Profit and Loss
What you must know
Establish the vocabulary: cost price (CP), selling price (SP), marked price (MP). Profit = SP − CP when SP > CP; loss is the reverse. Both profit% and loss% are conventionally calculated on CP unless the question says otherwise. Core relations: SP = CP(1 + p/100), CP = SP × 100/(100 + p), and for a loss CP = SP × 100/(100 − l).
Two powerful shortcuts deserve memorisation. First, if two articles are sold at the same price, one at a gain of x% and the other at a loss of x%, the overall result is always a loss of x²/100 percent. Second, dishonest-dealer problems: gain% = (error ÷ true value sold) × 100, where the error is the shortfall in weight or measure.
Traps
- Profit% on SP versus CP: if a question gives profit as a percentage of SP, convert before using standard formulas.
- Selling at a discount does not automatically mean a loss; compare discounted SP with CP.
- "Sold two articles at ₹500 each, one at +20% and one at −20%" gives a loss, not break-even; the equal-SP trap is the most common error.
- When CP changes (e.g., a man buys at 10% less), recompute both CP and SP, not just one.
Rule of thumb and example
When CP is unknown, set CP = 100 and work in rupees; the answer often falls out without algebra.
Worked example: A shopkeeper sells an item for ₹576 after a 20% profit on CP. CP = 576 × 100/120 = ₹480. If he had wanted 30% profit, SP = 480 × 1.30 = ₹624.