The term 'Percent' means for every hundred. In other words 'Percent' is a Fraction whose denominator is 100.
To convert percent into a fraction or number divide it by 100
Ex: 25% = 25/100
To convert a fraction or number into percent, multiply it by 100.
Ex: 5 as a percentage = 5 x 100 = 500%
Note:
     Percentage can be expressed as a Fraction or decimal number.
  Ex: 75% = 75/100 = 3/4 (or) 75/100 = 0.75%
Remember:
   100% = 1,    50% = 1/2,     25% = 1/4,     20% = 1/5,
   12 1/2% = 1/8,   6 1/4% = 1/16,   3 1/8% = 1/32,
   16 2/3% = 1/6,   33 1/3% = 1/3,   66 2/3% = 2/3,
   37 1/2% = 3/8,   87 1/2% = 7/8,   75% = 3/4,
   30% = 3/10,      40% = 2/3,   60% 3/5,     80% = 4/5.

'Per cent means' 'per hundred' it is given by '%' symbol.
Here x% means x per hundred or x/100. 
Thus, any percentage can be converted into an equivalent fraction by dividing 100. 
          eg,    20% = 20/100 = 1/5, 100% = 100/100 = 1, 150% = 150/100 = 3/2 
Also any fraction or decimal can be converted into its equivalent percentage by multiplying with 100.
          eg,    1/4 = 1/4 × 100 = 25%,
                 1/2 = 1/2 × 100 = 50%, 1 = 1 × 100 = 100%.           

Important Formulae

  I. Percentage increase= Increase/Original value × 100 
 II. Percentage decrease = Decrease/Original value × 100
III. If the price of commodity increases by r%, then the reduction in consumption so as not to increase
     the expenditure is [r/(100 + r) × 100]% 
 IV. If the price of commodity decrease by r%, then the increase in consumption so as not to decrease
     the expenditure is [r/(100 - r) × 100]%
  V. If A's income is r% more   than B's income then B's income is less than A's income by [r/(100 + r) × 100]%
 VI. If A's income is r% less than B's income then B's income is more than A's income by [r/(100 - r) × 100]%
VII. Let the population of the town be P now and suppose it increases at the rate of r% per annum, then
     (a) Population after n years = P[1 + (r/100)]n
     (b) population n years ago = P/[1 + (r/100)]n 
VIII. Let the present value of the machine be P and if it is depreciates at the rate of r% per annum, then 
      (a) Value of the machine after n years = P(1 - (r/100))n
      (b) Value of the machine after n years ago = P/(1 - (r/100))n    

Note: X as a percentage of Y?                   Note: Percent of Increase/Decrease
             (OR)                                           % Increase/Decrease = Increase/Decrease/initial value * 100
    What Percent of Y is X?                       Ex:The production of the company is increased from 125 units to 150 units      
Sol: (x/y)x100                                                then the % of increase
Ex: 75 is what percent of 150?                  Sol: Increase = 150 - 125 = 25 units
Sol: (75/150)x100 = 50%                                 % I = (25/125)x100 = 20%

Note:
      If a quantity is increased/decreased by K% then, we can find directly the new quantity.
In case of Increase K%
      New Quantity = Original Quantity x (100 + K/100)
In case of Decrease
      New Quantity = Original Quantity x (100 - K)/100
Ex: If the strength INDIANSTUDYHUB is Increased by 40% p.a. What will be the strength is 400?
Sol : New strength = 400 x 140/100 = 560

Example 1 : Express each of the following as a decimal
      (1) 20% = 20/100 = 0.2 
          (2) 0.4% = 0.4/100 = 0.004 
          (3) 0.008% = 0.008/100 = 0.00008 

Example 2 : Express each of the following as rate per cent 
      (1) 3/5 = [(3/5) × 100] = 60%
          (2) 0.001 = (0.001 × 100) = 0.1%
          (3) 1.5 = (1.5 × 100) = 150% 

Example 3 : What per cent of 60 is 15?
Solution : Required percentage = [(15/60) × 100]% = 25% 

Example 4 : What per cent of 8 is 144? 
Solution : Required percentage = [(144/8) × 100] = 1800%

Example 5 : What per cent of 5 kg is 25 g?
Solution : Required percentage = [(25/(5 ×1000)) × 100] = 0.5% 

Example 6 : What per cent of 4.5 L is 135 mL?
Solution : Required percentage = [(135/(4.5 × 1000)) × 100] = 3%

Solved Examples

Type 1

Based on Fundamental Concepts

Example 7 : ?% of 925 + 25% of 850 = 268 
Solution : ?% of 925 = 268 - 25% of 850 
=>    ?/100 × 925 = 268 - 25/100 × 850 
=>    ? = (268 - 212.5)/9.25 = 55.5/9.25 = 6 

Example 8 : 15.005% of 599.999 = ? 
Solution: ? ~ 15% of 600 ~ 15/100 × 600 ~ 90 

Example 9 : ? = 75% of 210 - ? = 62% of 249.65 
Solution : ? = 75% of 210 - 62% of 249.65 ~ 75/100 × 210 - 62/100 of 250 
         ~ 157.5 - 155 ~ 2.5  

Type 2

Words Problems

Example 10 : A's income is 25% more than B's income and 25% less than C's income. What is the B's 
income if C's income is Rs. 4500?
Solution: A's income = (100 - 25)% of 4500 = 75% of 
      4500 = 75/100 × 4500/1 = 3375 
Now, let B's income be x 
=>   125% of x  = 3375 
Npw let B's income be x 
=>  125% of x = 3375 
=>  125/100 x = 3375 
=> x = (3375 × 100)/125 = 2700 

Example 11 : in a  class of 50 students and 5 teachers each student got sweets that are 12% of the total 
number of students and each teacher got sweets that are 20% of the total number of students. How many 
sweets were there?
Solution : Total number of sweets = (12% of 50) × 50 + (20% of 50) × 5 
                   = [(12/100) × 50 × 50] + [(20/100) × 50 × 5]
                                   = 300 + 50 
                                   = 350