Step by step
The question
Question: A $500 TV goes up 10%, then down 10%.
What is the total percent change?
1 Up 10% of $500
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.
$500 divided by 10 is $50.
Up 10% means we add $50.
$500 plus $50 is $550.
2 Down 10% of $550
Now the price is $550.
The second 10% is not of the old number.
It is 10% of the new number, $550.
To find 10%, we divide by 10.
$550 divided by 10 is $55.
Down 10% means we take away $55.
$550 minus $55 is $495.
Careful!
We can't just add the percents.
The second percent is of a new number.
You might think up 10% and down 10% cancel out.
But we did not get back to $500.
The second 10% is of $550, a bigger number.
So it is $55, not $50 like the first 10%.
3 The total change
We started at $500 and ended at $495.
That is $5 less.
Percent means out of 100.
The percent change is the change over the start: 5 out of 500.
Smaller numbers are easier.
5 goes into 500: 5 times 100 is 500.
So 5 goes into both.
We divide the top and the bottom by 5.
That is 1/100.
1 out of 100 is 1%.
So it is 1% less than at the start.
So the total change is 1% less.
4 Check
Let's check another way: up 10% makes it 100% plus 10%.
That is 110% of the price.
110% is 110 out of 100: 1.1.
$500 times 1.1 is $550.
Then: down 10% leaves 100% minus 10%.
That is 90% of the price.
90% is 90 out of 100: 0.9.
$550 times 0.9 is $495.
And 1% of $500 is $5.
$500 minus $5 is $495, the same end.
So the answer is right!
Let's look back.
First: up 10% of $500.
Then: down 10% of $550.
Then: the total change.
Last: check.