Step by step
The question
Question: A $50 shirt goes up 20%, then down 20%.
What is the total percent change?
1 Up 20% of $50
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.
$50 divided by 10 is $5.
20% is 2 times 10%.
2 times $5 is $10.
Up 20% means we add $10.
$50 plus $10 is $60.
2 Down 20% of $60
Now the price is $60.
The second 20% is not of the old number.
It is 20% of the new number, $60.
To find 10%, we divide by 10.
$60 divided by 10 is $6.
20% is 2 times 10%.
2 times $6 is $12.
Down 20% means we take away $12.
$60 minus $12 is $48.
Careful!
We can't just add the percents.
The second percent is of a new number.
You might think up 20% and down 20% cancel out.
But we did not get back to $50.
The second 20% is of $60, a bigger number.
So it is $12, not $10 like the first 20%.
3 The total change
We started at $50 and ended at $48.
That is $2 less.
Percent means out of 100.
The percent change is the change over the start: 2 out of 50.
We want the bottom to be 100.
100 divided by 50 is 2: 50 times 2 is 100.
So we multiply the top and the bottom by 2.
That is 4/100.
4 out of 100 is 4%.
So it is 4% less than at the start.
So the total change is 4% less.
4 Check
Let's check another way: up 20% makes it 100% plus 20%.
That is 120% of the price.
120% is 120 out of 100: 1.2.
$50 times 1.2 is $60.
Then: down 20% leaves 100% minus 20%.
That is 80% of the price.
80% is 80 out of 100: 0.8.
$60 times 0.8 is $48.
And 4% of $50 is $2.
$50 minus $2 is $48, the same end.
So the answer is right!
Let's look back.
First: up 20% of $50.
Then: down 20% of $60.
Then: the total change.
Last: check.