Answer :
Sure Rajesh, let's go through the problem methodically. Remember the concepts of compound interest and how different compounding frequencies can affect the calculations.
### Part (a):
What do you understand by 590 tax deduction in the interest?
- The 590 tax deduction means that after calculating the interest earned, the bank deducts an amount of 590 RS as tax from the interest earned. This is a fixed deduction and is applied annually after the calculation of interest.
### Part (b):
What is the interest of the second year after deducting the tax?
To solve this, let's go through the problem step-by-step:
1. Calculate the amount after the first year with semi-annual compounding:
- Principal at the start: 400,000 RS
- Annual interest rate: 8% or 0.08
- Compounded semi-annually means it compounds twice a year.
Using the formula for compound interest:
[tex]\[ A = P \left(1 + \frac{r}{n}\right)^{n \cdot t} \][/tex]
where:
- [tex]\(P\)[/tex] is the principal amount (400,000 RS)
- [tex]\(r\)[/tex] is the annual interest rate (0.08)
- [tex]\(n\)[/tex] is the number of times interest is compounded per year (2 for semi-annual)
- [tex]\(t\)[/tex] is the time the money is invested (1 year)
Calculate the amount after one year:
[tex]\[ A_1 = 400,000 \left(1 + \frac{0.08}{2}\right)^{2 \cdot 1} = 400,000 \left(1 + 0.04\right)^2 = 400,000 \left(1.04\right)^2 = 432,640 \][/tex]
The interest earned for the first year before tax deduction:
[tex]\[ \text{Interest}_1 = 432,640 - 400,000 = 32,640 \][/tex]
Now, apply the tax deduction of 590 RS:
[tex]\[ \text{Interest}_1 \text{ after tax} = 32,640 - 590 = 32,050 \][/tex]
2. Calculate the updated principal for the second year:
The updated principal amount at the start of the second year is the amount after first year minus the tax:
[tex]\[ \text{Principal}_2 = 432,640 - 590 = 432,050 \][/tex]
3. Calculate the amount after the second year with quarterly compounding:
- Principal at the start of the second year: 432,050 RS
- Annual interest rate: 8%
- Compounded quarterly means it compounds four times a year.
Using the same compound interest formula:
[tex]\[ A_2 = 432,050 \left(1 + \frac{0.08}{4}\right)^{4 \cdot 1} = 432,050 \left(1 + 0.02\right)^4 = 432,050 \left(1.02\right)^4 = 467,664.814728 \][/tex]
The interest earned for the second year before tax deduction:
[tex]\[ \text{Interest}_2 = 467,664.814728 - 432,050 = 35,614.814728 \][/tex]
Now, apply the tax deduction of 590 RS:
[tex]\[ \text{Interest}_2 \text{ after tax} = 35,614.814728 - 590 = 35,024.814728 \][/tex]
So, the interest for the second year after tax is approximately 35,024.81 RS.
### Part (c):
What is the percentage difference between the interest of the second year and that of the first year after deducting tax?
The interest for the first year after tax is 32,050 RS, and for the second year after tax it is 35,024.81 RS.
To find the percentage difference, we use the formula:
[tex]\[ \text{Percentage Difference} = \left(\frac{\text{Interest}_2 - \text{Interest}_1}{\text{Interest}_1}\right) \times 100 \][/tex]
Substitute in the values:
[tex]\[ \text{Percentage Difference} = \left(\frac{35,024.81 - 32,050}{32,050}\right) \times 100 \approx 9.28\% \][/tex]
Therefore, the percentage difference between the interest of the second year and the first year after deducting the tax is approximately 9.28%.
### Part (a):
What do you understand by 590 tax deduction in the interest?
- The 590 tax deduction means that after calculating the interest earned, the bank deducts an amount of 590 RS as tax from the interest earned. This is a fixed deduction and is applied annually after the calculation of interest.
### Part (b):
What is the interest of the second year after deducting the tax?
To solve this, let's go through the problem step-by-step:
1. Calculate the amount after the first year with semi-annual compounding:
- Principal at the start: 400,000 RS
- Annual interest rate: 8% or 0.08
- Compounded semi-annually means it compounds twice a year.
Using the formula for compound interest:
[tex]\[ A = P \left(1 + \frac{r}{n}\right)^{n \cdot t} \][/tex]
where:
- [tex]\(P\)[/tex] is the principal amount (400,000 RS)
- [tex]\(r\)[/tex] is the annual interest rate (0.08)
- [tex]\(n\)[/tex] is the number of times interest is compounded per year (2 for semi-annual)
- [tex]\(t\)[/tex] is the time the money is invested (1 year)
Calculate the amount after one year:
[tex]\[ A_1 = 400,000 \left(1 + \frac{0.08}{2}\right)^{2 \cdot 1} = 400,000 \left(1 + 0.04\right)^2 = 400,000 \left(1.04\right)^2 = 432,640 \][/tex]
The interest earned for the first year before tax deduction:
[tex]\[ \text{Interest}_1 = 432,640 - 400,000 = 32,640 \][/tex]
Now, apply the tax deduction of 590 RS:
[tex]\[ \text{Interest}_1 \text{ after tax} = 32,640 - 590 = 32,050 \][/tex]
2. Calculate the updated principal for the second year:
The updated principal amount at the start of the second year is the amount after first year minus the tax:
[tex]\[ \text{Principal}_2 = 432,640 - 590 = 432,050 \][/tex]
3. Calculate the amount after the second year with quarterly compounding:
- Principal at the start of the second year: 432,050 RS
- Annual interest rate: 8%
- Compounded quarterly means it compounds four times a year.
Using the same compound interest formula:
[tex]\[ A_2 = 432,050 \left(1 + \frac{0.08}{4}\right)^{4 \cdot 1} = 432,050 \left(1 + 0.02\right)^4 = 432,050 \left(1.02\right)^4 = 467,664.814728 \][/tex]
The interest earned for the second year before tax deduction:
[tex]\[ \text{Interest}_2 = 467,664.814728 - 432,050 = 35,614.814728 \][/tex]
Now, apply the tax deduction of 590 RS:
[tex]\[ \text{Interest}_2 \text{ after tax} = 35,614.814728 - 590 = 35,024.814728 \][/tex]
So, the interest for the second year after tax is approximately 35,024.81 RS.
### Part (c):
What is the percentage difference between the interest of the second year and that of the first year after deducting tax?
The interest for the first year after tax is 32,050 RS, and for the second year after tax it is 35,024.81 RS.
To find the percentage difference, we use the formula:
[tex]\[ \text{Percentage Difference} = \left(\frac{\text{Interest}_2 - \text{Interest}_1}{\text{Interest}_1}\right) \times 100 \][/tex]
Substitute in the values:
[tex]\[ \text{Percentage Difference} = \left(\frac{35,024.81 - 32,050}{32,050}\right) \times 100 \approx 9.28\% \][/tex]
Therefore, the percentage difference between the interest of the second year and the first year after deducting the tax is approximately 9.28%.
