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Discounts and Price Increases
/in area/by Jill HackerHow 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.
Say a price of $25 increases by some percentage p. To find the increased price, multiply the original price by
. So in this case the new price is .)
Now that price decreases by percentage p, so you multiply by 
and the final price is \times(1-p/100).)
On the other hand, if the decrease comes first, the final price is\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.
How Many Steps?
/in statistics/by Jill Hacker