)[/tex] for a duration of [tex]\(T\)[/tex] years. It shows the effect of converting the annual rate into quarterly increments, thus applying compound interest every quarter.
### Part (b)
To determine how much annual compound interest Pemba will receive at the end of 1 year 6 months:
1. Principal (P): Rs 2,00,000
2. Annual Interest Rate (R): 16% (0.16 as a decimal)
3. Time (T): 1.5 years
Annual compound interest formula:
[tex]\[ A = P \left(1 + R \right)^{T} \][/tex]
Where:
- [tex]\(P = 2,00,000\)[/tex]
- [tex]\(R = 0.16\)[/tex]
- [tex]\(T = 1.5\)[/tex]
First, find the amount [tex]\(A\)[/tex]:
[tex]\[ A = 2,00,000 \left(1 + 0.16 \right)^{1.5} = 2,00,000 \left(1.16\right)^{1.5} \][/tex]
[tex]\[ A \approx 2,00,000 \times 1.24864 \][/tex]
[tex]\[ A \approx 2,49,871.65 \][/tex]
Now, to find the interest earned:
[tex]\[ \text{Interest} = A - P = 2,49,871.65 - 2,00,000 \approx 49,871.65 \][/tex]
So, the annual compound interest will be approximately Rs 49,871.65.
### Part (c)
To determine how much semi-annual compound interest Pemba will receive:
1. Principal (P): Rs 2,00,000
2. Annual Interest Rate (R): 16% (0.16 as a decimal)
3. Time (T): 1.5 years
4. Semi-annual period rate: [tex]\(0.16 / 2 = 0.08\)[/tex]
5. Number of periods: [tex]\(2 \times 1.5 = 3\)[/tex]
Semi-annual compound interest formula:
[tex]\[ A = P \left(1 + \frac{R}{2} \right)^{2T} \][/tex]
Where:
- [tex]\(P = 2,00,000\)[/tex]
- [tex]\(R/2 = 0.08\)[/tex]
- [tex]\(2T = 3\)[/tex]
First, find the amount [tex]\(A\)[/tex]:
[tex]\[ A = 2,00,000 \left(1 + 0.08 \right)^{3} = 2,00,000 \left(1.08\right)^{3} \][/tex]
[tex]\[ A \approx 2,00,000 \times 1.25971 \][/tex]
[tex]\[ A \approx 2,51,942.40 \][/tex]
Now, to find the interest earned:
[tex]\[ \text{Interest} = A - P = 2,51,942.40 - 2,00,000 \approx 51,942.40 \][/tex]
So, the semi-annual compound interest will be approximately Rs 51,942.40.
### Part (d)
To find the difference between semi-annual compound interest and annual compound interest:
[tex]\[ \text{Difference} = 51,942.40 - 49,871.65 \][/tex]
[tex]\[ \text{Difference} \approx 2,070.75 \][/tex]
So, the difference in interest is approximately Rs 2,070.75.
### Part (e)
To determine which interest system to suggest Dorje for borrowing:
- Annual Compound Interest: Rs 49,871.65
- Semi-Annual Compound Interest: Rs 51,942.40
The semi-annual compound interest yields a higher interest of Rs 2,070.75 more than annual compound interest.
Suggestion:
I would suggest that Dorje borrow under the annual compound interest system because it results in a lower total interest amount compared to the semi-annual compounding. This will make the repayment amount lower, saving Dorje money on interest.
