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Math · 3.4Combine and Reverse Percent Changes讲解 2/5

Multiply successive changes

An $80 item first increases in price by $25\%$, then receives a $20\%$ discount on its new price. What is the final price? Each percentage acts on the price immediately before that change, so the two bases differ.

1. 1
   $\displaystyle 1 + \frac{25}{100} = 1.25; 1 - \frac{20}{100} = 0.80$

   Keep the full price plus `25%` for the increase, and keep `80%` of the updated price for the discount.

2. 2
   $\displaystyle 80(1.25) = 100$

   The first multiplier raises the price to $100, which becomes the base for the following discount.

3. 3
   $\displaystyle 100(0.80) = 80$

   The discount removes $20 from the updated $100 price, leaving a final payment of $80.

4. 4
   $\displaystyle 1.25(0.80) = 1$

   These particular multipliers have product `1`, explaining why this chain returns exactly to its original price.

5. 5
   $\displaystyle 1.20(0.80) = 0.96$

   An equal `20%` increase and decrease instead leave `96%` of the original, because the decrease uses the larger updated base.

An increase keeps the original amount and adds a fraction; a decrease keeps only the remaining fraction. Use those multipliers in sequence, then compare with a $20\%$ increase followed by a $20\%$ decrease to see why opposite percentages need not cancel.