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Negative percent

A percentage calculation can easily result in a negative percentage instead of a positive percentage. You can also calculate a percentage from a negative number. The calculation gets a bit strange when you calculate the percentage change using a negative number, but that's not impossible either.

Let's start with a simple percentage calculation: How much is 50% of 60? The answer is 30 and it is calculated as follows: 50% * 60 = 0.5 * 60 = 30.

You can also calculate a percentage from a negative number

Next, let's use negative numbers and ask: How much is 50% of -60? The same formula shows: 50% * (-60) = 0.5 * (-60) = -30. The result of the percentage calculation is a negative number.

Percentage can also be negative

Percentage can also be negative. We can change the calculation described above and ask: How much is -50% of 60? The answer is (-50)% * 60 = -30.

What about how much is -50% of -60? The result is (-50)% * (-60) = (-0.5) * (-60) = 30. Here, as the product of two negative numbers, the answer is positive.

The following is a general formula for the calculation: how much is A% of B

X = A% * B

and since % is a hundredth, it can be written as

X = A * 0.01 * B

This actually shows that negative percentages can be used in calculations just like regular numbers.

Negative change

Most commonly, negative percentages are encountered when calculating the percentage change when the change is negative. For example, let's calculate the percentage change from 80 to 60.

The general formula for the percentage change (X) for numbers A and B is:

X = (B-A)/|A|

And this gives the change as (60-80)/80 = -20/80 = -25%.

The change is therefore negative.

Percentage change from a negative number

However, the percentage change can also be calculated using a negative number, and understanding it requires a bit more conceptual acrobatics.

Let's calculate the percentage change from -80 to 60. Using the formula described above, we get the answer as (60-(-80))/|-80| = (60+80)/80 = 140/80 = 1.75 = 175%.

So 60 is 175% greater than -80.

Of course, we can question such calculations. Is it even meaningful to go beyond the zero point in percentage calculations? After all, using zero may also make the calculation impossible, since you cannot divide by zero.

For example, we cannot calculate the percentage change from 0 to 60. The calculation would go as follows: (60-0)/0 = 60/0 and division by zero is not possible.

Instead, we can calculate the percentage change from -80 to 0, and the result is (0-(-80))/|-80| = (0+80)/80 = 80/80 = 1 = 100%.

In fact, the percentage change from any negative number to zero is always 100%.

This also brings us to our general formula

X = (B-A)/|A|,

where we have used the absolute value of A as the divisor. Of course, A could also be used in the formula, and we see a lot of formulas like this used. However, using only A would turn positive percentage changes into negative ones and negative percentage changes into positive ones if A is negative. So a change from -80 to 0 would be -100%, a change from -80 to -40 would be -50%, and our original calculation, i.e. a change from -80 to 60, would be -175%.

If the number goes from -80 to 60, then of course we would like to interpret the change as positive.

Back and forth over zero

When we go over zero and calculate percentage changes, i.e. from a negative number to a positive one or from a positive number to a negative one, the result is always either over 100% or under -100%. When a negative number increases to positive, the change is over 100%. When a positive number decreases to negative, the change is under -100%.

You can also visualize such a change by thinking that first we calculate the change from a negative number to zero. From that, we get either 100% or -100% as the result, depending on which direction we go. The rest of the change from zero to a positive number is obtained by how long the distance is from zero to a positive number compared to how long the distance is from zero to a negative number.

This kind of conceptual acrobatics makes it possible to calculate, for example, the percentage change in a company's profit if the result had first been negative and then positive.

Author:

Arkikoodi

Sources and additional information:

Wikipedia: Relative change

Furey, Edward Percentage Change Calculator / CalculatorSoup

Published: 25.11.2024

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Percentage and percentage point
A percentage means a hundredth and they are used to measure a share of something. The percentage point, on the other hand, is used when comparing percentages to each other or when referring to percentages of certain percentages.