To reverse a single discount, divide by the remaining multiplier. If an item costs $p after a 25% discount, the original price is p divided by 0.75, because 25% off leaves 75% of the original. For successive changes, divide the final amount by the product of all multipliers applied—for instance, a price that increased 25% then 8% to reach $351 means dividing by 1.25 × 1.08 = 1.35, yielding approximately $260.

A markup expresses profit as a percentage of the cost, while a margin expresses profit as a percentage of the selling price. If an item sells for $150 with a 25% margin, the profit is 0.25 × $150 = $37.50, making the cost $112.50. With a 25% markup instead, the cost × 1.25 = $150, so the cost is $120. The SAT frequently hides this distinction in wording, so identifying the base is essential.

The main traps include base switching (where the second percent applies to a new value, not the original), reverse percent errors (confusing x is what percent of y with the inverse relationship), percent of percent mistakes (treating 20% of 30% as an addition problem), and incorrect inversion (using the same percent to undo a change instead of dividing by the correct multiplier). A 20% decrease requires dividing by 0.80—not adding 20%—to return to the original amount.

Convert each operation to a multiplier and multiply in sequence: markup by m% multiplies by (1 + m), discount by d% multiplies by (1 - d), and tax by t% multiplies by (1 + t). For an $80 item marked up 35%, discounted 20%, and taxed 7.5%, the final price is 80 × 1.35 × 0.80 × 1.075. Tax typically applies to the post-discount price, not the original, and the order of factors does not affect the final result due to commutativity.

Two successive discounts never equal a single discount equal to their sum. For example, a 30% discount followed by a 20% discount leaves 0.70 × 0.80 = 0.56 of the original price, which is a 44% overall discount—not 50%. The correct approach is to multiply the remaining multipliers and convert back to a percent of the original.