Percentages

Core formulas:

x% = x/100
x% of y = y% of x
% change = change / original * 100
New after x% increase = old * (100 + x) / 100
New after x% decrease = old * (100 - x) / 100

Successive changes:

Net change for +a% and +b% = a + b + ab/100
Net change for +a% and -b% = a - b - ab/100

Fast examples:

20% increase then 20% decrease:
20 - 20 - 400/100 = -4%

10% decrease then 20% increase:
-10 + 20 - 200/100 = +8%

Reverse percentage:

If final = old after x% increase:
old = final * 100 / (100 + x)

If final = old after x% decrease:
old = final * 100 / (100 - x)

Speed tricks:

Use x% of y = y% of x: 18% of 50 = 50% of 18 = 9.
For close comparison, use percent difference on the required base.
If value becomes m/n of original, percentage change is (m - n) / n * 100.
If A is x% more than B, then B is x/(100 + x) * 100% less than A.
If A is x% less than B, then B is x/(100 - x) * 100% more than A.

Common traps:

Percentage change is not symmetric: 50 to 100 is +100%, but 100 to 50 is -50%.
"Percent more than" and "percent of" use different bases.
For population/salary/price, identify the original base before calculating.

Percentages​

Percentages