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Concept - Ratio & Proportion

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Ratio between two quantities is always expressed  in the same units

Multiplication or division of each term of a ratio by the same non-zero number doesn’t affect the ratio. But the same is not  true for addition or subtraction.
Eg:  2 : 5 is same as 4 : 10 or 20 : 50

 

if a:b = c:d
it can also be written as a:b ∷ c:d

The first and fourth  proportional are called the extreme terms
The second and the third proportional are called mean terms

 

Product of extremes = product of means.

 

Inverse Ratio:-
If the ratio between two numbers is a:b, then inverse ratio is 1/a : 1/b i.e. b:a

 

Direct Proportion:-
When one quantity is increased (or decreased), the other quantity also increases (or decreases) in the same ratio.
Eg: Number of pens and total cost of pens.

 

Indirect or Inverse Proportion:-
When one quantity is increased the other quantity decreases in the same ratio and vice-versa.

Eg: Number of workers and days taken to do a work.
Speed and time taken to cover a distance.

 

Mean Proportional:-
If  ‘x’ is mean or  second proportion of ‘a’ and ‘b’, then a : x = x : b

x=$\sqrt{\mathrm{ab}}$

 

 

Third Proportional:-
If ‘x’ is third proportion of ‘a’ and ‘b’, then a:b = b:x

 

 

Fourth Proportional:-
If ‘x’ is fourth proportion of a,b and c, then a:b = c:x
If a:b = m :n and b:c = p:q then a:c = m/n x p/q   x=bc/a
The rule can be extended for any number of ratios.
If ‘x’  times of a number is equal to ‘y’ times of second number, then ratio between two number is 1/x : 1/y = y:x

 

What must be subtracted from each of four numbers a,b,c and d, so that the resultant numbers are in proportion:

 

What must be added to each of the four numbers a,b,c and ‘d’ so that the resultant numbers are in proportion.

If sum of two numbers is ‘x’ and their difference is ‘y’ then the numbers are in the ratio of
(x+y) : (x-y)

 

Compound Ratio:-

Ratios are compounded by multiplying together the antecedents for a new antecedent, and the consequents for a new consequent.

 

Compound ratio of a:b, b:c, c:d is

Duplicate ratio of a:b is a² b²
Triplicate ratio of a:b is a³ b³
Sub duplicate ratio of a:b is $\sqrt{a}$:$\sqrt{b}$
Sub Triplicate ratio of a:b is   $3\sqrt{a}$:$3\sqrt{b}$

 

Theorem:
The incomes of two persons are in the ratio a:b and their expenditures are in the ratio c:d.  If each of them saves Rs.X,then

 

Theorem:
A number which, when added to the terms of the ratio a:b makes it equal to the ratio

A number which, when subtracted from the terms of the ratio a:b makes it equal to the ratio

 

Previous Exam Questions:

 

Practice following examples

 

1)An amount of Rs.8400 is distributed among A,B & C in the ratio of 1:2:4. The difference between the shares of A & C is:

1) Rs.4800 2)Rs.2400 c)Rs.1200 d) Rs.3600

 

2)An amount of Rs.7,200 is distributed among P,Q & R in the ratio of 2:7:3. The share of Q exceeds the share of P by

1) Rs.2000 2)Rs.3000 3)Rs.1500 4)s.2800

 

3)An amount is distributed among P,Q & R in the ratio of 2:3:8. The difference between the shares of P&R is Rs.2400. What is the share of Q?

1)Rs.1200 2)Rs.1800 3) Rs.1600 4) Rs.2400

 

4)If a:b = 7:9 and b:c = 15:7, then what is a:c

1)5:3 2)3:5 3)7:21 4)7:15

 

5)The ratio of two numbers is 3:8 and their difference is 115. Then the smaller of the two numbers is

1)184 2)194 3)69 4)59

 

Answers:

1) Rs.3600

2) Rs.3000

3) Rs.1200

4) 5:3

5) 69

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