2. Markup, Discounts, and Sale Prices

Retailers rarely sell an item for exactly what they paid for it. They add a markup to cover overhead and profit, and later apply discounts to move slow inventory or attract customers during a sale. Both operations use the same multiplier idea: a percent change of \(p\%\) either scales a price up by \(\left(1 + \dfrac{p}{100}\right)\) or down by \(\left(1 - \dfrac{p}{100}\right)\), depending on whether the store is increasing or decreasing the price.

The reason this works is that "100%" represents the whole original price. Adding \(p\%\) means the new price is the original 100% plus an extra \(p\%\), so the multiplier is \(1 + \dfrac{p}{100}\). Subtracting \(p\%\) means the new price keeps only \((100-p)\%\) of the original, giving the multiplier \(1 - \dfrac{p}{100}\).

Worked example (markup): A store buys a jacket for $40 and marks it up 60%. The selling price is
\[ 40 \times (1 + 0.60) = 40 \times 1.60 = 64 \text{ dollars.} \]

Worked example (discount): A $90 pair of shoes is 25% off. The sale price is
\[ 90 \times (1 - 0.25) = 90 \times 0.75 = 67.50 \text{ dollars.} \]

Working backward: a shirt is on sale for $48 after a 20% discount. If \(x\) is the original price, then \(0.80x = 48\), so \(x = \dfrac{48}{0.80} = 60\) dollars.

Case study: A furniture store buys a dining table wholesale for $220. To cover shipping, showroom rent, and profit, they mark it up 85% for the sales floor. The listed price is \(220 \times 1.85 = 407\) dollars. Six months later, the table hasn't sold, so the manager marks it down 30% for a clearance sale. The clearance price is \(407 \times 0.70 = 284.90\) dollars — still well above the original $220 wholesale cost, which shows why a store can survive even a fairly deep clearance discount as long as the initial markup was large enough.

Worked example (two markups compared): Two stores buy the same $50 lamp. Store A marks it up 40%, listing it at \(50 \times 1.40 = 70\) dollars. Store B marks it up 40% and then adds a 10% "premium display" markup on top of that new price: \(70 \times 1.10 = 77\) dollars. Even though both stores describe their pricing with round percentages, stacking a second markup on top of the first — rather than adding the two percents together — produces a higher final price than a single 50% markup would (\(50 \times 1.50 = 75\)).

Common mistakes to avoid:

• Finding the dollar amount of the discount or markup and forgetting to add or subtract it from the original price — the multiplier method in this lesson skips that extra step and gives the final price directly.
• Assuming a markup followed by an equal-percent discount returns you to the original price. It does not: marking up 50% and then discounting 50% leaves a price lower than the start, because the second percent is taken from a larger number.