Concepts of Percentages and its Important Shortcuts

Let’s understand the concepts of the Percentage in detail:

1.0 Concept – One:

1. A+B+AB/100 When A and B both are a positive change
2. A-B-AB/100 When A is positive change and B is a negative change
3. -A+B-AB/100 When A is negative change and B is a positive change.
4. -A-B+AB/100 When A and B both are a negative change.

Important: There is no need to remember the above formulas, you have to just remember:

•  ±A ± B ±AB/100 and put the sign of change, if negative, then (-) and positive then, (+) but keep in mind that sign of AB is the product of signs of A and B.

Example1: The price of a book is reduced by 10% and the sale of the book is increased by 15%. Find the net effect on revenue.

Example 2: If the length and breadth of a rectangle is increased by 5% and 8% respectively. Find the % change in area of the rectangle.

2.0 Concept – Two:

New solution × new % = old solution × old %
This formula is applicable for the commodity which is constant in the solution or mixture, its quantity doesn’t change after mixing in solution.

Example 3:  A mixture of sand and water contains 20% sand by weight. Of it 12 kg of water is evaporated and the mixture now contains 30% sand.
Solution: In this sand is constant in the mixture. So we will apply this formula on sand not on water.
(a)Find the original mixture.
Let the original mixture is P kg, So new mixture = (P-12) kg
old% = 20 and new % = 30
new solution × new % = old solution × old %
 (P-12)× 30% = P × 20%
 3P – 36 = 2P
 P = 36 Kg.
(b) Find the quantity of sand and water in the original mixture.
     Quantity of sand in original mixture = 20% of 36 = 7.2 Kg
     Quantity of water in original mixture = 80% 0f 36 = 28.8 Kg
OR = Quantity of mixture – quantity of sand = 36 -7.2 = 28.8 Kg

Example 4: 30 litres of a mixture of alcohol and water contains 20% alcohol. How many litres of water must be added to make the alcohol 15% in the new mixture?
Solution: only water is added to the mixture so, there is no change in alcohol. We will apply the above formula on alcohol. Let water added is P litres.
Old mixture = 30 litres, old % of alcohol = 20%
New mixture = 30+P litres, new % of alcohol = 15
using, new solution × new % = old solution × old %
(30+P) × 15% = 30 × 20%
P = 10 litres, hence 10 litres of water is added.

3.0 Concept – Three:

Example 5: If the price of milk increased by 25%, by how much per cent must Rahul decrease his consumption, so as his expenditure remains the same.
Solution: Let the price of milk is 20 Rs/ litre and Rahul consumes 1-litre milk.
Expenditure of Rahul = price × consumption
Now the price of milk is increased by 25%, so the new price is (125/100)× 20 = 25 Rs.
but his expenditure remains the same
So, new consumption × new price = old price × old consumption
      new consumption × 25 = 20 × old consumption
      new consumption  =(20/25)  × old consumption
      new consumption% = (20/25)× old consumption × 100
      new consumption% = 80% of old consumption
decrease in consumption = 20 %
Using the above trick:  Given, price % is increased so sign will be (+) and consumption % will decrease.
Decrease in consumption =(25/125) × 100 = 20%

Example 6: If the price of milk decreased by 25%, by how much per cent must Rahul increase his consumption, so as his expenditure remains the same.
Solution: Let the price of milk is 20 Rs/ litre and Rahul consumes 1-litre milk.
Expenditure of Rahul = price × consumption
Now the price of milk is decreased by 25%, so the new price is × 20 = 15 Rs.
but his expenditure remains the same
So, new consumption × new price = old price × old consumption
      new consumption × 15 = 20 × old consumption
      new consumption  =  (20/15)× old consumption
      new consumption% = (20/15)× old consumption × 100
      new consumption% = 133(1/3)% of old consumption
     increase in consumption = 33(1/3) %
Using the above trick: given price, % is decreased so sign will be (-) and consumption % will increase.
Increase in consumption = (25/75) x  100 = 33(1/3)%

4.0 Concept – Four:

Example 7: The population of a town is 6000. It increases 10% during the 1^st year, increases 25% during the 2^nd year and then again decreases by 10% during the 3^rd year. What is the population after 3 years.
Example 8: The population of a village increases by 10% during the first year, decreased by 12% during 2^nd year and again decreased by 15% during the 3^rd year. If the population at the end of the 3^rd year is 2057.
 

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