# How to Calculate Percent Change

Three Parts:Sample Percent Change CalculatorCalculating Percent Change in General CasesFinding Percent Changes in Special Cases

In mathematics, the concept of percent change is used to describe the relationship between an old value or quantity and a new value or quantity. Specifically, the percent change expresses the difference of the old and new values as a percentage of the old value. In general cases where *V _{1}* represents an old or initial value and

*V*represents the new or current value, percent change can be found with the equation

_{2}**((**. Note that this quantity is expressed as a

*V*-_{2}*V*)/_{1}*V*) × 100_{1}**percentage**. See Step 1 below for a breakdown of this process.

## Steps

### Sample Percent Change Calculator

Percent Change Calculator

### Part 1 Calculating Percent Change in General Cases

- 1
**Find old and new values for a certain variable.**As noted in the intro, the purpose of most percent change calculations is to determine the*change*in a variable over time. Thus, to make a percent change calculation for a certain variable, you'll need to start with two pieces of information - an old (or "starting") value and a new (or "ending") value. The percent change equation finds the percent change*between these two points*.- One example of this is in the world of retail. When a product receives a discount, this is often expressed as the product being "
*x*% off" - in other words, as the percent change from the old price. Let's consider a pair of pants that were previously sold for $50 and are now sold for $30. In this example,**$50**is our "old" value and**$30**is our "new" value. In the next few steps, we'll find the percent change between these two prices.

- One example of this is in the world of retail. When a product receives a discount, this is often expressed as the product being "
- 2
**Subtract the old value from the new value.**The first step in finding the percent change between two values is to find the*difference*between them. The difference between two numbers is found by subtracting one from the other. The reason that we subtract the old from the new (and not vice versa) is that this conveniently gives us a negative percentage as our final product if the variables value decreases and a positive value if it increases.- For instance, in our example, we would start with $30, our new value, and subtract $50, our old value. 30 - 50 =
**-$20**.

- For instance, in our example, we would start with $30, our new value, and subtract $50, our old value. 30 - 50 =
- 3
**Divide your answer by your old value.**Next, take the answer you obtained by subtracting your old value from your new value and divide it by your old value. This gives you the proportional relationship of the change in values to the old starting value, expressed as a decimal. In other words, this represents the total change in value over the starting value for your variable.- In our example, dividing the difference between our new and old values (-$20) by our old value ($50) gets us -20/50 =
**-0.40**. Another way of thinking of this is that the $20 change in value is 0.40 of the $50 starting point, and that the change of value was in the negative direction.

- In our example, dividing the difference between our new and old values (-$20) by our old value ($50) gets us -20/50 =
- 4
**Multiply your answer by 100 to get a percentage.**Values for percent change are (appropriately) expressed as percentages, rather than decimals. To convert your decimal answer to a percentage, multiply it by 100. After this, all that's left to do is to add a percentage sign. Congratulations! This final number represents the percent change between your old and new values.- In our example, to get our final percentage, we would multiply our decimal answer (-0.40) by 100. -0.40 × 100 =
**-40%**. This answer means that the new price of $30 for the pants is*40% less than*the old price of $50. In other words, the pants are "40% off". Another way to think of this is that the $20 difference in price is 40% of the initial price of $50 - since this price difference results in a*lower*final price, we give it a negative sign. - Note that a positive answer for your final percentage implies an increase in the value of your variable. For instance, if the final answer for our example problem was not -40% but instead 40%, this would mean that the new price of the pants was $70 - 40%
*more*than the initial price of $50.

- In our example, to get our final percentage, we would multiply our decimal answer (-0.40) by 100. -0.40 × 100 =

### Part 2 Finding Percent Changes in Special Cases

- 1
**When dealing with variables with more than one change in value, find the percent change only for the two values you wish to compare.**Finding the percent change for a certain variable that changes value more than once over time can seem intimidating, but don't let the multiple value changes lead you to over-complicate things. The percent change equation only compares*two values at a time*. This means that if you're asked to calculate percent change in a situation involving a variable with multiple value changes, only calculate the percent change between the two values specified.*Don't*calculate the percent changes between every value in the sequence, then average or add up the percent changes. This is not the same as finding the percent change between two points and can easily give nonsensical answers.- For example, let's say that a pair of pants has an initial price of $50, is discounted to $30, is then marked up to $40, and is finally discounted again to $20. The percent change formula can find us the percent change between any two of these values; the two leftover values are not needed. For instance, to find the percent change between the initial price and the final price, use $50 and $20 as your "old" and "new" values, respectively. Solve as follows:
- ((
*V*-_{2}*V*)/_{1}*V*) × 100_{1} - ((20 - 50)/50) × 100
- (-30/50) × 100
- -0.60 × 100 =
**-60%**

- ((

- For example, let's say that a pair of pants has an initial price of $50, is discounted to $30, is then marked up to $40, and is finally discounted again to $20. The percent change formula can find us the percent change between any two of these values; the two leftover values are not needed. For instance, to find the percent change between the initial price and the final price, use $50 and $20 as your "old" and "new" values, respectively. Solve as follows:
- 2
**Divide the new value by the old value and multiply by 100 to find the absolute relationship between the two values.**A process that is similar (but not identical) to the process used to find percent change is that used to find the absolute percentage relationship between the "old" and "new" values. To do this, simply divide the old value by the new value and multiply by 100 - this gives you a percentage that directly compares the new value to the old, rather than expressing the change between the two.- Note that subtracting %100 from this answer will give the percent change.
- For example, let's use this process for the discounted pants example above. If the pants started at $50 and ended at $20, dividing 20/50 and multiplying by 100 gives us 20/50 × 100 =
**40%**. This tells us that $20 is 40% of $50. Note that subtracting by 100% gives us 40 - 100 = -60%, the percent change we found above. - This process can give answers over 100% For instance, if $50 is our old price and
*$75*is our new price, 75/50 ×100 =**150%**. This means that 75$ is 150% of 50$.

- 3
**Generally, use**The language surrounding the process of calculating percent change can get somewhat tricky when the two values being compared are themselves percentages. In these cases, it's important to distinguish between the percent change and the*absolute change*when dealing with two percentages.*absolute change*. The latter is the exact number of percentage points by which the new value differs from the old value -*not*the familiar percent change concept we've covered.- For example, let's say that a pair of pants is offered at 30% off (a percent change of -30% from their old price). If the discount is increased to 40% off (a percent change of -40% from their old price), it's not inaccurate to say that the percent change of this discount is ((-40 - -30)/-30) × 100 =
**33.33%**. In other words, the pants are 33.33% "more" discounted than before. -
**However,**this is usually instead stated as the discount having a**"10 percent increase"**. In other words, we usually refer to the*absolute change*between the two percentages rather than the percent change.

- For example, let's say that a pair of pants is offered at 30% off (a percent change of -30% from their old price). If the discount is increased to 40% off (a percent change of -40% from their old price), it's not inaccurate to say that the percent change of this discount is ((-40 - -30)/-30) × 100 =

## Tips

- If the normal price of an item is $50.00 and you bought it on sale for $30.00 then the percentage change is:
- ($50.00 - $30.00)/$50.00 × 100 = 20/50 × 100 = 40%

The price you bought it for was less than the original price so this is a percentage decrease of 40 percent. Hence you saved 40% of the original price.

- ($50.00 - $30.00)/$50.00 × 100 = 20/50 × 100 = 40%
- Now let's consider that you would like to sell the pants you just purchased. For example if you purchased the pants for $30 and later sold them for $50 then the change is $50 - $30 = $20. The starting value was $30 so the percentage change is:
- ($50.00 - $30.00)/$30.00 × 100 = 20/30 × 100 = 66.7%

Thus the value of the pants increased by 66.7% of the original price, a 66.7% increase.

- ($50.00 - $30.00)/$30.00 × 100 = 20/30 × 100 = 66.7%
- When the pants changed in value from $50 down to $30 they depreciated in value by 40%. When the pants changed in value again from $30 back up to $50 they appreciated in value by 66.7%. However, it's important to note that the
*profit percentage*when the pants were sold for $50 was still only 40% because it is based on the $20 increase unlike the appreciation value.

## Tips 2

- ($50.00 - $30.00)/$50.00 × 100 = 20/50 × 100 = 40%

## Sources and Citations

- Math Goodies -- Percent Change
- Wikipedia -- Relative change and difference
- Khan Academy -- Growing by a percentage
- Clemson Phoenix Tutorials -- Physics Tutorial: % Error & % Difference

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