Sure, I'll provide a detailed, step-by-step solution. Let's address each part of the question in the order they were asked:
### (a) Formulae for Depreciation and Increment
For depreciation:
[tex]\[ P_T = P \left(1 - \frac{R}{100}\right)^T \][/tex]
- [tex]\(P_T\)[/tex]: Price at time [tex]\(T\)[/tex]
- [tex]\(P\)[/tex]: Initial price
- [tex]\(R\)[/tex]: Rate of depreciation per annum (in percentage)
- [tex]\(T\)[/tex]: Time in years
For increment:
[tex]\[ P_T = P \left(1 + \frac{R}{100}\right)^T \][/tex]
- [tex]\(P_T\)[/tex]: Price at time [tex]\(T\)[/tex]
- [tex]\(P\)[/tex]: Initial price
- [tex]\(R\)[/tex]: Rate of growth per annum (in percentage)
- [tex]\(T\)[/tex]: Time in years
### (b) Price of 1 Aana land before 1 year
1. Initial price 3 years ago: Rs. 3,200,000 per Ropani. Since there are 16 Aana in 1 Ropani, the initial price per Aana is:
[tex]\[ \frac{3,200,000}{16} = 200,000 \, \text{Rs per Aana} \][/tex]
2. Depreciation over the first 2 years:
[tex]\[ P_2 = 3,200,000 \times \left(1 - \frac{10}{100}\right)^2 \][/tex]
This gives us the price after 2 years.
3. Increment in the third year:
[tex]\[ P_3 = P_2 \times \left(1 + \frac{20}{100}\right) \][/tex]
This gives us the price at the end of the third year.
4. Price of one Aana before one year:
[tex]\[ \text{Price per Ropani 1 year ago} = 3,110,400 \, \text{Rs} \][/tex]
[tex]\[ \text{Price per Aana 1 year ago} = \frac{3,110,400}{16} = 194,400 \, \text{Rs per Aana} \][/tex]
### (c) Current price of 4 Aana
1. Price of 1 Aana before 1 year: Rs 194,400
2. Price of 4 Aana:
[tex]\[ \text{Current price of 4 Aana} = 194,400 \times 4 = 777,600 \, \text{Rs} \][/tex]
### (d) Analysis of the land price over the past 3 years
1. Initial price 3 years ago:
[tex]\[ 3,200,000 \, \text{Rs per Ropani} \][/tex]
2. Price before 1 year:
[tex]\[ 3,110,400 \, \text{Rs per Ropani} \][/tex]
3. Total depreciation or increment over 3 years:
[tex]\[ \text{Percentage change} = \left( \frac{\text{Initial price} - \text{Current price}}{\text{Initial price}} \right) \times 100 \][/tex]
[tex]\[ = \left( \frac{3,200,000 - 3,110,400}{3,200,000} \right) \times 100 = 2.8\% \, \text{(decrease)} \][/tex]
So the land, which was initially Rs. 3,200,000 per Ropani, has decreased in value by 2.8% over the last 3 years. The current price is Rs. 3,110,400 per Ropani.
