**In Mathematics, a percentage is a number or ratio expressed as a fraction of 100. It is denoted by “%”.The term “percent” is derived from the Latin per centum, meaning “by the hundred”.**

**For example – The percentage of marks obtained by a student in the exams is 70%.**

**11% is same as 11/100 or 0.11.Percentages are used as ratios to express how large or small one quantity is relative to another quantity.**

**Why study about Percentages ?**

**– It forms the part of other topics such as Profit & Loss , Ratio & Proportion , Time & Work , Time Distance Speed.**

**– It helps in expressing the same value in the different ways as decimals , fractions, and conversion between them.**

**– It forms a powerful tool for comparison of data such as marks of students , the growth of population, profits, expenditures ,etc.**

**Percentages are meaningless unless information about the base over which it is calculated is known.**

**I urge you all to memorize the table given so that you can form calculations at a faster rate.**

**Q.1) Prem scores 25 marks in an examination out of 30 , while Praveen scores 40 marks out of 70. Who has performed better?**

**Solution : Very important to note down, the percentage depends on the base value (out of) quantity.**

**Thus, Prem’s percentage = (25/30) x 100 =83.33 %. Praveen’s percentage =57.14%.**

**Therefore, Prem’s performance is better than Praveen.**

**Percentage change (Percentage increase/decrease) :**

**Absolute value change: Actual change in the quantity.**

**For example :- If Rohan gets 20 marks in first I.A and 25 marks in the second I.A, then the absolute value of change =25-20=5.**

**Percentage Change :**

**= ( Absolute Value Change/Original Value ) x 100**

**For above example, percentage change = (5/20) x 100 =25%.**

**Calculation of %age – Concept of successive addition.**

**Q.2) 35 % of 40 = ?**

**We will simplify the calculations as follows.**

**30% of 40 = (3)(4) => 12 .**

**5% of 40 = (1/20) x 40 = 2**

**Adding above 30%and 5%, we get**

**Therefore, 35% = 12+2 = 14.**

**Q.3) 45 % of 50 = ?**

**We will simplify the calculations as follows.**

**10 % of 50 => 5**

**40% of 50 = (4)(5) => 20**

**5% of 50 = (5/2) = >2.5**

**Adding above 40%and 5%, we get**

**Therefore, 45% = 20 +2.5 = 22.5 .**

**Note : You can interchange the two values in the percentage calculations.**

**To understand, go through the example below.**

**Q.4) 84% of 150 = ?**

**Solution :**

**150% of 84 =?**

**100% of 84 =84**

**50% of 84 = (84/2) =42.**

**Adding the above two values,we get**

**Therefore,150% of 84 =84+42 =126, which is same as that of 84% of 150.**

**Successive Changes in the value :**

**Successive change in the value is commutative, the order of increase or decrease in the percentage value of quantity or number does not change the final value .**

**For example :**

**If a number is increased by 10 % and 20 % successively, then what is the percentage change in the value of a number ?**

**Solution :**

**Normal Method :**

**10 —- 10% inc—> 11**

**11—–20% inc —->(11+2.2)—->13.2**

**Shortcut Method:**

**Using the above formula,we get as**

**10 +10 +[(10 x 10)/100]**

**= 21 %**

**Q.5) A salesman gives successive discounts of 10%, 20% ,30% respectively.What is the net discount percentage ? ( Or single discount which he can give? )**

**Solution :**

**: **

**Since it is discount the value of original price decreases .**

**Therefore a and b are taken ‘Negative value”in the formula,we get**

**There are three discounts,we will take two discounts and perform the calculations using the above formula , then this value is used with the third discount and the process is repeated**

**-10 -20 +[(-10)(-20)/100]**

**= -30+2**

**=-28 %**

**This discount value is used with -30%, we get as follows.**

**-28-30+[(-28)(-30)/100]**

**= -58+8.4**

**=-49.6%**

**Therefore, the overall discount of 10% ,20% an 30% = 49.6%.**

**Q.6) If the product of 10 x 10 x 10 is changed to 11 x 12 x 13.Find the percentage change in the product.**

**Solution: We can do it directly or the other approach as mentioned earlier.**

**By observing the percentage change in each number,we get a=10% ,b=20%,c=30%.**

**10+20+[(10)(20)/100]**

**=32%.**

**Again**

**32+30+[(32)(30)/100]**

**=71.6%.**

**Time Saving Results: **

**1.) If the price of a quantity is increased by p% add then increased by k%,then the resultant increase in its price is given by **

** **

**We need to choose the sign of p & k depending whether it is the increase (+ve) or decrease(-ve).**

** For example: **

** Q.1) If the price of milk is increased by 40% and consumption is decreased by 20%.What is the %age change in expenditure?**

**Solution :Applying the above formula, we get 40-20+[(40)(-20)/100]**

**Therefore, the percentage change in the expenditure = 12%.**

** Q.2) A car increases its speed by 25%, after that, it again increases its speed by 20%.By what %age is the car’s final speed greater than its original speed ?**

** Solution : Applying the above TSR, we get **

** = 25+20 +[(25)(20)/100]**

** = 50%.**

**Q.3 ) The length and the breadth of a rectangle are changed by 20% and -10% respectively. What is percentage change in the area of the rectangle ?**

** Solution : Applying the above TSR, we get **

** = 20-10 +[(20)(-10)/100]**

** Percentage change in the area of rectangle = 8%.**

**Q.4) If the side of a square is increased by 30%, then the percentage change in the area of the square is ____.**

**Solution : Applying the above TSR, we get **

** = 30+30 +[(30)(30)/100]**

** = 69 %.**

**Q.5) If the side of the cube is increased by 10%, then what is the percentage change in the value of cube ?**

**Solution : Volume = Side x Side x Side **

** = 10% x 10% x 10%**

**Applying the TSR twice, we get**

** 10 +10+[(10)(10)/100] => 21**

**21+10+[(21)(10)/100] **

**=31+2.1**

**=33.1%**

**Therefore, Percentage change in the volume of cube = 33.1 %**

**Q.6) The length ,breadth & height of a room in the shape of a cuboid are increased by 10%,20% & 50% respectively.Find the percentage change in the volume of the cuboid.**

**Solution**

** : **

**Applying the above formula twice, we get**

**10+20+[(10)(20)/100]**

**= 32%**

**Again, 32 +50+[(32)(50)/100] => 98%**

**Therefore, % age change in the volume of the room = 98%.**

**Q.7) If the radius of a cylinder increases by 10% and height increases by 30% .What is the percentage change in the volume of the cylinder ?**

**Solution: **

** We know that , Volume of Cylinder = pi x radius^2 x height**

**Therefore, applying the above TSR, we get**

** 10 +10+[(10)(10)/100] => 21**

** 21+30+[(21)(30)/100] **

** =51+6.3**

** =57.3 %**

**Therefore, Percentage change in the volume of cylinder = 57.3 %**

**Q.8) ****The population of a village is 1,00,000.The rate of increase is 10% p.a .Find the population at the start of the third year.**

**Solution : Applying the above TSR, we get **

**10+10+[(10)(10)/100]**

**= 21%**

**Again, applying the same TSR, we get**

**21+10+[(21)(10)/100] => 33.1 %**

**Population at the start of third year =(33.1 x 100000)/100 => 100000+3310 =103310.**

**Q.9) The price of a commodity is increased by 15% and its sale gets decreased by 10%.What will be the percentage effect on the income ?**

**Solution : Applying TSR.1 , we get **

**% effect = 15-10 +[(15)(-10)/100]**

**= 3.5 % **

**Therefore, his income is increased by 3.5 %.**

**Q.10) ****The price of a commodity is increased by 10% and its sale gets increased by 25% What will be the percentage effect on the price ?**

**Solution : Applying TSR.1 , we get **

**% effect = -10+25+ [(-10)(25)/100]**

**= 12.5**

**Therefore, his income is increased by 12.5%**

**Q.11) Two successive discounts of 10% and 20% are equal to a single discount of ___.**

**Solution: Applying the same TSR, we get **

**-10-20+[(-10)(-20)/100] => -30+2 = 28% **

**Q.12) The population of a town increased by 10%, 20% and then decreased by 30%. The**

**new population is what % of the original ?**

**Solution : Applying TSR, we get **

**= 10+20+[(10)(20)/100] => 32% increase**

**Again applying the TSR with 32% and -30% , we get **

**= 32-30+[(32)(-30)/100]=> 7.6 decrease.**

**Therefore, new population % of the original is 100%-7.6 % = 92.4%.**

**2) In measuring the sides of a rectangle, one side is taken p% in excess and the other k% in deficit. The percentage in the area calculated from the measurement is **

**p-k +[p.k/100] in excess or deficit.**

**3) If the sides of triangle,rectangle,square,circle,rhombus ( any two-dimensional figure)are increased by x% , its area is increased by **

**For example**

**Q.1 If the radius of a circle is increased by 10%,find the percentage increase in its area.**

**Solution : Applying the above TSR.3 , we get **

**= 2 (10)+[ 10^2 /100] **

=**21 %**

**Therefore, the percentage increase in the area of the circle is 21 %.**

**Q.2 If the length and breadth of rectangle are increased by 20% and 30% respectively, then the perimeter of rectangle increases by ___% **

** a) 56 b) 50 c)-56 d) Data insufficient **

**Answer : Data insufficient .**

**Q.3) In an election of two candidates, the candidate who gets 41% is rejected by a majority of 2412 votes.Find the total number of votes polled.**

**Solution: (59% -41%) x total no of votes = 2412**

**=> 18% =2412**

**Therefore, 100% = (2412/ 18) x 100 = 13400.**

**Q.4) Mr.Praveenkumar salary was decreased by 50% and subsequently increased by 50% .How much percent does he lose?**

**Solution: **

**Applying the above formula to the given question,we get **

**=** **-50+50+[(-50)(50)/100]**

**=25% loss**

**TSR : If two values are respectively x% and y% more than the third number, then the first is [(100+x)/(100+y)] x 100% of the second.**

**For example**

** Q.1) Two numbers are 25% and 50% more than third number.What is the percentage is the first of the second ?**

**Solution: From the above TSR, we have **

**= [(100+25)/(100+50)] x 100% **

**=[(125)/(150)] x 100%**

**=(5/6) x 100**

**= 83.33%**

**Q.2) Two numbers are respectively 12.5% and 25% more than third number.What is the first number as percentage of the second number.**

**Solution: Applying the above TSR, we have **

**= [(100+12.5)/(100+25)] x 100**

**=[(112.5)/125)] x 100**

**= 90%.**

**Q.3) Two numbers are respectively 20% and 50% more than a third number. What is the percentage is the first to the second ?**

**Solution: Applying the TSR, we get **

**=[(100+20)/(100+50)] x 100**

**=[(120)/(150)] x 100**

**=80%.**

** Note: If the above problem is changed “more than ” to “less than” then change the sign in the formula and use it.**

**Commodity Increase/ Decrease:**

**If the price of a commodity increases by R%, then the reduction in consumption so as not to increase the expenditure is**

**If the price of the commodity decreases by **R%**, then to maintain the same expenditure by increasing the consumption is:
**

**Concept : Population Increase/Decrease **

**Let the population of the town be P now and suppose it increases at the rate of R% per annum, then **

**I’m Greater ,No I’m Lesser **

**If A is **R%** more than B, then B is less than A by**

**If A is R% less than B , then B is more than A by
**

**More Concepts & Questions will be added soon. **