Profit and Loss

Profit and Loss is a fundamental topic in aptitude exams. It deals with evaluating the financial outcome of buying and selling items. The core ideas revolve around understanding cost, selling, marking, discount, and profit/loss percentage.

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Intuitive Understanding

• You buy something at a price (Cost Price or CP)
• You sell it at another price (Selling Price or SP)
• If SP > CP → Profit
• If SP < CP → Loss

The real challenge is in how these terms interact — especially when percentages, marked prices, discounts, or multiple transactions come into play.

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Key Formulas and Concepts

Basic Terms

• Cost Price (CP): Price at which the item is bought
• Selling Price (SP): Price at which it is sold
• Profit or Gain = $SP - CP$
• Loss = $CP - SP$

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Profit/Loss Percentages

$$\text{Profit\%} = \frac{\text{Profit}}{\text{CP}} \times 100 = \frac{SP - CP}{CP} \times 100$$ $$\text{Loss\%} = \frac{\text{Loss}}{\text{CP}} \times 100 = \frac{CP - SP}{CP} \times 100$$

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SP and CP with Percentages

$$SP = CP \times \left(1 + \frac{\text{Profit\%}}{100}\right)$$ $$SP = CP \times \left(1 - \frac{\text{Loss\%}}{100}\right)$$ $$CP = SP \div \left(1 + \frac{\text{Profit\%}}{100}\right) \quad \text{or} \quad CP = SP \div \left(1 - \frac{\text{Loss\%}}{100}\right)$$

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Marked Price (MP) and Discounts

• Marked Price (MP): Listed or printed price (before discount)
• Discount = $MP - SP$

$$\text{Discount\%} = \frac{\text{Discount}}{MP} \times 100$$ $$SP = MP \times \left(1 - \frac{\text{Discount\%}}{100}\right)$$

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Successive Profit or Loss (Two Transactions)

When profit/loss occurs successively:

$$\text{Net Change\%} = a + b + \frac{ab}{100}$$

Where:

• $a$% and $b$% are successive gain/loss (positive for profit, negative for loss)

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Conceptual Tricks and Shortcuts

1. Selling at x% profit/loss is same as selling at:

$$SP = CP \times \left(1 \pm \frac{x}{100}\right)$$

2. Profit is always calculated on Cost Price, but Discount is always on Marked Price

3. If same item is sold at profit and loss (same %):

$$\text{Net Loss} = \frac{\text{Profit\%}^2}{100}$$

(Only valid if CP is same in both cases.)

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Visual Explanations

Profit and Loss Line Diagram

      Loss         CP         Profit
<------------|-------------|------------>
             SP (when loss)     SP (when profit)

Discount Flow:

        MP
         ↓   - Discount%
        SP
         ↓   - Cost (CP)
      Profit or Loss

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Common Mistakes and Tips

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Real-Life Applications

• Retail pricing
• Online shopping discounts
• Bargaining in real-life purchases
• Stock market and trade
• Business and sales analytics

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Examples

Example 1: Simple Profit

A man buys a chair for ₹500 and sells it at ₹600. Find profit %.

$$\text{Profit} = 600 - 500 = \text{Rs.~}100 \Rightarrow \frac{100}{500} \times 100 = 20\%$$

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Example 2: Loss from Discount

A shopkeeper marks a product at ₹1,000 and gives 20% discount. His cost is ₹700. Find profit/loss %.

$$SP = 1000 \times 0.8 = \text{Rs.~}800 \Rightarrow \text{Profit} = 800 - 700 = \text{Rs.~}100 \Rightarrow \text{Profit\%} = \frac{100}{700} \times 100 = 14.29\%$$

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Example 3: Net Successive Gain

A trader gains 10% on first sale and 20% on second. Find overall gain %.

$$\text{Net Gain} = 10 + 20 + \frac{10 \times 20}{100} = 32\%$$

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Example 4: Equal Profit & Loss → Always Loss

A man sells an item at 20% profit and again at 20% loss. Find net result.

$$\text{Net Loss\%} = \frac{20^2}{100} = 4\%$$

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Example 5: Finding Marked Price

SP = ₹800 after 20% discount. Find MP.

$$MP = \frac{SP}{1 - \frac{20}{100}} = \frac{800}{0.8} = \text{Rs.~}1000$$