## Discounts and Price Increases

Say you want to buy a jacket. You find one online that you like for $25, but by the time you make up your mind the price has gone up 20%. You aren’t ready to pay that much, so you hesitate, and while you wait the jacket goes on sale for 20% off. How much do you have to pay for it (before tax)? Well, the original price was$25, and then there was a price increase, and then an equal price decrease, so the final price must be $25. Right? Let’s see. The price increase is 20% of$25. That’s $5. So the increased price is$30. Then the sale takes the price down 20% of $30. That’s$6. So the final price is $30 –$6 = $24. So no, the final price is not the same as the original price. I fibbed when I said the price decrease is equal to the increase. The decrease is more. Why? Because the decrease is the same percent — 20% — of a larger number. The increase is 20% of$25 and the decrease is 20% of $30. What if the sale comes first and then the price increase? Let’s see. The price decrease is 20% of$25. That’s $5. So the sale price is$20. Then inflation (or whatever) takes the price up 20% of $20. That’s$4. So the final price is $20 +$4 = $24. Same result. Same final price. Is that a fluke, or does it never make a difference whether you increase or decrease first? Say a price of$25 increases by some percentage p. To find the increased price, multiply the original price by $1+p/100$. So in this case the new price is $25\times(1+p/100).$ Now that price decreases by percentage p, so you multiply by $1-p/100,$ and the final price is $100\times(1+p/100)\times(1-p/100).$

On the other hand, if the decrease comes first, the final price is $100\times(1-p/100)\times(1+p/100).$

The only difference is that the terms get multiplied in a different sequence. But in multiplication, changing the sequence does not change the result.

Bottom line: If a percentage increase is followed by the same percentage decrease, the final amount will not equal the original amount — but it will equal what you get if you switch the order and do the decrease first.