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# Percentage Difference

A Percentage Difference is a difference shown as a percentage.

## Two Meanings!

"Percentage Difference" can have two meanings, but if you are comparing an old value to a new value people commonly use this method:

## Comparing Old to New

 Difference means to subtract one value from another. For example the difference from 5 to 3 is: 5-3 = 2. Percentage Difference means to show that difference as a percent of the old value ... so divide by the old value and make it a percentage: So the percentage difference from 5 to 3 is: 2/5 = 0.4 = 40%

This is also called "Percentage Change"

## How to Calculate

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

### Method 1

 Step 1: Calculate the difference (subtract one value form the other) Step 2: Divide that Difference 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 cal it a decrease.

## Examples

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

• 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.

• Step 1: Divide new value by old value: \$6/\$5 = 1.2
• Step 2: Convert to percentage: 1.2×100 = 120% (ie \$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 difference?

 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.

## The Formula

You could also just 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.

## How to Reverse a Rise or Fall

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

 For example, 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), which equals 99 (not the 100 we started with) 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.

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...

 Or use this handy-dandy calculator (just type in a value and click in the other box) Percent Rise: <=> Percent Fall: