Overview
A percentage is always relative to a base. When that base changes, equal-looking percentages no longer represent equal amounts. Successive changes are therefore best modeled with multipliers, while a percent increase requires dividing the change by the specified original amount. We will apply this idea to discounts, recovery after a loss, changing rates and a profit constraint.
Apply successive changes to the current amount
An $80 pass receives a 15% discount and then a 5% tax on the discounted price. The discount leaves 85% of the price, so the first amount is 80·0.85 = 68. Tax then multiplies this amount by 1.05, giving $71.40.
Subtracting the two rates and taking a net 10% reduction would give $72, but the discount and tax use different bases. The tax is five percent of 68, not five percent of 80. Multipliers preserve that sequence automatically: each factor acts on the amount reached at its stage. Labeling the base also explains why reversing operations may or may not matter in a particular context; the written tax rule, rather than an assumed rule about all purchases, determines what is taxed.
Distinguish percentage points from relative increase
A success rate rising from 40% to 50% increases by ten percentage points. Relative to the original rate, the increase is 10/40 = 1/4, or 25%. The answer 10 names the absolute difference between percentage values rather than the requested percent increase.
The same base distinction explains recovery after a 20% loss. Eighty percent of the original amount remains, and restoring the missing twenty percent requires an increase of 20/80 = 25% of what remains. A 20% increase would act on the smaller base and would not undo the loss. Write “change divided by starting amount for this comparison” before substituting numbers. The correct starting amount may be an original total, a current depleted total or a rate; the question’s wording decides which one.
Preserve profit rather than only revenue
A workshop charges $20 per item and spends $12 to produce it, leaving $8 profit per unit. A proposed price cut increases units sold by 25%, or a factor of 5/4, while unit cost stays fixed. To preserve total profit, the new margin must be at least 8/(5/4) = $6.40. Thus the new price must be at least $18.40.
The greatest permitted cut is $1.60 from $20, which is 8%. A 20% price cut can preserve revenue under the sales multiplier, but revenue includes money needed to cover production costs. It therefore does not preserve profit. This problem requires choosing the correct quantity before applying percentages: price minus unit cost is the relevant margin, and that margin is then multiplied by units sold. The fixed cost per unit is a modeling condition, not an incidental detail.
You can make the profit condition precise without knowing the original number of units sold. Call that number q. Original profit is 8q, and new profit is (p − 12)(5q/4). The positive factor q cancels when the profits are compared. This explains why the question has enough information: it asks for a proportional threshold, so the unknown sales volume does not need to be supplied.
Reverse a two-stage change with the correct final base
A quantity rises by p% and then falls by p%, ending at 96% of its start. Let r = p/100. The combined multiplier is (1 + r)(1 − r) = 1 − r² = 0.96, so r = 0.20 under the positive restriction. The first increased value was 1.20 times the original, while the final value is 0.96 times the original. To return from the final value to the first increased value requires 1.20/0.96 − 1 = 0.25, or 25%.
The distractor 24 is the difference measured against the original base, not against the current final amount. For transfer, write a chain of labeled amounts and multipliers. Before every percent comparison, identify its denominator explicitly. In practice, explain whether a wrong choice used an old base, subtracted rates, confused percentage points or modeled the wrong economic quantity.
Apply the method to a new case
A store pays 80 dollars for an item and sells it at a price that includes a 25% markup on cost. A platform then takes a fee of 10% of the selling price. Find the store's profit after that fee. The marked-up price is 80·1.25 = 100 dollars. The fee is 10 dollars, so the store receives 90 and its profit is 90 − 80 = 10 dollars. Subtracting ten percentage points from the 25% markup and applying 15% to cost would give 12 dollars, but the fee and markup use different bases. Check by expressing net receipt as 80·(5/4)·(9/10) = 90. Define the base of each percentage before combining multipliers; 'profit' also requires subtracting cost from the net receipt rather than reporting the amount paid by the customer.