Cost Price Calculation: SP & Profit Changes

This solution explains how to determine the percentage increase in cost price when selling price and profit percentage change.

Initial Conditions Setup

Let the original Cost Price be denoted as CP.

The initial profit was 20%. The initial Selling Price (SP1) is calculated using the formula:

$ SP1 = CP \times (1 + \text{Profit Percentage}) $

Substituting the initial profit percentage:

$ SP1 = CP \times (1 + 0.20) = 1.20 \times CP $

Final Conditions Calculation

The selling price increases by 25%. The new Selling Price (SP2) is:

$ SP2 = SP1 \times (1 + 0.25) = 1.25 \times SP1 $

Substitute the expression for SP1:

$ SP2 = 1.25 \times (1.20 \times CP) = 1.50 \times CP $

The new profit is 30%. Let the new Cost Price be CP'.

The new Selling Price (SP2) is also expressed in terms of the new Cost Price:

$ SP2 = CP' \times (1 + \text{New Profit Percentage}) $

$ SP2 = CP' \times (1 + 0.30) = 1.30 \times CP' $

Relating Cost Prices

Equate the two derived expressions for SP2:

$ 1.50 \times CP = 1.30 \times CP' $

Solve for the new Cost Price (CP') relative to the original Cost Price (CP):

$ CP' = \frac{1.50}{1.30} \times CP = \frac{15}{13} \times CP $

Percentage Increase in Cost Price

Calculate the percentage increase using the formula:

$ \text{Percentage Increase} = \frac{\text{New CP} - \text{Original CP}}{\text{Original CP}} \times 100 $

$ \text{Percentage Increase} = \frac{CP' - CP}{CP} \times 100 $

Substitute the expression for CP':

$ \text{Percentage Increase} = \frac{\left(\frac{15}{13} \times CP\right) - CP}{CP} \times 100 $

Simplify the expression:

$ \text{Percentage Increase} = \left( \frac{15}{13} - 1 \right) \times 100 $

$ \text{Percentage Increase} = \left( \frac{15 - 13}{13} \right) \times 100 $

$ \text{Percentage Increase} = \frac{2}{13} \times 100 $

$ \text{Percentage Increase} = \frac{200}{13} \% \approx 15.38 \% $

The cost price increased by approximately 15.38%.