## Percentages-1

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:

TSR for most of the percentage related questions

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.