4.3 Q-16

Question Statement

A milkman sells 500 liters of milk at Rs. 12.50 per liter and 700 liters at Rs. 12.00 per liter. Assuming the relationship between the sale price and the quantity of milk sold is linear, find how many liters of milk the milkman can sell at Rs. 12.25 per liter.

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Background and Explanation

This problem requires knowledge of linear equations. Specifically, we are using the formula for the equation of a straight line. To solve it, we need:

1. Linear Equation Formula:
   The general form of a straight line is:
   $P - P_1 = m (\ell - \ell_1)$
   where $P$ is the price, $\ell$ is the number of liters of milk sold, and $m$ is the slope of the line.

2. Slope Calculation:
   The slope $m$ is given by:
   $m = \frac{P_1 - P_2}{\ell_1 - \ell_2}$
   where $(P_1, \ell_1)$ and $(P_2, \ell_2)$ are two known points on the line.

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Solution

1. Identify Known Points:
   From the problem, we have the following two points:

   • $\left(\ell_1, P_1\right) = (500, 12.50)$
   • $\left(\ell_2, P_2\right) = (700, 12.00)$

2. Calculate the Slope $m$:
   The formula for the slope is:
   $m = \frac{P_1 - P_2}{\ell_1 - \ell_2} = \frac{12.50 - 12.00}{500 - 700} = \frac{0.50}{-200} = \frac{-1}{400}$

3. Find the Equation of the Line:
   Using the point-slope form of the equation $P - P_1 = m (\ell - \ell_1)$, we substitute $P_1 = 12.50$, $\ell_1 = 500$, and $m = \frac{-1}{400}$:
   $P - 12.50 = \frac{-1}{400} (\ell - 500)$
   Simplifying:
   $P - 12.50 = \frac{-1}{400} (\ell - 500)$
   Multiply both sides by 400 to eliminate the fraction:
   $400(P - 12.50) = - (\ell - 500)$
   Therefore, the equation becomes:
   $\ell = 500 - 400(P - 12.50)$

4. Substitute the Price $P = 12.25$:
   Now, to find the number of liters sold at Rs. 12.25 per liter, substitute $P = 12.25$ into the equation:
   $\ell = 500 - 400(12.25 - 12.50)$
   $\ell = 500 - 400(-0.25)$
   $\ell = 500 + 100$
   $\ell = 600$

   So, the milkman can sell 600 liters of milk at Rs. 12.25 per liter.

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Key Formulas or Methods Used

• Slope Formula:
  $m = \frac{P_1 - P_2}{\ell_1 - \ell_2}$

• Equation of a Line:
  $P - P_1 = m (\ell - \ell_1)$

• Substitution: Substituting the value $P = 12.25$ into the equation to find $\ell$.

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Summary of Steps

1. Identify the two points: $(500, 12.50)$ and $(700, 12.00)$.
2. Calculate the slope $m$.
3. Write the equation of the line using the point-slope form.
4. Substitute $P = 12.25$ into the equation to find $\ell$.
5. The result is $\ell = 600$ liters.