Alright, let's break down this problem step by step.
### Question (a):
What does [tex]\( R \)[/tex] indicate in the formula [tex]\( P_{T} = P \left(1 + \frac{R}{100}\right)^{T} \)[/tex]?
In this formula, [tex]\( P_T \)[/tex] denotes the price after [tex]\( T \)[/tex] years, [tex]\( P \)[/tex] is the initial price, and [tex]\( R \)[/tex] is the rate of increase or decrease in price per year. Therefore, [tex]\( R \)[/tex] represents the annual percentage rate of change (either increase or decrease) in the price.
### Question (b):
What will be the price of the land after 2 years?
Let's start with the initial price of the land:
[tex]\[
P_{\text{land\_initial}} = 80,00,000 \text{ Rs}
\][/tex]
The annual rate of increase in the price of the land is:
[tex]\[
R_{\text{land}} = 20\%
\][/tex]
The number of years,
[tex]\[
T = 2
\][/tex]
The formula to calculate the price after [tex]\( T \)[/tex] years is:
[tex]\[
P_{\text{land\_2\_years}} = P_{\text{land\_initial}} \times \left(1 + \frac{R_{\text{land}}}{100}\right)^{T}
\][/tex]
Plugging the values into the formula:
[tex]\[
P_{\text{land\_2\_years}} = 80,00,000 \times \left(1 + \frac{20}{100}\right)^{2}
\][/tex]
[tex]\[
= 80,00,000 \times (1.20)^{2}
\][/tex]
[tex]\[
= 80,00,000 \times 1.44
\][/tex]
[tex]\[
= 1,15,20,000 \text{ Rs}
\][/tex]
Thus, the price of the land after 2 years is:
[tex]\[
1,15,20,000 \text{ Rs}
\][/tex]
### Question (c):
What will be the price of the house after 2 years?
Let's start with the initial price of the house:
[tex]\[
P_{\text{house\_initial}} = 2,70,00,000 \text{ Rs}
\][/tex]
The annual rate of decrease in the price of the house is:
[tex]\[
R_{\text{house}} = -20\%
\][/tex]
The number of years,
[tex]\[
T = 2
\][/tex]
The formula to calculate the price after [tex]\( T \)[/tex] years is:
[tex]\[
P_{\text{house\_2\_years}} = P_{\text{house\_initial}} \left(1 + \frac{R_{\text{house}}}{100}\right)^{T}
\][/tex]
Plugging the values into the formula:
[tex]\[
P_{\text{house\_2\_years}} = 2,70,00,000 \times \left(1 - \frac{20}{100}\right)^{2}
\][/tex]
[tex]\[
= 2,70,00,000 \times (0.80)^{2}
\][/tex]
[tex]\[
= 2,70,00,000 \times 0.64
\][/tex]
[tex]\[
= 1,72,80,000 \text{ Rs}
\][/tex]
Thus, the price of the house after 2 years is:
[tex]\[
1,72,80,000 \text{ Rs}
\][/tex]
### Question (d):
Will the prices of the land and house be the same after 2 years? If not, in how many years will the prices of the land and house be equal?
From parts (b) and (c) above, the price of the land after 2 years is [tex]\( 1,15,20,000 \text{ Rs} \)[/tex] and the price of the house after 2 years is [tex]\( 1,72,80,000 \text{ Rs} \)[/tex]. Clearly, these are not equal. Thus, the prices of the land and house will not be the same after 2 years.
To find out when they will be equal, we use the formula for both land and house:
1. For land: [tex]\( P_{\text{land}} = P_{\text{land\_initial}} \left(1 + \frac{R_{\text{land}}}{100}\right)^T \)[/tex]
2. For house: [tex]\( P_{\text{house}} = P_{\text{house\_initial}} \left(1 + \frac{R_{\text{house}}}{100}\right)^T \)[/tex]
We set these two equations equal to each other to find [tex]\( T \)[/tex]:
[tex]\[
80,00,000 \times (1.20)^T = 2,70,00,000 \times (0.80)^T
\][/tex]
Solving for [tex]\( T \)[/tex]:
[tex]\[
(1.20)^T = \frac{2,70,00,000}{80,00,000} \times (0.80)^T
\][/tex]
[tex]\[
(1.20)^T = 3.375 \times (0.80)^T
\][/tex]
Taking the natural logarithm on both sides to solve for [tex]\( T \)[/tex]:
[tex]\[
T \approx 3
\][/tex]
Thus, the prices of the land and house will be equal after approximately 3 years.
