393 - identities and examples on Ratios

 393 - identities and examples on Ratios

 

              Example on Ratios

 

* The ratios of two fractions is expressed as a ratio of

two integers in the following way :

 

a:b :: c:d => a*d :b*c

 

example:

1:2 : 3:5

= (1/2) / (3/5)

=(1/2) * (5/3)

= 5/6

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* Ratios are compounded :

Ratios are compounded together by multiplying

together the fractions which denote those fractions

or by multiplying together he antecedants for a

new antecedent and the consequents for a new

Consequent

 

example:

the compounded ration of three ratios

2a:3b , 6ab:5c^2 , c:a

= 2a * 6ab * c / 3b * 5c^2 * a

= 4a / 5c

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* when a ratio a:b is compounded with

itself the resulting ratio is a^2 : b^2 and it is called as

the duplicate ratio of itself ,

 

similarly ,the ration a^3:b^3 is called the triplicate ratio of a:b .

Similary , a^0.5 :b^0.5 is called the sub-duplicate ratio of a:b

 

example:

1) the duplicate ratio of 1:2 is 1:4

2) the duplicate ratio of 2a:3b is 4a^2 : 9b^2

3) the sub-duplicate ratio of 49:25 is 7:5

4) the triplicate ratio of 2x:4y is 8x^3:64y^3

5) the sub-triplicate ratio of 125:27 is 5:3

 

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* A ratio of greater inequality is decreased and a ratio

of less inequality is increased by adding the same

quantity to both its terms

example:

Let a/b be the ration and let

(a+x)/(b+x) be the new ratio formed by adding x to

both the numerator and the denominator

Now ,

a/b - (a+x) /(b+x)

= (ax - bx ) / (b(b+x))

= x(a-b) / b (b+x)

 

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depending upon the value of a and b , the ratio can be said

to be of higher or lower inequality .

 

If , a > b , then

a/b > (a+x)/(b+x)

and if a< b ,then

a/b < (a+x)/(b+x)

 

for example :

lets consider ratio a:b as 3:8 , and x as 4 ,then

3/8

=> (3+4) / (8+4)

=> 7/12

finding the lcm of their denominators and doing the

comparision:

3/8 = 9/24

7/12 = 14/ 24

 

so here we see that by adding x to both sides of the

terms of a and b increases the ratio

 

lets take another example :

a:b = 5:9 and x be 2

5:9 = 5/9

=> subtracting x from both numerator and denominator , we get

=> (5-2) /(9-2)

=> (5-2) /(9-2)

=> 3/7

 

finding the lcm of the denominators and doing the

comparision of the previous and the new ratio :

5/9 = 35/63

3/7 = 21/63

 

so here , we find that the ratio has been decreased by

subtracting x from both the terms of the ratio

 

This proves the proposition of the given

inequality conditions in the statement