PERCENTAGES PART 2

6.SUCCESSIVE  PERCENTAGE CHAINGE

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)%
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:
Answers:
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%



7.Product Constancy Ratio

In this article, we deal with topic of Product Constancy Ratio. This is essentially an application of percentages and can be quite useful while solving problems.
What is the Product Constancy Ratio?
In this concept, we essentially refer to the practice wherein two or more quantities make up a third quantity. With the variation in the numbers of one quantity, the other quantities need to undergo change in order to maintain the same product.
Let’s take an example.
Let there be two quantities A and B that multiply to form a quantity P. We can say:
A × B = P
Now if A is increased by a certain percentage, then B is required to be decreased by a certain percentage to keep the product P stable.
The following table illustrates the varying values of A and B that will maintain the same product P.
 
Application of Product Constancy: Expenditure Questions
If the price of a commodity increases or decreases by a%, then, the percentage decrease or increase in consumption, so as not to increase or decrease the expenditure is equal to:
(a/100+a) x 100%
Example: Length of a rectangle is increased by 33.33%. By what percentage should the breadth be decreased so that area remains constant?
Solution: Using the table above:
Since length is increased by 33.33%, the breadth will decrease by 25% to keep area constant.
Let’s make these calculations also.
Let original length= L
Original Breadth= B
Increase length= 4/3 L
Since the area remains same, we can say
L x B = Increased Breadth x 4/3L
Therefore,
Increased breadth = ¾ Original Breadth = 25% reduction in breadth.
Example: When speed of a car is increased by 25%, time taken reduces by 40 minutes in covering a certain distance.What is the actual time taken to cover the same distance by actual speed?
Solution: We have Speed × Time = Distance
Since speed has been increased by 25%, time will reduce by 20%.
Now, 20% of actual time = 40 min
Actual time = 200 min

8.Compound Growths.
Typically compound growths are used in investment growth analysis (compound interest) or in population growth (things like cattle population, steel production output growth). In this section, we will be primarily concerned with compound growth related to population.

If P is the population of a country and it grows at r % per annum, then the population after n years will be:
A = P [(100+r) / 100]n