Ratio proportion and variation formula with aptitude tricks

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

$\frac{x}{y} \ = \ \frac{ax}{ay} \ = \ \frac{bx}{by}$

$\frac{x}{y} \ = \ \frac{x/a}{y/a} \ = \ \frac{x/b}{y/b}$

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

$\frac{a}{b} \ : \ \frac{p}{q} = \ \frac{a/b}{p/q} \ = \ \frac{aq}{bp}$

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

$k = \frac{a}{b} \ = \ \frac{c}{d} \ = \frac{p}{q} \ = \frac{r}{s} \ = \frac{a+c+p+r}{b+d+q+s} \$

For example : If a : b = c : d = e : f = 4 : 5 then find value of $\frac{ap+cq+er}{bp+dq+fr} = ?$

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

$\frac{ap+cq+er}{bp+dq+fr} = \frac{4}{5}$

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

$\frac{p}{q} = \frac{p+a}{q+b}$ if and only if $\frac{p}{q} = \frac{a}{b}$

Also

if $\frac{p}{q} \ > \ \frac{a}{b}$ then $\frac{p}{q} \ > \ \frac{p+a}{q+b}$

if $\frac{p}{q} \ < \ \frac{a}{b}$ then $\frac{p}{q} \ < \ \frac{p+a}{q+b}$

Property – 6

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

a : b :: b : c are in proportion

b² = ac ⇒ $b = \sqrt{ac}$

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 = a² : b²

Property – 8

If the four quantities a, b, c & d are in proportion, ( i.e a : b :: b : c or $\frac{a}{b} \ = \ \frac{c}{d}$ ) then

1. Alternando Ratio $\frac{a}{c} \ = \ \frac{b}{d}$

2. Invertendo Ratio $\frac{b}{a} \ = \ \frac{d}{c}$

3. Componendo ratio $\frac{a +b}{b} \ = \ \frac{c + d}{d}$

4. Dividendo ratio $\frac{a -b}{b} \ = \ \frac{c -d}{d}$

5. Componendo and Dividendo ratio $\frac{a +b}{a-b} \ = \ \frac{c+d}{c-d}$

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 – 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