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Percentage change at a glance
The use of I, (1+I), (1-I) can be summarised in a table:
|Increase by I%
|Decrease by I%
Using these formulas you should now find it very easy to work out the effect of a sales tax or value added tax, VAT, on nett price. If the current VAT rate is V% then the gross price can be calculated as:
gross price = Nett*(1+V)
Similarly if the discount on a price is D% then the discounted price is given by
discounted price = price*(1-D)
What percentage change?
Another interesting question is how much percentage increase or decrease has occurred when a quantity changes.
For example, if the price of a product has increased from $2.50 to $3.00 what is the percentage increase. To work this out all you need to do is solve:
new value = old value *(1+I)
for I. That is:
new value = old value +old value * I
new value - old value = old value *I
(new value -old value)/old value = I
I= (new value - old value)/old value
or if you want a more compact version:
I= new value/old value -1
The only change needed for a percentage decrease is to put a minus sign in front of the expression:
I= 1- new value/old value
In the case of the $2.50 to $3.00 increase the percentage increase is 20% i.e. 2.5/3.0-1.0=0.2..
Nett from gross
Now to the question of undoing the effect of increasing or decreasing a quantity by a percentage.
If you know the Gross price including tax, VAT say, and want to know the Nett price without it, how do you work it out?
An attractive, but incorrect, line of reasoning is to say that as the Nett price (i.e. without tax) was increased by V% to give the Gross price (i.e. with tax) then the Gross price should be decreased by V% to get back to Nett. So Nett=Gross*(1-V).
This is misguided because the amount add to Nett, i.e. Nett*V, isn’t the same as the amount subtracted from Gross i.e. Gross*V. In general if you increase something by I% and then decrease the result by I% you do not get back to where you started!
Figure 3 - Increasing 200 by 50% and then decreasing it by 50% gets you back to 150
If you would like a demonstration that this method is incorrect then create the small spreadsheet shown in Figure 3. The value entered into A2 is increased by the percentage in B1 and then decreased by the same percentage. You can see in the figure that in the case of 200 and 50% the result is 150 which is very clearly not 200!
If you experiment with this spreadsheet what you will find is that for small values and small percentages the difference between the original value and the calculated value can be quite small and this sometimes leads people to believe that the answer is in fact correct except for a small arithmetic error. This is of course not true but on occasion the approximation can be useful.
The correct answer depends on a simple rearrangement of the formula
to give the Nett in terms of the Gross:
That is, if you multiply by (1+V) to get the Gross you have to divide by (1+V) to get back to the Nett.
- if you know that a value has been increased by I% and you want to calculate the original value divide by (1+I)
- if you know that a value has been decreased by I% and you want to calculate the original value divide by (1-I)