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.

0 replies

Leave a Reply

Want to join the discussion?
Feel free to contribute!

Leave a Reply