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 1^{st} proportion

‘q’ is called 2^{nd} proportion

‘r’ is called 3^{rd} proportion

‘s’ is called 4^{th} 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.

**Direct variations**

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.

**Inverse variation**

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 example

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

Similarly

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

Also

if then

if then

**Property – 6**

If the three quantities a, b & c are in continued proportion, then

a : b :: b : c are in proportion

b^{2} = ac ⇒

Here

‘a’ is called 1^{st} proportion

‘b’ is called mean

‘c’ is called 2^{nd} 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 = a^{2} : b^{2}

**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 – p^{2} : q^{2} is called duplicate ratio of p : q.

Triplicate Ratio – p^{3} : q^{3} is called triplicate ratio of p : q.

Sub -Duplicate Ratio – p^{1/2} : q^{1/2} is called sub-duplicate ratio of p : q

Sub – triplicate Ratio – p^{1/3} : q^{1/3} 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

** Continued…**

Ratio proportion and variation problems with solutions Click here

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