4.3 Q-18

Question Statement

A house was purchased for Rs. 1 million in 1980. Its value increased to Rs. 4 million in 1996. Assuming that the value of the house increased by the same amount each year, find an equation that gives the value of the house after $t$ years from the purchase date. Also, calculate the value of the house in 1990.

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Background and Explanation

This problem involves finding the equation of a line to model the increase in the house’s value over time. Key concepts involved are:

1. Linear Equation: Since the house value increases at a constant rate, we can model the value with a linear equation of the form:
   $P = P_1 + m(t - t_1)$
   where:

   • $P$ is the house value at time $t$,
   • $P_1$ is the initial value of the house,
   • $m$ is the slope of the line (rate of change of value per year),
   • $t_1$ is the starting year (1980 in this case).

2. Slope Calculation: The slope $m$ represents the rate at which the value increases each year, and it can be found by:
   $m = \frac{P_2 - P_1}{t_2 - t_1}$

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Solution

1. Identify the Known Points:
   From the problem, we are given:

   • $P_1 = 1$ million in 1980, so $t_1 = 1980$,
   • $P_2 = 4$ million in 1996, so $t_2 = 1996$.

2. Calculate the Slope $m$:
   Using the formula for the slope $m$: $m = \frac{P_2 - P_1}{t_2 - t_1} = \frac{4 - 1}{1996 - 1980} = \frac{3}{16}$
   So, the slope of the line is $m = \frac{3}{16}$.

3. Write the Equation of the Line:
   Using the point-slope form of the line, $P - P_1 = m(t - t_1)$, and substituting $P_1 = 1$, $m = \frac{3}{16}$, and $t_1 = 1980$, we get: $P - 1 = \frac{3}{16}(t - 1980)$
   Expanding this: $P = 1 + \frac{3}{16}(t - 1980)$
   This is the equation that gives the value of the house after $t$ years since 1980.

4. Find the Value of the House in 1990:
   To find the value of the house in 1990, substitute $t = 1990$ into the equation: $P = 1 + \frac{3}{16}(1990 - 1980)$
   Simplifying: $P = 1 + \frac{3}{16}(10) = 1 + \frac{30}{16} = 1 + 1.875 = 2.875 \text{ million}$
   Therefore, the value of the house in 1990 was Rs. 2.875 million.

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Key Formulas or Methods Used

• Slope Formula:
  $m = \frac{P_2 - P_1}{t_2 - t_1}$

• Equation of a Line:
  $P = P_1 + m(t - t_1)$

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Summary of Steps

1. Identify the two points: $(1980, 1)$ and $(1996, 4)$.
2. Calculate the slope $m$ using the formula: $m = \frac{3}{16}$.
3. Write the equation of the line: $P = 1 + \frac{3}{16}(t - 1980)$.
4. To find the value in 1990, substitute $t = 1990$ into the equation: $P = 2.875$ million.