In this article, we deal with the concept of successive percentage change. This is a problem type in Percentages and using the formula in this article, you can easily solve questions based on this concept in matter of seconds.

What is Successive Percentage Change?

The concept of successive percentage change deals with two or more percentage changes applied to quantity consecutively. In this case, the final change is not the simple addition of the two percentage changes (as the base changes after the first change).

Formula for Percentage Change:

Suppose a number N undergoes a percentage change of x % and then y%, the net change is:
New number = N × (1 + x/100) × (1 + y/100)
Now, (1 + x/100) × (1 + y/100) = 1 + x/100 + y/100 + xy/10000
If we say that x + y + xy/100 = z, then (1 + x/100) × (1 + y/100) = 1 + z/100
Here, z is the effective percentage change when a number is changed successively by two percentage changes.

Various cases for Percentage Change:
Both percentage changes are positive:
x and y are positive and net increase = (x+y+xy/100) %.

One percentage change is positive and the other is negative:
x is positive and y is negative, then net percentage change = (x-y-xy/100)%

Both percentage changes are negative:
x and y both are negative and imply a clear decrease= (-x-y+xy/100)%
Percentage Change involving three changes:
If value of an object/number P is successively changed by x%, y% and then z%, then final value. Example 1: The capacity of a ground was 100000 at the end of 2012. In 2013, it increased by 10% and in 2014, it decreased by 18.18%. What was the ground’s capacity at the end of 2014?

Solution:
When One percentage change is positive and the other is negative:
x is positive and y is negative, then net percentage change = (x-y-xy/100)%
Final Percentage Change over the original value = 10-18.18 – (10 × 18.18/100)= -9.998
(the difference above is cause by using exact values).
So the capacity of the ground is decreased by 9.998%
Hence, net capacity = 90002

Example 2: A’s salary is increased by 10% and then decreased by 10%. The change in salary is
Solution:
Percentage change formula when x is positive and y is negative = {x – y – (xy/100)}%
Here, x = 10, y = 10
= {10 – 10 – (10 x 10)/100} = -1%
As negative sign shows a decrease, hence the final salary is decreased by 1%.

Example 3: A number is first increased by 10% and then it is further increased by 20%. The original
number is increased altogether by:
Percentage change formula when both x and y are positive ={x + y + (xy/100)}%
Here, x = 10 and y = 20
Hence net percentage change == {10 + 20 + (10 x 20)/100} = 32%

### Percentages: The Complete Lesson

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