**RATIO AND PROPORTION**

*** SOME IMPORTANT SUTRAS**

1). If the ratio of any three quantities P , Q and R is a:b:c , then P = aK , Q = bK and R = cK.

Where K is any constant.

2). Inverse ratio of x , y and z is = 1/x : 1/y : 1/z =

yz : zx : xy

3). If x is to be divided in the ratio of a:b:c , then first part = ax/a+b+c , second part = bx/a+b+c and third part = cx/a+b+c.

4). If x is the mean proportion between numbers a and b , then x = √(ab).

5). If x and y are the two numbers then their duplicate ratio is = x² : y²

6). If x and y is the numbers then their sub duplicate ratio is = √x : √y

7). If x and y is the two numbers then their triplicate ratio is = x³ : y³

8). If x and y is the two numbers then their sub triplicate ratio = x^1/3 : y^1/3

9). If x and y is the two numbers and their ratio is x:y , then their inverse ratio is = y:x

10). If a:b , c:d and e:f are the three ratio then their compound ratio is = ace : bed

11). If the antecedent is greater than the consequent , then the ratio is known as ratio of greater inequality.

12). If the antecedent is smaller than the consequent then the ratio is known as the ratio of less inequality.

13). If x is the third proportional to a and b then,x = b^2/a

14). If x is the fourth proportional to a , b and c then , x = bc/a

15). If a:b = c:d then their inverse ratio is also equal that is = (b:a = d:c)

16). If (a/b) = (c/d) , that is , a:b = c:d , then

* Invertendo = (b/a) = (d/c)

* Alternendo = (a/c) = (b/d)

* Componendo = a+b/b = c+d/d

* Dividendo = a-b/b = c-d/d

* Componendo and dividendo =

(a+b/a-b) = (c+d/c-d)

17). If a/b = c/d = i/j = k, then

a+c+i+ …./b+d+j+… = K

18). If A:B = a:b and B:C = m:n then

A:B:C =am:mb:nb and A: C = am : nb

19). If A:B = a:b , B:C = c:d and C:D = e:f then

A:B:C:D = ace:bce:bde:bdf

20). If A is to be divided into ratio a:b , then after division , the difference between the first part and the second part is = (a-b/a+b).A

21). If sum of two numbers is A and their differences is a , then the ratio of the numbers is=

A+a/A-a

22). If the ratio of diagonals any two square is a:b, then the ratio of their areas = a² : b²

23). If the ratio of any two numbers is a:b , if each number is increased by x then their ratio become c:d , then the sum of two numbers is =

x(a+b)(c-d)/(ad-bc)

*Difference between the two numbers is =

x(a-b)(c-d)/(ad-bc)

* The two numbers is = xa(c-d)/(ad-bc) and

xb(c-d)/(ad-bc)

24). If the ratio of income of two persons is a:b , and the ratio of their expenditure is c:d , if each saves X, then-

* Their income = xa(d-c)/(ad-bc) and

xb(d-c)/(ad-bc)

* Their expenditure is = xc(b-a)/(ad-bc) and

xd(b-a)/(ad-bc)

25). . If the ratio of any two numbers is a:b , if each number is decreased by x then their ratio become c:d , then the two numbers is=

xa(c-d)/(ad-bc) and xb(c-d)/(ad-bc)