## Study notes on Ratio and Proportion for DSSSB Exam

Here we are providing you Study notes on Ration and Proportion which will be helpful for Upcoming CTET and DSSSB Exams

Ratio:-

Ratio is used to compare the two quantities of same unit. If given quantities are not in same unit, Ratio of those quantities cannot be calculated. Commonly ratio is expressed in the form of fraction.

For example :- If A and B are two quantities of same unit then their ratio will be   or A : B.
where A is known as antecedent (1st term) and B is called consequent (2nd term).

Note:-  If each term of ratio is multiplied or divided by the same non - zero number, in that case ratio does not get affected. It remains same.
For example, 3 : 5 is the same as (3 x 2)/(5 x 2)=6/10 =3/5

Expression of Ratio in Percentage form:-
Ratio can be expressed as percentage. To express the value of a ratio as a percentage, we multiply the ratio by 100.
Example :-Convert ratio 3/5 in percentage form.
Solution:- (3/5) x 100=60 %

Types of Ratio:-
(1) Duplicate Ratio :- When both terms of a ratio are multiplied by themselves or squared, then resulted ratio is known as duplicate ratio. If a and b are two numbers then their ratio will be a : b and duplicate ratio will be a2 : b2.
Example :- Duplicate ratio of 8 : 9 = 82 : 92 = 64 : 81

(2) Sub-duplicate Ratio :- This is the ratio of square roots of two numbers. If two numbers are a and b, then sub-duplicate ratio between them will be √a : √b
Example :- Sub - duplicate ratio of 4 : 9 = √4 : √9 =2:3

(3) Triplicate Ratio :- The ratio of cubes of two numbers is known as triplicate ratio. If a and b are two numbers then their triplicate ratio will be a3 : b3.
Example :- Triplicate ratio of    3 : 4 = 33 : 43= 27 : 64

(4) Sub-triplicate Ratio :- The ratio of cube roots of two numbers is known as sub-triplicate ratio. If a and b are two numbers then sub-triplicate ratio of a and b will be

(5) Inverse Ratio :- Inverse Ratio of two numbers refers to the ratio of their reciprocal. If a and b are two numbers then the inverse ratio of a and b is (1/a): (1/b) =b:a
Example:- Inverse ratio of 3 and 4=(1/3): (1/4)=4:3

(6) Compound Ratio :- Compound ratio of two ratios refers to the ratio made after multiplying their antecedents and consequents. The compound ratio of a : b and c : d will be
a x c: b x d= ac:bd

Proportion:- The equality of two ratios is called proportion. If a, b, c and d are in proportion then we can write them
as a : b = c : d or a : b : : c : d.
In proportion, we call ‘first and fourth terms’ as extremes and ‘second and third terms’ as means.

Note :- In a proportion, product of extremes = product of means. If a : b : : c : d then ad = bc.

Types of Proportion:-

(1) Continued Proportion :- Four quantities a, b, c and d are said to be in a continued proportion, if
a :b =b:c=c:d
or (a/b) =(b/c) or b2=ac
In this relationship, b is said to be the mean proportional between ‘a and c’ and c is said to be a third proportional to ‘a and b’.

(2) Direct proportion:-
If A is directly proportional to B then as A increases, B also increases proportionally.

For example:- The relationship between speed, distance and time, is as follows - “speed is directly proportional to distance when time is constant”. It means if speed is doubled, distance travelled will also be doubled when time is kept constant.

(3) Inverse proportion:- A is inversely proportional to B means if A increases, B decreases proportionally.

For example:- If speed is doubled, time taken to cover the same distance is reduced to half.

Some properties on Ratio and Proportion:-

(1) Rule of Invertendo-
If a : b :: c : d, then b : a :: d : c

(2) Rule of Alternendo-
If a : b :: c : d, then a : c :: b : d

(3) Rule of Componendo-
If a : b :: c : d, then its componendo will be
(a+b) : b :: (c+d) : d.

(4) Rule of Dividendo-
If a : b :: c : d then (a-b) : b :: (c-d) : d.

(5) Rule of Componendo and Dividendo-
If a : b :: c : d, then its componendo and dividendo will be
=(a+b) : (a-b) :: (c+d) : (c-d)
(a+b)/ (a-b)  =(c+d)/ (c-d)

Some Memorable Points :-

(1) If a : b :: b : c, then c is called the 3rd proportional to a and b. c will be calculated as following :-
a : b :: b : c
a /b= b /c
b2=ac
then
c=b2/a

(2) If a : b :: c : d, then d is called the 4th proportional to a, b and c. d will be calculated as following.
a : b :: c : d
a /b= c /d
d=bc/a

(3) Mean proportional between a and b is  √ab
If mean proportional is ‘x; then
a /x= x /c
x2=ac
x=√ab

# Study notes on Mensuration for CTET and DSSSB Exams

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