PERCENTAGE CHANGE

Percentage Change

Subtract the old from the new, then divide by the old value. Show that as a Percentage.

Comparing Old to New

	Change: subtract old value from new value. Example: You had 5 books, but now have 7. The change is: 7-5 = 2.
	Percentage Change: show that as a percent of the old value ... so divide by the old value and make it a percentage: So the percentage change from 5 to 7 is: 2/5 = 0.4 = 40%

Percentage Change is all about comparing old to new values. See percentage change, difference and error for other options.

How to Calculate

Here are two ways to calculate a percentage change, use whichever method you prefer:

Method 1

Step 1: Calculate the change (subtract old value from the new value)

Step 2: Divide that change by the old value (you will get a decimal number)

Step 3: Convert that to a percentage (by multiplying by 100 and adding a "%" sign)

Note: if the new value is greater then the old value, it is a percentage increase, otherwise it is a decrease.

Method 2

Step 1: Divide the New Value by the Old Value (you will get a decimal number)

Step 2: Convert that to a percentage (by multiplying by 100 and adding a "%" sign)

Step 3: Subtract 100% from that

Note: if the result is positive it is a percentage increase, if negative, just remove the minus sign and call it a decrease.

Examples

Example: A pair of socks went from $5 to $6, what is the percentage change?

Answer (Method 1):

• Step 1: $5 to $6 is a $1 increase
• Step 2: Divide by the old value: $1/$5 = 0.2
• Step 3: Convert 0.2 to percentage: 0.2×100 = 20% rise.

Answer (Method 2):

• Step 1: Divide new value by old value: $6/$5 = 1.2
• Step 2: Convert to percentage: 1.2×100 = 120% (i.e. $6 is 120% of $5)
• Step 3: Subtract 100%: 120% - 100% = 20%, and that means a 20% rise.

Another Example: There were 160 smarties in the box yesterday, but now there are 116, what is the percentage change?

		Answer (Method 1): 160 to 116 is a decrease of 44. Compared to yesterday's value: 44/160 = 0.275 = 27.5% decrease.
		Answer (Method 2): Compare today's value with yesterday's value: 116/160 = 0.725 = 72.5%, so the new value is 72.5% of the old value. Subtract 100% and you get -27.5%, or a 27.5% decrease.

Why Compare to Old Value?

Because you are saying how much a value has changed.

Example: Milk was $2, now it is $3 ... did it rise $1 compared to $2 or $3 ?

We compare to the original $2 value, so we say the change is $1/$2 = 0.5 which is a50% increase.

The Formula

You can also put the values into this formula:

New Value - Old Value	× 100%
|Old Value|

(The "|" symbols mean absolute value, so negatives become positive)

Example: There were 200 customers yesterday, and 240 today:

240 - 200	× 100% = (40/200) × 100% = 20%
|200|

A 20% increase.

Example: But if there were 240 customers yesterday, and 200 today we would get:

200 - 240	× 100% = (-40/240) × 100% = -16.6...%
|240|

A 16.6...% decrease.

How to Reverse a Rise or Fall

Some people think that a percentage increase can be "reversed" by the same percentage decrease.But no!

Example: 10% of 100

A 10% increase from 100 is an increase of 10, which equals 110 ...

... but a 10% reduction from 110 is a reduction of 11 (10% of 110 is 11)

So we ended up at 99 (not the 100 we started with)

What happened? 10% took us up 10 Then 10% took us down 11 Because the percentage rise or fall is in relation to the old value: The 10% increase was applied to 100. But the 10% decrease was applied to 110.

How to do it properly

To "reverse" a percentage rise or fall, use the right formula here:

To Reverse:	Use this Percent:	Example 10%
An "x" percent rise:	x/(1+x/100)	10/(1+10/100) = 10/(1.1) = 9.0909...
An "x" percent fall:	x/(1-x/100)	10/(1-10/100) = 10/(0.9) = 11.111...

Percentage Difference,
Percentage Error,
Percentage Change

They are very similar ...

They all show a difference between two values as a percentage of one (or both) values

Use Percentage Change when comparing an Old Value to a New Value
Use Percentage Error when comparing an Approximate Value to an Exact Value
Use Percentage Difference when both values mean the same kind of thing (one value is not obviously older or better than the other).