Ratio and Proportion

 

Ratio and Proportion

 

RATIO

A ratio is a comparison of two quantities by division. Since, ratio is an abstract number, the two quantities that are being compared must be expressed in the same unit. e.g., if A earns Rs. 10 and B earns Rs. 5, then A’s earning, consists of two times the earning of B or B’s earning consists of \[\frac{1}{2}\text{th}\] of A’s earning. The ratio of A to B is written as \[A:B=A\div B=\frac{A}{B}.\]

 

Simplest Form of Ratio

If there is no common factor between the prior and post terms of a ratio, then the ratio is in its simplest form. e.g., Simplest form of 24 : 36 is 2 : 3.

If each term of ratio is multiplied or divided by a non-zero number, then the ratio remains unaltered.

i.e.,       \[a:b=am:bm;\,\,\,\,a:b=\frac{a}{m}:\frac{b}{m}\]

 

Types of Ratio

Compound Ratio If \[\frac{a}{b},\]\[\frac{c}{d}\]and \[\frac{e}{f}\] are three ratios, then their compound ratio is \[\frac{a\times c\times e}{b\times d\times f}.\]

Duplicate Ratio If \[a:b\] is a ratio, then its duplicate ratio will be \[{{a}^{2}}:{{b}^{2}}.\]

 

Sub-duplicate Ratio If \[a:b\]is a ratio, then its sub-duplicate ratio will be \[\sqrt{a}:\sqrt{b}.\]

 

Triplicate Ratio If a : b is a ratio, then its triplicate ratio will be \[{{a}^{3}}:{{b}^{3}}.\]

 

Sub-triplicate Ratio If a : b is a ratio, then its sub-triplicate ratio is \[\sqrt[3]{a}:\sqrt[3]{b}.\]

 

Inverse Ratio or Reciprocal Ratio If y : x is a given ratio, then its inverse ratio will be x : y.

If the ratio between P, Q and R is a : b : c, then

P = ak; Q = bk; R = ck, where k is the common ratio.

 

PROPORTION

A proportion is a statement that two ratios are equal. Thus it involves four quantities. These four quantities are said to be in proportion when the ratio of the first two is equal to the ratio of the last two. e.g., 2, 3, 6 and 9 are in proportion. as \[\frac{2}{3}=\frac{6}{9}.\] This proportion written as 2 : 3 :: 6 : 9 where :: denotes proportion i.e., 2 : 3 = 6 : 9.

 

Fourth Proportional

If four numbers or quantities a, b, c and d are in proportion, then d is known as the fourth proportional of a, b, c and d will be calculated as below a : b :: c : d

\[\Rightarrow \]   a : b = c : d

\[\Rightarrow \]   \[a\times d=b\times c\]

\[\Rightarrow \]   \[d=\frac{dc}{a}\]

 

Continued Proportion

The non-zero numbers a, b and c will be in continued proportion, if \[\frac{a}{b}=\frac{b}{c}\]

i.e.,       \[{{b}^{2}}=ac\]

Here, b is known as mean proportional and c is known as third proportional.

 

Basic Operations on Proportion

If four non-zero quantities a, b, c and d are in proportion i.e.,            \[a:b::c:d,\]then

(i) Invertendo \[=b:a::d:c\]

(ii) Alternendo \[=a:c::b:d\]

(iii) Componendo \[=(a+b):b::(x+d):d\]

(iv) Dividendo \[=(a-b):b::(c-d):d\]

(v) Componendo and dividendo

            \[=(a+b):(a-b)::(c+d):(c-d)\]

 

Quicker One

Ø     If A : B = a : b, B : C = m : n, then A : B : C

Ø     If A : B = a : b, B : C =m : n and C : D = r : s, then A : B : C : D is equal to amr : bmr : bmr : bns.

Ø     If a quantity P is distributed in the ratio a : b : c, then First part \[=\frac{a}{a+b+c}\times p;\] Second part \[=\frac{b}{a+b+c}\times p;\] third part \[=\frac{c}{a+b+c}\times p\]

Ø     If two numbers are in the ratio a : b and on adding x to each term, the ratio become c : d, then the numbers will be \[\frac{xa\,(c-d)}{ad-bc}\] and \[\frac{xb\,(c-d)}{ad-bc}\] and sum and difference is \[\frac{x(a+b)(c-d)}{ad-bc}\] and sum add difference is \[\frac{x(a+b)(c-d)}{ad-bc}\] and \[\frac{x(a-b)(c-d)}{ad-bc},\] respectively.