Step by step
The question
Question: A $200 jacket goes down 50%, then up 50%.
What is the total percent change?
1 Down 50% of $200
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.
$200 divided by 10 is $20.
50% is 5 times 10%.
5 times $20 is $100.
Down 50% means we take away $100.
$200 minus $100 is $100.
2 Up 50% of $100
Now the price is $100.
The second 50% is not of the old number.
It is 50% of the new number, $100.
To find 10%, we divide by 10.
$100 divided by 10 is $10.
50% is 5 times 10%.
5 times $10 is $50.
Up 50% means we add $50.
$100 plus $50 is $150.
Careful!
We can't just add the percents.
The second percent is of a new number.
You might think down 50% and up 50% cancel out.
But we did not get back to $200.
The second 50% is of $100, a smaller number.
So it is $50, not $100 like the first 50%.
3 The total change
We started at $200 and ended at $150.
That is $50 less.
Percent means out of 100.
The percent change is the change over the start: 50 out of 200.
Smaller numbers are easier.
50 goes into 200: 50 times 4 is 200.
So 50 goes into both.
We divide the top and the bottom by 50.
That is 1/4.
We want the bottom to be 100.
100 divided by 4 is 25: 4 times 25 is 100.
So we multiply the top and the bottom by 25.
That is 25/100.
25 out of 100 is 25%.
So it is 25% less than at the start.
So the total change is 25% less.
4 Check
Let's check another way: down 50% leaves 100% minus 50%.
That is 50% of the price.
50% is 50 out of 100: 0.5.
$200 times 0.5 is $100.
Then: up 50% makes it 100% plus 50%.
That is 150% of the price.
150% is 150 out of 100: 1.5.
$100 times 1.5 is $150.
And 25% of $200 is $50.
$200 minus $50 is $150, the same end.
So the answer is right!
Let's look back.
First: down 50% of $200.
Then: up 50% of $100.
Then: the total change.
Last: check.