PROPERTIES OF PROPORTION

Property 1 : 

An equality of two ratios is called a proportion.

Four quantities a, b, c, d are said to be in proportion if

a : b = c : d

(also written as a : b :: c : d)

That is, if a/b = c/d.

Property 2 :

The quantities a, b, c, d are called terms of the proportion; a, b, c and d are called its first, second, third and fourth terms respectively.

First and fourth terms are called extremes (or extreme terms). Second and third terms are called means (or middle terms). 

Property 3 :

In a proportion,

product of extremes = product of means.

If a : b = c : d are in proportion, ad  =  bc.

This is called cross product rule.

Property 4 : 

Three quantities a, b, c of the same kind (in same units) are said to be in continuous proportion.

Now, we can write a, b, c in proportion as given below. 

a : b  =  b : c

Using cross product rule, we have b²  =  ac

Property 5 :

If a, b, c are in continuous proportion, then the middle term b is called the mean proportional between a and c, a is the first proportional and c is the third proportional.

Thus, if b is mean proportional between a and c, then

b²  =  ac

b  =  √ac

Property 6 :

In a proportion a : b = c : d, all the four quantities need not be of the same type. The first two quantities should be of the same kind and last two quantities should be of the same kind.

Property 7 :

Invertendo :

If a : b = c : d, then b : a = d : c

Property 8 :

Alternendo :

If a : b = c : d, then a : c = b : d

Property 9 :

Componendo :

If a : b = c : d, then (a+b) : b  =  (c+d) : d

Property 10 : 

Dividendo :

If a : b = c : d, then (a-b) : b  =  (c-d) : d

Property 11 :

Componendo and Dividendo :

If a : b = c : d, then (a+b) : (a-b)  =  (c+d) : (c-d)

Property 12 : 

Addendo :

If a : b = c : d = e : f =.........., then each of these ratios is equal

(a + c + e + ........) : (b + d + f + ........)

Property 13 :

Subtrahendo :

If a : b = c : d = e : f =.........., then each of these ratios is equal

(a - c - e - ........) : (b - d - f - ........)

Solving Problems Using Properties of Proportion

Problem 1 :

If a : b = c : d = 2.5 : 1.5, what are the values of ad : bc and a + c  :  b + d ?

Solution : 

In the given proportion a : b and c : d, applying cross product rule, we get

ad  =  bc

Dividing by bc on both sides, we get  

ad/bc  =  1

ad/bc  =  1/1

ad : bc  =  1 : 1

Given : a : b  =  c : d  =  2.5 : 1.5 ------ (1)

In the given proportion a : b and c : d, applying the property addendo, we get

a : b  =  c : d  =  (a+b) : (c+d) ------ (2)

From (1) and (2), we get  

(a+b) : (c+d)  =  2.5 : 1.5

(a+b) : (c+d)  =  (2.5x10) : (1.5x10)

(a+b) : (c+d)  =  25 : 15

(a+b) : (c+d)  =  (25/5) : (15/5)

(a+b) : (c+d)  =  5 : 3

Problem 2 :

If a : 3  =  b : 4  =  c : 7, then, find the value of (a+b+c) : c

Solution : 

In the given proportion a : 3  =  b : 4  =  c : 7, applying the property addendo, we get

a : 3  =  b : 4  =  c : 7  =  (a + b + c) : (3 + 4 + 7)

a : 3  =  b : 4  =  c : 7  =  (a + b + c) : 14

Now, all the above four ratios are equal.

Taking the last two ratios, we get

c : 7  =  (a + b + c) : 14

c / 7  =   (a + b + c) / 14

14 / 7  =  (a + b + c) / c

2  =  (a + b + c) / c

or

 (a + b + c) / c  =  2/1

(a + b + c) : c  =  2 : 1

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