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Sagot :
Sure, let's tackle each problem one by one.
### Problem 6
#### Given:
- Principal amount: ₹ 5,000
- Compound interest rate for the first year: 5% per annum
- Compound interest rate for the second year: 6% per annum
#### Solution:
Step 1: Calculate the amount after the first year
For the first year, the interest rate is 5%.
[tex]\[ A_1 = P \times \left(1 + \frac{r_1}{100}\right) \][/tex]
Where:
- [tex]\( P \)[/tex] is the principal amount
- [tex]\( r_1 \)[/tex] is the rate of interest for the first year
[tex]\[ A_1 = 5000 \times \left(1 + \frac{5}{100}\right) \][/tex]
[tex]\[ A_1 = 5000 \times 1.05 \][/tex]
[tex]\[ A_1 = 5250 \][/tex]
So, the amount after the first year is ₹ 5,250.
Step 2: Calculate the amount after the second year
For the second year, the interest rate is 6% on the amount after the first year.
[tex]\[ A_2 = A_1 \times \left(1 + \frac{r_2}{100}\right) \][/tex]
Where:
- [tex]\( A_1 \)[/tex] is the amount after the first year
- [tex]\( r_2 \)[/tex] is the rate of interest for the second year
[tex]\[ A_2 = 5250 \times \left(1 + \frac{6}{100}\right) \][/tex]
[tex]\[ A_2 = 5250 \times 1.06 \][/tex]
[tex]\[ A_2 = 5565 \][/tex]
So, the amount payable by Nirmal after 2 years is ₹ 5,565.
### Problem 7
#### Given:
- Present value of the machine: ₹ 1,00,000
- Annual depreciation rate: 10%
#### Solution:
Step 1: Calculate the value of the machine after one year
For the first year, the depreciation rate is 10%.
[tex]\[ V_1 = V_0 \times \left(1 - \frac{d}{100}\right) \][/tex]
Where:
- [tex]\( V_0 \)[/tex] is the present value of the machine
- [tex]\( d \)[/tex] is the rate of depreciation
[tex]\[ V_1 = 100000 \times \left(1 - \frac{10}{100}\right) \][/tex]
[tex]\[ V_1 = 100000 \times 0.90 \][/tex]
[tex]\[ V_1 = 90000 \][/tex]
So, the value of the machine after one year is ₹ 90,000.
Step 2: Calculate the value of the machine after the second year
For the second year, the depreciation rate is again 10%.
[tex]\[ V_2 = V_1 \times \left(1 - \frac{d}{100}\right) \][/tex]
Where:
- [tex]\( V_1 \)[/tex] is the value of the machine after the first year
[tex]\[ V_2 = 90000 \times \left(1 - \frac{10}{100}\right) \][/tex]
[tex]\[ V_2 = 90000 \times 0.90 \][/tex]
[tex]\[ V_2 = 81000 \][/tex]
So, the value of the machine after two years is ₹ 81,000.
### Problem 6
#### Given:
- Principal amount: ₹ 5,000
- Compound interest rate for the first year: 5% per annum
- Compound interest rate for the second year: 6% per annum
#### Solution:
Step 1: Calculate the amount after the first year
For the first year, the interest rate is 5%.
[tex]\[ A_1 = P \times \left(1 + \frac{r_1}{100}\right) \][/tex]
Where:
- [tex]\( P \)[/tex] is the principal amount
- [tex]\( r_1 \)[/tex] is the rate of interest for the first year
[tex]\[ A_1 = 5000 \times \left(1 + \frac{5}{100}\right) \][/tex]
[tex]\[ A_1 = 5000 \times 1.05 \][/tex]
[tex]\[ A_1 = 5250 \][/tex]
So, the amount after the first year is ₹ 5,250.
Step 2: Calculate the amount after the second year
For the second year, the interest rate is 6% on the amount after the first year.
[tex]\[ A_2 = A_1 \times \left(1 + \frac{r_2}{100}\right) \][/tex]
Where:
- [tex]\( A_1 \)[/tex] is the amount after the first year
- [tex]\( r_2 \)[/tex] is the rate of interest for the second year
[tex]\[ A_2 = 5250 \times \left(1 + \frac{6}{100}\right) \][/tex]
[tex]\[ A_2 = 5250 \times 1.06 \][/tex]
[tex]\[ A_2 = 5565 \][/tex]
So, the amount payable by Nirmal after 2 years is ₹ 5,565.
### Problem 7
#### Given:
- Present value of the machine: ₹ 1,00,000
- Annual depreciation rate: 10%
#### Solution:
Step 1: Calculate the value of the machine after one year
For the first year, the depreciation rate is 10%.
[tex]\[ V_1 = V_0 \times \left(1 - \frac{d}{100}\right) \][/tex]
Where:
- [tex]\( V_0 \)[/tex] is the present value of the machine
- [tex]\( d \)[/tex] is the rate of depreciation
[tex]\[ V_1 = 100000 \times \left(1 - \frac{10}{100}\right) \][/tex]
[tex]\[ V_1 = 100000 \times 0.90 \][/tex]
[tex]\[ V_1 = 90000 \][/tex]
So, the value of the machine after one year is ₹ 90,000.
Step 2: Calculate the value of the machine after the second year
For the second year, the depreciation rate is again 10%.
[tex]\[ V_2 = V_1 \times \left(1 - \frac{d}{100}\right) \][/tex]
Where:
- [tex]\( V_1 \)[/tex] is the value of the machine after the first year
[tex]\[ V_2 = 90000 \times \left(1 - \frac{10}{100}\right) \][/tex]
[tex]\[ V_2 = 90000 \times 0.90 \][/tex]
[tex]\[ V_2 = 81000 \][/tex]
So, the value of the machine after two years is ₹ 81,000.
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