Worked examples
The question
Question: A $50 shirt goes up 20%, then down 20%.
What is the new price?
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.
10% of $50 = $50 ÷ 10 = $5
10% = ÷ 10
20% is 2 times 10%.
2 times $5 is $10.
20% of $50 = 2 × $5 = $10
Up 20% means we add $10.
$50 plus $10 is $60.
new price = $50 + $10 = $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.
10% of $60 = $60 ÷ 10 = $6
20% is 2 times 10%.
2 times $6 is $12.
20% of $60 = 2 × $6 = $12
Down 20% means we take away $12.
$60 minus $12 is $48.
new price = $60 − $12 = $48
Careful!
We can't just add the percents.
The second percent is of a new number.
Careful: not +20% − 20% = 0%: $50 ✗
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%.
not back to the start: 20% of $60 = $12, 20% of $50 = $10
So the new price is $48.
3 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.
100% + 20% = 120% = 1.2: $50 × 1.2 = $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.
The same.
So the answer is right!
100% − 20% = 80% = 0.8: $60 × 0.8 = $48
Let's look back.
First: up 20% of $50.
Then: down 20% of $60.
Last: check.
Answer not back to the start: 20% of $60 = $12, 20% of $50 = $10
▶ Watch it solved
The question
Question: A $400 laptop goes down 25%, then up 25%.
What is the new price?
1 Down 25% of $400
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.
$400 divided by 10 is $40.
10% of $400 = $400 ÷ 10 = $40
10% = ÷ 10
20% is 2 times 10%.
2 times $40 is $80.
20% of $400 = 2 × $40 = $80
5% is half of 10%.
Half of $40 is $20.
5% of $400 = $40 ÷ 2 = $20
25% is 20% plus 5%.
$80 plus $20 is $100.
25% of $400 = $80 + $20 = $100
Down 25% means we take away $100.
$400 minus $100 is $300.
new price = $400 − $100 = $300
2 Up 25% of $300
Now the price is $300.
The second 25% is not of the old number.
It is 25% of the new number, $300.
To find 10%, we divide by 10.
$300 divided by 10 is $30.
10% of $300 = $300 ÷ 10 = $30
20% is 2 times 10%.
2 times $30 is $60.
20% of $300 = 2 × $30 = $60
5% is half of 10%.
Half of $30 is $15.
5% of $300 = $30 ÷ 2 = $15
25% is 20% plus 5%.
$60 plus $15 is $75.
25% of $300 = $60 + $15 = $75
Up 25% means we add $75.
$300 plus $75 is $375.
new price = $300 + $75 = $375
Careful!
We can't just add the percents.
The second percent is of a new number.
Careful: not −25% + 25% = 0%: $400 ✗
You might think down 25% and up 25% cancel out.
But we did not get back to $400.
The second 25% is of $300, a smaller number.
So it is $75, not $100 like the first 25%.
not back to the start: 25% of $300 = $75, 25% of $400 = $100
So the new price is $375.
3 Check
Let's check another way: down 25% leaves 100% minus 25%.
That is 75% of the price.
75% is 75 out of 100: 0.75.
$400 times 0.75 is $300.
100% − 25% = 75% = 0.75: $400 × 0.75 = $300
Then: up 25% makes it 100% plus 25%.
That is 125% of the price.
125% is 125 out of 100: 1.25.
$300 times 1.25 is $375.
The same.
So the answer is right!
100% + 25% = 125% = 1.25: $300 × 1.25 = $375
Let's look back.
First: down 25% of $400.
Then: up 25% of $300.
Last: check.
Answer not back to the start: 25% of $300 = $75, 25% of $400 = $100
▶ Watch it solved
The question
Question: A $80 coat goes up 25%, then down 20%.
What is the new price?
1 Up 25% of $80
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.
$80 divided by 10 is $8.
10% of $80 = $80 ÷ 10 = $8
10% = ÷ 10
20% is 2 times 10%.
2 times $8 is $16.
20% of $80 = 2 × $8 = $16
5% is half of 10%.
Half of $8 is $4.
5% of $80 = $8 ÷ 2 = $4
25% is 20% plus 5%.
$16 plus $4 is $20.
25% of $80 = $16 + $4 = $20
Up 25% means we add $20.
$80 plus $20 is $100.
new price = $80 + $20 = $100
2 Down 20% of $100
Now the price is $100.
The second 20% is not of the old number.
It is 20% of the new number, $100.
To find 10%, we divide by 10.
$100 divided by 10 is $10.
10% of $100 = $100 ÷ 10 = $10
20% is 2 times 10%.
2 times $10 is $20.
20% of $100 = 2 × $10 = $20
Down 20% means we take away $20.
$100 minus $20 is $80.
new price = $100 − $20 = $80
Careful!
We can't just add the percents.
The second percent is of a new number.
Careful: not +25% − 20% = +5%: $84 ✗
So it is back to $80.
3 Check
Let's check another way: up 25% makes it 100% plus 25%.
That is 125% of the price.
125% is 125 out of 100: 1.25.
$80 times 1.25 is $100.
100% + 25% = 125% = 1.25: $80 × 1.25 = $100
Then: down 20% leaves 100% minus 20%.
That is 80% of the price.
80% is 80 out of 100: 0.8.
$100 times 0.8 is $80.
The same.
So the answer is right!
100% − 20% = 80% = 0.8: $100 × 0.8 = $80
Let's look back.
First: up 25% of $80.
Then: down 20% of $100.
Last: check.
Answer Careful: not +25% − 20% = +5%: $84 ✗
▶ Watch it solved
