A businessman sells 2000 items per month at a price of Rs. 10 each. It is estimated that monthly sales will increase by 250 items for each Re. 0.25 reduction in price. Find the demand function corresponding to this estimate.

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Introduction

In economics and business mathematics, the demand function shows the relationship between the price of a product and the quantity demanded. In this problem, we are asked to construct a demand function based on the sales behavior of a businessman. We are given how quantity sold changes with price and need to develop a linear equation connecting these variables.

Given Information

Initial price per item = Rs. 10
Initial quantity sold = 2000 items
For every Re. 0.25 reduction in price, monthly sales increase by 250 items

Step 1: Let Variables

Let p be the price per item
Let x be the number of items sold per month

We need to express x in terms of p. This is our demand function: x = f(p)

Step 2: Understand the Relationship

We are told that:

At price Rs. 10, sales = 2000 items
Each time the price decreases by Rs. 0.25, sales go up by 250 items

This is a linear relationship. Let’s find the slope first.

Step 3: Use Two Points to Form the Equation

We are given two points on the (p, x) plane:

Point A: (10, 2000)
Point B: (9.75, 2250) — since reducing price by Rs. 0.25 increases sales by 250

We will now find the slope (m) of the demand function:

m = (x₂ – x₁)/(p₂ – p₁) = (2250 – 2000)/(9.75 – 10) = 250 / -0.25 = -1000

Step 4: Use Point-Slope Form

We know:

x – x₁ = m(p – p₁)

Using point (10, 2000) and m = -1000:

x – 2000 = -1000(p – 10)

⇒ x = -1000p + 10000 + 2000 = -1000p + 12000

Step 5: Final Demand Function

x = -1000p + 12000

This is the required demand function.

Interpretation

The negative slope (-1000) means that as price increases, demand decreases — which aligns with the law of demand.
The constant 12000 is the intercept — the estimated maximum number of items that could be sold at a price of Rs. 0.

Verification

Let’s verify the function for price = Rs. 10:

x = -1000×10 + 12000 = -10000 + 12000 = 2000 ✅

For price = Rs. 9.75:

x = -1000×9.75 + 12000 = -9750 + 12000 = 2250 ✅

Conclusion

The demand function that relates the number of items sold per month (x) to the price per item (p) is:

x = -1000p + 12000

This linear function helps the businessman understand how changes in price affect the number of items sold. It is a valuable tool in pricing strategy and revenue optimization in business mathematics.