Percentage: Successive Percentage Increase/Decrease

 In this article we will learn how to deal with successive percentage increase/ Decrease.


Problem based on successive percentage increase/ Decrease are very frequent in competitive examinations.

Understanding Successive percent increase/decrease is very important. Since successive percentage increase/decrease is used while solving problems related to Profit, Loss, Discount & Compound Interest problems.

Let us understand first concept of Multiplying Factor (M.F.)


Multiplying Factor (M.F.)                                                  

Case 1:  When a quantity is increased by certain percentage.

A) Suppose we have to increase a value 120 by 10%. What will be the final value?

Final Value= Initial Value + 10% of Initial Value
                  = 120 + 10% of 120

                  =120 (1+ 10%)

                  =120(1+10/100)

                  =120(1.1)

                  =120 x 1.1=132

So now we can say for 10% increase if we multiply initial value by 1.1 we will get our final value.

So, here we call 1.1 as Multiplying Factor (M.F.)

Note: For 10% increase Multiplying Factor =1.1

 B) Suppose we have to increase a value 120 by 20%. What will be the final value?

Final Value= Initial Value + 20% of Initial Value

                  = 120 + 20% of 120

                  =120 (1+ 20%)

                  =120(1+20/100)

                  =120(1+0.2)

                  =120 x 1.2=144

So now we can say for 20% increase if we multiply initial value by 1.2 we will get our final value.

So, here we call 1.2 as Multiplying Factor (M.F.)

Note: For 20% increase Multiplying Factor =1.2

Let us generalize the above case.

So, if we increase our initial value by x%. What will be our final value?

Final Value= Initial Value + x% of Initial Value

                  = Initial Value (1+ x %)

                  = Initial Value (1+ x/100)

So, if we increase a value by x%.

To get Final Value we need to Multiply Initial value by Multiplying Factor .

Multiplying Factor= 1+x/100.

Case 2:  When a quantity is decreased by certain percentage.       

A) Suppose we decrease 120 by 10%. What will be the final value?

Final Value= Initial Value - 10% of Initial Value

                  = 120 - 10% of 120

                  =120 (1 - 10%)

                  =120(1-10/100)

                  =120(1- 0.1)

                  =120 x 0.9=108

So now we can say for 10% decrease if we multiply initial value by 0.9, we will get our final value.

So, here we call 0.9 as Multiplying Factor (M.F.)

Note: For 10% decrease Multiplying Factor = 0.9

 B) Suppose we have to decrease a value 120 by 20%. What will be the final value?

Final Value= Initial Value - 20% of Initial Value

                  = 120 - 20% of 120

                  =120 (1 - 20%)

                  =120(1-20/100)

                      =120(1-0.2)

                  =120 x 0.8=96

So now we can say for 20% decrease if we multiply initial value by 0.8 we will get our final value.

So, here we call 0.8 as Multiplying Factor (M.F.)

Note: For 20% decrease Multiplying Factor =0.8

Let us generalize the above case.

So, if we decrease our initial value by x%. What will be our final value?

Final Value= Initial Value - x% of Initial Value

                  = Initial Value (1- x %)

                  = Initial Value (1 - x/100)

So, if we decrease a value by x%.

To get Final Value we need to Multiply Initial value by Multiplying Factor .

Multiplying Factor= 1-x/100.

Summary:

Case 1: for x% increase Multiplying Factor= 1+ x/100

Case 2: for x% decrease Multiplying Factor= 1- x/100

Let us make a table for Case 1 & 2

If initial Value is increased by x%: 

X% increase
M.F. for X% INCREASE= 1+x/100
Final Value= Initial Value x Multiplying Factor
5%
1+5/100       = 1.05
Initial Value x 1.05
9%
1+9/100       = 1.09
Initial Value x  1.09
10%
1+10/100     = 1.1
Initial Value x  1.1
15%
1+15/100     = 1.15
Initial Value x  1.15
20%
1+20/100     = 1.20
Initial Value x  1.20
25%
1+25/100     = 1.25
Initial Value x  1.25
30%
1+30/100     = 1.30
Initial Value x  1.30
50%
1+50/100     = 1.50
Initial Value x   1.50
75%
1+75/100     = 1.75
Initial Value x  1.75
95%
1+95/100     = 1.95
Initial Value x  1.95
100%
1+100/100   = 2
Initial Value x  2
150%
1+ 150/100  = 2.5
Initial Value x  2.5
200%
1+200/200   = 3
Initial Value x 3

If initial Value is decreased by x%:

X% decrease
M.F. for X% decrease= 1-x/100
Final Value= Initial Value x Multiplying Factor
5%
1 - 5/100      = 0.95
Initial Value x  0.95
9%
1 - 9/100      = 0.91
Initial Value x  0.91
10%
1 - 10/100    = 0.9
Initial Value x  0.9
15%
1 - 15/100    = 0.85
Initial Value x  0.85
20%
1 - 20/100    = 0.80
Initial Value x  0.80
25%
1 - 25/100    = 0.75
Initial Value x  0.75
30%
1 - 30/100    = 0.70
Initial Value x  0.70
50%
1 - 50/100    = 0.50
Initial Value x   0.50
75%
1 - 75/100    = 0.25
Initial Value x  0.25
95%
1 - 95/100    = 0.05
Initial Value x  0.05
100%
1 - 100/100  = 0
Initial Value x  0
150%
1 - 150/100  = -0.5
Initial Value x  -0.5
200%
1 - 200/200  = -1
Initial Value x -1

Note: So, from now onwards your thought process should be

10% increase ==> multiply by 1.1

10% decrease==> multiply by 0.9

5% increase ==> multiply by 1.05

5% decrease ==> multiply by 0.95


So, thinking in terms of multiplying factor saves time during calculations.

Let us take an example:
Question: Population of a city increases by 10% in 1st year, it increases by 20% in next year. If initial population in 10,000. Calculate the Population after two years.

A] 11900          B] 13000       c] 13200           D] None

Solution:

Initial Population=10000.

Multiplying Factor for 1st Year = 1.1 [since 10% increase]

Population after one year = 10000 x Multiplying Factor

                                            = 10000 x 1.1 = 11000

Multiplying Factor for second Year = 1.2 [since 20% increase]

Population after 2nd Year = 11000 x 1.2

                                            = 13200.

Hence [C] option is correct.

Though Process during Exam





 (OR)


Final Population = Initial Population x M.F. of first year x M.F. of 2nd Year

                           = 10000 x 1.1 x 1.2

                           = 13200

Note: In successive % increase/Decrease individual multiplying factors gets multiplied.

Question: Population of a city increases by 10% in 1st year, it increases by 20% in next year. Calculate the equivalent net % increase?

A] 28%          B] 30%            C] 32%            D] NONE

Final Population= Initial Population x M.F of 1st Year x M.F. of 2nd Year

                             = Initial Population x 1.1 x 1.2

 Final Population= Initial Population x 1.32

 Final Population = Initial Population x M.F.

So here net M.F. =1.32 ==> which means 32% increase

Hence option C is correct

Note:  To calculate equivalent or net % increase or decrease you need to multiply individual Multiplying factor.

Summary:  From now onwards, if you see 10% increase in question, it should automatically come in your mind that it means multiplication by 1.1, if you see decrease of 15% it means multiplication by 0.85 and so on…….


Question 1:  If population increases by 20% in first year and decreases by 30% in second year. Calculate net percentage change?


A] 16% increase B] 16% decrease C] 10% decrease D] NONE

Solution:
Let us take initial population P.

Population is first increased by 20% ===> Multiplying Factor = 1.2

so population at the end of 1 year= 1.2 P

In second year population is decreased by 30% ===> Multiplying Factor=0.7

So, population at the end of second year= 1.2 P x 0.7 =  0.84 P

so population P becomes 0.84 P ===> which means 16% decrease.


Hence option B is correct.

Question 2:  If population decreases by 20% in first year and decreases by 30% in second year. Calculate net percentage change?

A] -44 %     B] -50%          C] +16%          D] None




Hence option A is correct.

Question 3: If population increases by 10% in first year, increases 20% in second year, increases 30% in third year. What will be net percentage increase?

A] 60%     B] 65%          C] 71.6%          D] None




Hence option C is correct.

Question 4: If radius of circle is increased by 10%. By what percent area of circle increases?

A] 19%             B] 20%               C] 21%               D] None





Hence option C is correct.


Question 5: If radius of circle is decreased by 10%. By what percent area of circle decreases?

A] 19%         B] 20%            C] 21%           D] None

Solution:  Solve it. Similar to previous question. A option is correct.

Question 6: If side of square is increased by 20%. By what percent area of square increases?

A] 36%            B] 40%               C] 44%              D] None





 Hence option C is correct.


Question 7:  If length of rectangle is increased by 10% and width is decreased by 20%. By what percent area of rectangle decreases.

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