1. Stacked discounts are not added
A 30% discount plus another 20% discount is not the same as 50% off. The second discount applies to the reduced price, not the original price. If a $100 item is discounted by 30%, it becomes $70. Another 20% off removes $14, so the final price is $56. The total discount is 44%, not 50%.
The shortcut is to multiply what remains. After 30% off, 70% remains. After another 20% off, 80% remains. 0.70 × 0.80 = 0.56, so you pay 56% of the original price.
2. Tax usually applies after the discount
In many shopping situations, sales tax is applied to the discounted price rather than the original ticket price. If an item costs $100, has a 25% discount, and tax is 8%, the discounted price is $75. Tax is then $6, making the total $81. If you tax the original $100 first, you get a different result.
Receipt check: calculate the discounted subtotal first, then apply tax, unless the receipt or local rule states otherwise.
3. A decrease and increase of the same percentage do not cancel
If a price falls by 20% and then rises by 20%, it does not return to the starting price. A $100 item that drops 20% becomes $80. A 20% increase from $80 adds $16, so the new price is $96. The base changed.
To return from $80 to $100, the increase must be 25%, because $20 is 25% of $80. This is why investment losses require larger gains to recover: a 50% loss needs a 100% gain to get back to the starting value.
Percentage points are not percentages
A percentage point and a percentage change are two different things. If an interest rate rises from 4% to 6%, it increased by 2 percentage points. But as a percentage change, that is a 50% increase — because 2 is 50% of the original 4. News stories, financial reports, and political arguments regularly confuse the two.
In practical terms, a drug trial that improves a success rate from 20% to 25% gained 5 percentage points. It also achieved a 25% relative improvement (5 is 25% of 20). Both statements are accurate; they measure different things. The question to ask is always: “relative to what original value?”
Watchword: When a headline says something “increased by 3%,” verify whether it means 3 percentage points (absolute) or a 3% relative change. They are rarely the same number and the distinction changes the meaning significantly.
Reverse percentages: working backwards
Sometimes you know the result of a percentage change but not the original value. If a jacket costs $85 after a 15% discount, the sale price is 85% of the original. Divide $85 by 0.85 to get $100. This is the reverse percentage formula: original = final ÷ (1 − discount rate). It works for any percentage decrease or increase.
The same logic applies to tip and tax problems. If a restaurant bill including 20% tax is $72, divide by 1.20 to find the pre-tax total: $60. Reverse percentages are useful whenever you need to undo a known rate.
The clean formula
For most percentage questions, ask: “percentage of what?” That “what” is the base. In discount stacking, the base changes after each discount. In tax calculations, the base is usually the discounted subtotal. In increase/decrease problems, the old value is the base for the change.
Use a calculator when precision matters
Mental math is useful for estimates, but exact shopping, budgeting, tax, grades, and finance decisions are easier with the Percentage Calculator. For a broader guide, read Understanding Percentages in Everyday Life. If the problem is a restaurant bill, the Tip Calculator handles the split and total cleanly.
The base number is the whole problem
Most percentage mistakes come from using the wrong base. A 30% discount followed by another 20% discount is not 50% off because the second discount applies to the reduced price. A 20% drop followed by a 20% increase also does not return to the original value because the increase starts from the smaller number.
When a percentage problem feels confusing, write one sentence before calculating: “the percent applies to ___.” That blank is the base. If the base changes halfway through the problem, calculate each step separately.
Quick audit: In shopping, grades, taxes, and markups, identify the original amount, the changed amount, and whether the question asks for a percent, a percentage-point change, or a final total.
When to use the calculator
The Percentage Calculator is best for exact totals, reverse percentages, and percentage change. Use mental math for a fast estimate, then use the calculator when a receipt, grade, invoice, or budget needs a number you can verify.