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Successive Percentage Change

Suppose we have to increase a quantity successively by 20% and 30%. Ideally this should be done by taking 100 as the initial value and then changing this initial value first by 20% and then by 30%. It can be seen below:
100  20%  120  30%↑→ 156
So, net percentage increase = 56%

This is known as the straight line method of solving the problems.


Alternatively, we can use successive change formula also: Description: 2973.png
 
Using this formula, 20 + 30 + Description: 2982.png = 56%

This kind of calculation is very much important in DI questions, especially in pie chart which can be seen with the help of following example.

The following figure shows the sales figure of Due North Inc. for the year 2002-03 and 03-04. It is also given that the share of north India in total sales figure is 20% and 24% in respective years.
 
Description: 2-1.tif
2002-03
Description: 2-2.tif
2003-04
 
Total sales for the year 2002-03 is Rs 1202 crores and for the year 2003-04, it is Rs 1381 crores.
 
Now the question is: What is the percentage growth in the sales value of north India in 2003-04 over the sales value of 2002-03?
 
Normal way of doing this question is:
Sales value of north India in 2002-03 = Rs 240.4 crores
Sales value of north India in 2003-04 = Rs 331.4 crores
So, growth = Rs 91 crore
So, the percentage growth =Description: 2991.png

Alternatively, percentage increases in two factors in the given pie-charts viz., share is increasing from 20% to 24% and then total sales value is increasing from Rs 1202 crores to Rs 1381 crores.

Rather than calculating the sales value of individual years, we can directly calculate the net percentage change by calculating percentage changes independently in the above-mentioned two factors.

Percentage growth in share = 20%
Description: 3000.png 

Percentage growth in total value  Approx. 15%
So, 100  20%  120  15%↑→ 138, or
100  15%↑→ 115  20%↑→ 138
So, net percentage growth = 38%
 
Example
The price of petrol is increased by 20%. But expenses increase only by 10%. What is the percentage increase or decrease in consumption?
Solution
If consumption remains constant, then the expenses should have also increased by 20%. But since expenses increase by only 10%, so consumption has been reduced.
 

These kinds of problems can be done in three ways:
  1. Straight line method
     
    100 (Initial expenditure) 20% in price120 x% in consumption110 (Final expenditure), x% = Description: 3009.png
  2. 10 × 10 method
     
    We know, Price × Consumption = Expenditure
     
    Assume Initial price = Initial consumption = Rs 10
     
    So, Initial expenditure = Rs 100
     
        Price   Consumption   Expenditure
    Initially… 10 × 10 = 100
    Finally… 12 × C = 110 
     
    So, C = 110/12 = Rs 9.16
    So, percentage reduction = 8.33%
  3. Formula method
     
    Here a = 20%, b is not known and final percentage change = 10%
     
    20 + b + 20b/100 = 10
     
     20 + b + b/5 = 10
     
     b = 8.33%
     
    So, there is a decrease of 8.33% in the consumption.




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