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Percent of Change (6-6)

Percent of Change

Vocabulary

Percent of Change- the amount, stated as a percent, that a number increases or decreases.

Percent of Increase- if the amount goes up.

Percent of Decrease- if the amount goes down.

Examples

To calculate the percentage increase:

First: work out the difference (increase) between the two numbers you are comparing.
Increase = New number - Original Number
Then:  divide the increase by the original number and multiply the answer by 100.
% increase = Increase ÷ Original Number × 100.
If your answer is a negative number then this is a percentage decrease.

To calculate percentage decrease:

First: work out the difference (decrease) between the two numbers you are comparing.
Decrease = Original Number - New Number
Then: divide the decrease by the original number and multiply the answer by 100.
% Decrease = Decrease ÷ Original Number × 100
If your answer is a negative number then this is a percentage increase.
If you wish to calculate the percentage increase or decrease of several numbers then we recommend using the first formula.  Positive values indicate a percentage increase whereas negative values indicate percentage decrease.

Examples - Percentage Increase and Decrease

In January Dylan worked a total of 35 hours, in February he worked 45.5 hours – by what percentage did Dylan’s working hours increase in February?
To tackle this problem first we calculate the difference in hours between the new and old numbers.  45.5 - 35 hours = 10.5 hours.  We can see that Dylan worked 10.5 hours more in February than he did in January – this is his increase.  To work out the increase as a percentage it is now necessary to divide the increase by the original (January) number:
10.5 ÷ 35 = 0.3  
Finally, to get the percentage we multiply the answer by 100.  This simply means moving the decimal place two columns to the right.
0.3 × 100 = 30

Dylan therefore worked 30% more hours in February than he did in January.
In March Dylan worked 35 hours again – the same as he did in January (or 100% of his January hours).  What is the percentage difference between Dylan’s February hours (45.5) and his March hours (35)?  You may think that as there was a 30% increase between Dylan’s January hours (35) and February (45.5) hours that there will be a 30% decrease between his February and March hours.  This assumption is incorrect – let’s calculate the difference.
First calculate the decrease in hours, that is: 45.5 - 35 = 10.5
Then divide the decrease by the original number (February hours) so:
10.5 ÷ 45.5 = 0.23 (to two decimal places).
Finally multiply 0.23 by 100 to give 23%.

 Dylan’s hours were 23% lower in March than in February.
Sometimes it is easier to show percentage decrease as a negative number – to do this follow the formula above to calculate percentage increase – your answer will be a negative number if there was a decrease.  In Dylan’s case the decrease works out at -15.5.  -10.5 ÷ 45.5 = -0.23.  -0.23 × 100 = -23%.
Dylan's hours could be displayed in a data table as:
Month Hours
Worked		 Percentage
Change
January 35
February 45.5 30%
March 35 -23%

Calculating Values Based on Percentage Change

Sometimes it is useful to be able to calculate actual values based on the percentage increase or decrease.  It is common to see examples of when this could be useful in the media.
You may see headlines like:
UK rainfall was 23% above average this summer.
Unemployment figures show a 2% decline.
Banker’s bonuses slashed by 45%.
Such headlines give an idea of a trend – where something is increasing or decreasing, but often no actual data.
Without data percentage change figures can be misleading.
Ceredigion, a county in West Wales has a very low violent crime rate.
Police reports for Ceredigion in 2011 showed a 100% increase in violent crime. This is a startling number, especially for those living or thinking about moving to Ceredigion.
However, when the underlying data is examined it shows that in 2010 one violent crime was reported in Ceredigion. So an increase of 100% in 2011 meant that two violent crimes were reported.
When faced with the actual figures, perception of the amount of violent crime in Ceredigion changes significantly.

In order to work out how much something has increased or decreased in real terms we need some actual data.
Take the example of “UK rainfall this summer was 23% above average” – we can tell immediately that the UK experienced almost a quarter (25%) more rainfall than average over the summer. However, without knowing either what the average rainfall is or how much rain fell over the period in question we cannot work out how much rain actually fell.
Calculating the actual rainfall for the period if the average rainfall is known.
If we know the average rainfall is 250mm then we can work out the rainfall for the period using:  the sum 250 + 23%.
First work out 1% of 250, 250 ÷ 100 = 2.5 then multiply the answer by 23, because there was a 23% increase in rainfall.
2.5 × 23 = 57.5.

Rainfall for the period in question was therefore 250 + 57.5 = 307.5mm.
Calculating the average rainfall if the actual amount is known.
If the news report states the new measurement and a percentage increase, “UK rainfall was 23% above average... 320mm of rain fell…”.
In this example we know the total rainfall amount of 320mm, we also know that this is 23% above the average.  To calculate the average we divide the total (320) by 1.23.
320 ÷ 1.23 = 260.1626.  Rounded to one decimal place, the average rainfall is 260.2.
The difference between the average and the actual rainfall can now be calculated:
320 - 260.2 = 59.8.
We can conclude that 59.8 is 23% of the average rainfall amount (260.2), and that in real terms 59.8mm more rain than average fell.
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