Bike Cost Price Calculation: Equal Selling Prices

This problem requires finding the cost price (CP) of the first bike when the total cost price of two bikes is known, and they are sold at different profit/loss percentages but achieve the same selling price (SP).

Problem Analysis

• Total Cost Price (CP₁ + CP₂) = ₹1,00,000
• Bike 1: Sold at 20% profit
• Bike 2: Sold at 20% loss
• Condition: Selling Prices are equal (SP₁ = SP₂)
• Goal: Find the Cost Price of the first bike (CP₁)

Calculating Selling Prices Relation

We use the standard formulas:

• SP = CP $\times$ (1 + Profit%/100)
• SP = CP $\times$ (1 - Loss%/100)

For Bike 1 (20% profit): SP₁ = CP₁ $\times$ (1 + 20/100) = CP₁ $\times$ 1.20

For Bike 2 (20% loss): SP₂ = CP₂ $\times$ (1 - 20/100) = CP₂ $\times$ 0.80

Given that SP₁ = SP₂:

$ CP_1 \times 1.20 = CP_2 \times 0.80 $

Cost Price Ratio Derivation

Rearrange the equation to find the relationship between the cost prices:

$ \frac{CP_1}{CP_2} = \frac{0.80}{1.20} $

Simplify the fraction:

$ \frac{CP_1}{CP_2} = \frac{8}{12} = \frac{2}{3} $

This means the ratio CP₁ : CP₂ is 2 : 3. We can express CP₂ in terms of CP₁:

$ CP_2 = \frac{3}{2} CP_1 $

Final Cost Price Calculation

Use the total cost price equation and substitute the relation found:

CP₁ + CP₂ = 1,00,000

Substitute CP₂ = $\frac{3}{2}$ CP₁:

$ CP_1 + \frac{3}{2} CP_1 = 1,00,000 $

Combine the CP₁ terms:

$ \left(1 + \frac{3}{2}\right) CP_1 = 1,00,000 $

$ \left(\frac{5}{2}\right) CP_1 = 1,00,000 $

Solve for CP₁:

$ CP_1 = 1,00,000 \times \frac{2}{5} $

$ CP_1 = 40,000 $

Therefore, the cost price of the first bike is ₹40,000.