Step by step
The question
Question: A $1,000 painting goes up 30%, then down 30%.
What is the new price?
1 Up 30% of $1,000
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.
$1,000 divided by 10 is $100.
30% is 3 times 10%.
3 times $100 is $300.
Up 30% means we add $300.
$1,000 plus $300 is $1,300.
2 Down 30% of $1,300
Now the price is $1,300.
The second 30% is not of the old number.
It is 30% of the new number, $1,300.
To find 10%, we divide by 10.
$1,300 divided by 10 is $130.
30% is 3 times 10%.
3 times $130 is $390.
Down 30% means we take away $390.
$1,300 minus $390 is $910.
Careful!
We can't just add the percents.
The second percent is of a new number.
You might think up 30% and down 30% cancel out.
But we did not get back to $1,000.
The second 30% is of $1,300, a bigger number.
So it is $390, not $300 like the first 30%.
So the new price is $910.
3 Check
Let's check another way: up 30% makes it 100% plus 30%.
That is 130% of the price.
130% is 130 out of 100: 1.3.
$1,000 times 1.3 is $1,300.
Then: down 30% leaves 100% minus 30%.
That is 70% of the price.
70% is 70 out of 100: 0.7.
$1,300 times 0.7 is $910.
The same.
So the answer is right!
Let's look back.
First: up 30% of $1,000.
Then: down 30% of $1,300.
Last: check.