Discount and Markup — Explained
Detailed Explanation
Which price is the percentage calculated on?
Markup and discount problems use several prices with different roles. Cost price is the seller's acquisition cost under the assumptions of the exercise. Marked price is the stated price before the specified discount. Selling price is the amount received after that discount, unless taxes or other charges are separately introduced. Profit is selling price minus cost price. Profit percentage uses cost price as its denominator, while a stated discount normally uses marked price.
Start by naming these quantities. A 20 per cent markup and a 20 per cent discount usually act on different bases, so they do not cancel. The apparent symmetry of the words is not a mathematical symmetry. Multipliers preserve the bases: a markup of m per cent multiplies cost by 1 + m/100; a discount of d per cent multiplies marked price by 1 - d/100.
What does a complete worked example look like?
Take an original hypothetical item costing ₹800. A 25 per cent markup gives a marked price of ₹1,000. A 10 per cent discount then reduces the amount paid to ₹900. Profit is ₹100, which is 12.5 per cent of the ₹800 cost. The ₹100 discount is 10 per cent of marked price, but the same rupee amount is 12.5 per cent of cost price.
Subtracting ten from twenty-five would suggest a 15 per cent profit and would be wrong. The correct multiplier is 1.25 multiplied by 0.90, giving 1.125 times cost. This calculation is a useful compact method, but it should be supported by the labelled prices when the question also asks for a rupee amount. A percentage answer without its base is incomplete information.
How do successive discounts combine?
Successive discounts apply one after another. A 20 per cent discount followed by a 10 per cent discount leaves 0.80 multiplied by 0.90 = 0.72 of the original marked price. The equivalent single discount is 28 per cent. On a marked price of ₹1,000, the first discount removes ₹200 and the second removes ₹80 from the remaining ₹800. The total reduction is ₹280, not ₹300.
Reversing these two pure percentage discounts gives the same final price because multiplication commutes. That does not mean every promotional sequence is interchangeable. A fixed coupon, a minimum-spend condition or a tax calculated at a particular stage can make order relevant. Read the rule supplied by the problem and build the price sequence in that order. Do not transfer the commutative shortcut beyond the conditions that justify it.
How can the required markup be recovered?
Work backwards from the desired selling price. Suppose a seller wants a 20 per cent profit on a cost of ₹750 while offering a 25 per cent discount. The desired selling price is ₹900. Since the customer pays 75 per cent of marked price, the marked price must be ₹900 divided by 0.75, or ₹1,200. The markup is ₹450 on ₹750, which is 60 per cent.
Adding the desired profit percentage and discount percentage to obtain 45 per cent is tempting but incorrect. The discount applies after markup, so it removes a share of the enlarged price. The general relationship is selling-price multiplier = markup multiplier multiplied by discount multiplier. Rearranging that relationship avoids trial and error and makes clear why a larger discount may require a substantially larger markup to preserve the same profit.
How should fixed coupons and percentage offers be compared?
Suppose an eligible ₹1,600 purchase can use either a ₹250 coupon or an 18 per cent discount, but not both. The percentage offer saves ₹288, which exceeds the coupon saving by ₹38. The break-even marked price solves 0.18 times price = ₹250, giving approximately ₹1,388.89. Above that threshold the percentage saving is larger, subject to any cap or eligibility condition specified in the exercise.
If the percentage discount is capped at ₹200, the fixed coupon becomes better for the same purchase. A threshold, maximum benefit or exclusion is part of the mathematical rule, not incidental advertising language. Compare final amounts payable under each complete offer. A larger advertised percentage does not determine the better offer when it applies to fewer items, a different base or a capped benefit.
How do taxes and additional costs change the model?
Only introduce a tax when the question specifies its rate and base. In a hypothetical exercise with a 10 per cent tax on a discounted price of ₹900, the final customer payment is ₹990. That does not by itself establish that the seller's profit is ₹190 on an ₹800 cost: the tax component is not necessarily retained as revenue. Separate the arithmetic of customer payment from the exercise's definition of seller profit.
Likewise, transport, repair or packaging costs belong in the cost base only when the problem requires them. If an item costs ₹800 and an additional ₹50 is explicitly included as selling cost, a ₹900 sale yields ₹50 profit on total cost of ₹850. The corresponding percentage differs from the earlier 12.5 per cent. State the adopted base rather than silently switch between acquisition cost and total cost.
Which checks make the result trustworthy?
Reconstruct the forward chain after solving backwards. In the required-markup example, ₹750 increased by 60 per cent becomes ₹1,200; reducing this by 25 per cent gives ₹900; the ₹150 surplus is 20 per cent of ₹750. Each stage confirms the denominator used at the preceding stage. A single final number without this chain can conceal a compensating pair of mistakes.
Finally, distinguish profit from margin. Profit as a percentage of cost and profit as a percentage of selling price are different ratios. For the ₹800-cost, ₹900-sale example they are 12.5 per cent and about 11.11 per cent respectively. Record which measure is requested. A clear sequence of cost, marked price, discount and selling price makes these questions predictable without assuming that percentages can simply be added or subtracted.