Profit, Loss & Price Changes

• The CP Base Principle: In standard financial mathematics, profit and loss percentages are always calculated using the Cost Price as the denominator. This is because the CP represents the initial investment against which performance is measured.

• Percentage Formulas: Profit and Loss percentages are expressed as:

$Profit \% = \left( \frac{Profit}{CP} \right) \times 100$ $Loss \% = \left( \frac{Loss}{CP} \right) \times 100$

• Proportionality: If the CP and SP are both multiplied by the same factor, the profit or loss percentage remains unchanged. This allows for scaling calculations to simpler numbers during problem-solving.

The Multiplier Method

• Calculating SP from CP: To find the Selling Price when a profit percentage ($P\%$) is known, use the multiplier $(1 + \frac{P}{100})$. For a loss ($L\%$), use $(1 - \frac{L}{100})$.
• Formula: $SP = CP \times \frac{100 \pm \text{Percentage}}{100}$. This method is more efficient than calculating the absolute profit/loss first and then adding/subtracting it.

Successive Price Changes

• Net Change Formula: When a price is changed by $x\%$ and then by $y\%$, the effective net percentage change is given by:

$Net \% = x + y + \frac{xy}{100}$

• Application: Use positive values for increases (mark-ups) and negative values for decreases (discounts). This formula is critical for finding the overall effect of multiple price adjustments without calculating intermediate values.

• Marked Price (MP): Also known as the list price or tag price, this is the price set by the seller before any discounts are applied. It is usually higher than the CP to allow for both a discount and a profit.

• Mark-up: The difference between the Marked Price and the Cost Price ($MP - CP$). The Mark-up Percentage is calculated on the CP: $\frac{Mark-up}{CP} \times 100$.

• Discount: A reduction offered on the Marked Price to arrive at the Selling Price ($MP - SP$). The Discount Percentage is always calculated on the Marked Price:

$Discount \% = \left( \frac{Discount}{MP} \right) \times 100$

• It is vital to distinguish which base is used for different percentage calculations to avoid significant errors.

• Successive Discounts: Two successive discounts of $10\%$ and $10\%$ do NOT equal a $20\%$ discount. The second discount is applied to the already reduced price, resulting in a net discount of $19\%$ ($10 + 10 - \frac{10 \times 10}{100}$).

• The '100' Assumption: When no absolute values are given, assume the Cost Price is $100$. This simplifies percentage calculations into direct additions or subtractions.

• Sanity Check: Always verify if the SP is higher than CP for profit and lower for loss. If a discount is applied, the SP must be lower than the MP.

• Common Trap: Watch for questions that ask for profit percentage based on the Selling Price. Unless explicitly stated, always default to the Cost Price as the base.

• Reverse Calculation: If given the SP and Profit %, find CP by dividing: $CP = \frac{SP}{1 + \text{profit decimal}}$. For example, if SP is $120$ at $20\%$ profit, $CP = \frac{120}{1.2} = 100$.