In this article discussed about basic concepts, Important Formulas, Properties with Quantitative aptitude shortcuts & tricks of ratio proportion and variation
Ratio Proportion and Variation aptitude formulas for all Competitive Exams
What is Ratio
When compare any two numbers, some times it is necessary to find out how many times number is greater or less than other .
In other words, we often need to express one number as a fraction of other.
In general, the ratio of a number P to a number Q is defined as the quotient of the number P and Q
Ratio is a comparison in division method P, Q ratio is P : Q
Here P is called Antecedent and Q is called Consequent.
For example: Find consequent value for the ratio of 5 : 4 and antecedent is 20
Sol: ratio is 5 : 4
For antecedent 5 gives 20 then
Consequent 4 gives 20 x 4 / 5 = 16
What is Proportion
When two ratios are equal, the four quantities comparing them are said to be proportionals. Thus if p/q = r/s, then p,q,r & s are proportionals.
The proportion is represented as
p : q :: r : s or p : q = r : s
It is read as p : q is as r : s
Here ‘p’ is called 1st proportion
‘q’ is called 2nd proportion
‘r’ is called 3rd proportion
‘s’ is called 4th proportion
‘p’ & ‘s’ are called extreams the and ‘q’ & ‘r’ are called the means
Product of extreams = Product of means
i.e p x s = q x r
What is Variation
Variation deals with how one quantity varies with respect to one or more other quantities.
A quantity ‘A” is said to vary directly as another ‘B’ when the two quantities depend upon each other in such a manner that if B is changed , A is changed in same ratio.
The symbol ‘ ∝ ‘ is used to denote variation.
Essentially there are two types of variations known as direct variation and inverse variation.
If Quantity ‘A’ varies directly with Quantity ‘B’ then they are said to be in direct variation
i.e If A increased then B is also increased. (Logical variation)
If A increased by 20% then B is also increased by 20%. (Calculation implication)
In the direct variation the ratio of A/B is constant
Direct variations can be expressed as A = kB, where, k is called the constant of proportionality.
If Quantity ‘A’ varies inversely with Quantity ‘B’ then they are said to be in inverse variation
i.e If A increased then B is decreased. (Logical variation)
If A increased by 10% then B is decreased by 9%. (Calculation implication)
In the direct variation the product of A x B is constant
Inverse variation can be expressed as A = k/B where, k is called the constant of proportionality
Impotent formulas and portieres of Ratios and Proportions
Property : 1 ( Multiply or divide of ratio)
If we multiply or divide the numerator and denominator of a ratio by same number, the ratio remains unchanged.
For the ratio of x : y and ‘a’ & ‘b’ are the real number then
Property :2 ( Multiplication and Comparison of two ratios)
The ratio of two fractions can be expressed as a ratio of two integers.
For example a / b and p / q are two fractions then
Now compare the values of aq and bp
If aq > bp then a/b > p/q or
If aq < bp then a/b < p/q
Property – 3 (Properties of Equal Ratios )
For the ratios of a/b = b/c = p/q = r/s = k
then each ratio is equal to
For example : If a : b = c : d = e : f = 4 : 5 then find value of
Solution: a/b = c/d = e /f = 4/5 then
Now according to property – 1
pa/pb = qc/qd = re /rf = 4/5
Now by the property of Equal Ratios
Property – 4
If the ratio p /q > 1 ( i.e A ratio is called greater inequality) and if N is a positive number then
( p + N ) / ( q + N ) < p/q & ( p – N ) / ( q – N ) > p/q
If the ratio p /q < 1 and if N is a positive number then
( p + N ) / ( q + N ) > p/q & ( p – N ) / ( q – N ) < p/q
Property – 5 (Adding number to numerator and denominator)
The ratio of the addition should be equal to the original ratio to maintain equality of ratios when two different numbers are added the numerator and denominator.
For example : If p/ q is the ratio then
if and only if
Property – 6
If the three quantities a, b & c are in continued proportion, then
a : b :: b : c are in proportion
b2 = ac ⇒
‘a’ is called 1st proportion
‘b’ is called mean
‘c’ is called 2nd proportion
Property – 7
If three quantities quantities are proportionals the first is the third is the duplicate ratio of the first to the second.
a : b :: b : c are in proportion then
a : c = a2 : b2
Property – 8
If the four quantities a, b, c & d are in proportion, ( i.e a : b :: b : c or ) then
1. Alternando Ratio
2. Invertendo Ratio
3. Componendo ratio
4. Dividendo ratio
5. Componendo and Dividendo ratio
Types of Ratios: ( Kinds of Ratios)
When the ratio p/ q is compounded with it self, then
Duplicate Ratio – p2 : q2 is called duplicate ratio of p : q.
Triplicate Ratio – p3 : q3 is called triplicate ratio of p : q.
Sub -Duplicate Ratio – p1/2 : q1/2 is called sub-duplicate ratio of p : q
Sub – triplicate Ratio – p1/3 : q1/3 is called sub-triplicate ratio of p : q
Mathematical Applications of Ratios
1. Calculating the ratio by percentage and decimal values
Ratio can be expressed as a percentage. To express the value of a ratio as a percentage, we multiply the ratio by 100.
For the ratio of 5 : 4
Thus, 5/4 = 5 *100 / 4 = 125 %
The ratio 5 /4 has a percentage value of 125% and it has a decimal value of 1.25
Ratio proportion and variation problems with solutions Click here
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