Step by step
The question
Question: A $100 bag goes down 10%, then up 10%.
What is the new price?
1 Down 10% of $100
Percent means out of 100.
So 10% is 10 out of 100.
That is the same as 1 out of 10: one tenth.
To find 10%, we divide by 10.
$100 divided by 10 is $10.
Down 10% means we take away $10.
$100 minus $10 is $90.
2 Up 10% of $90
Now the price is $90.
The second 10% is not of the old number.
It is 10% of the new number, $90.
To find 10%, we divide by 10.
$90 divided by 10 is $9.
Up 10% means we add $9.
$90 plus $9 is $99.
Careful!
We can't just add the percents.
The second percent is of a new number.
You might think down 10% and up 10% cancel out.
But we did not get back to $100.
The second 10% is of $90, a smaller number.
So it is $9, not $10 like the first 10%.
So the new price is $99.
3 Check
Let's check another way: down 10% leaves 100% minus 10%.
That is 90% of the price.
90% is 90 out of 100: 0.9.
$100 times 0.9 is $90.
Then: up 10% makes it 100% plus 10%.
That is 110% of the price.
110% is 110 out of 100: 1.1.
$90 times 1.1 is $99.
The same.
So the answer is right!
Let's look back.
First: down 10% of $100.
Then: up 10% of $90.
Last: check.