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Grades 6–8 · Everyday Life

Why don’t successive discounts simply add?

A percentage is always a percentage of a base amount. Successive discounts do not usually add as percentages because the second discount can use a different base from the first. Its size depends on what amount the stated rule applies to.

J10 visual explanationFictional exact-dollar price model: start at 100 dollars; subtract 20 percent of the original, 20 dollars, leaving 80. Then subtract 25 percent of the reduced price, with base 80 dollars and discount 20. Final price 60, a 40 percent reduction from the original. Sequential factors are four-fifths times three-fourths, leaving three-fifths; same-original-base subtraction leaves eleven-twentieths. No taxes, rounding or fees. This is arithmetic, not buying or investment advice.Which price is the percentage of?Fictional arithmetic · no purchase recommendationOriginal price: $100First: 20% offBase: $100Remove $20; left $80Then: 25% offBase: reduced $80Remove $20; left $60Final $60: 40% off the original $100.

In the diagramWhich price is the percentage of?Fictional arithmetic · no purchase recommendationOriginal price: $100First: 20% offBase: $100Remove $20; left $80Then: 25% offBase: reduced $80Remove $20; left $60Final $60: 40% off the original $100.

The second discount is 25% of the reduced $80, not the original $100. The final $60 is a 40% reduction overall.

Starting from the fictional $100 price, 20% off removes $20 and leaves $80. A further 25% off the reduced price removes one quarter of $80, or another $20. The final price is $60. The total reduction is $40 out of the original $100, which is 40%, not 45%.

The remaining fractions explain the same result. The first discount leaves four-fifths of the original price. The second leaves three-fourths of that remainder. Multiplying four-fifths by three-fourths gives three-fifths, so 60% remains and 40% has been removed.

A different explicit rule could take both discounts from the original $100. Then the reductions would be $20 and $25, leaving $55. That is a different calculation because its second base is different, not because the meaning of 25% changed. These fictional amounts have no taxes, fees or rounding; they explain percentage bases rather than establish an actual offer or purchase recommendation.

About this explanation

Starting knowledge: Find 20% and 25% of whole-number dollar amounts, subtract reductions, and identify the whole that a percent refers to.

Sources

  • OpenStax, Rice University — 6.2 Discounts, Markups, and Sales Tax
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