Ratio And Proportion


 1.   In ratio a : b (or   a  ), a (numerator) is called antecedent and b (denominator) is called consequent.
b

 2.   Qualities to be expressed as ratio should have same units (like both should be in cm or in kg, but not one in kg and other in gram).

 3.   Ratio has no units.

 4.   Value of ratio (a : b) is not changed when both a and b are multiplied (or divide) by same number.

 5.   a/b   =   a   ×   d
c/d b c

 6.   Compounded ratio is result of multiplication of two or more ratios.

 7.   Duplicate ratio of (a : b) = (  a  )2.
b

 8.   Sub-Duplicate ratio of (a : b) = (  a  )1/2.
b

 9.   Triplicate ratio of (a : b) = (  a  )3.
b

10.   Sub-triplicate ratio of (a : b) = (  a  )1/3.
b

11.   a   =   (c + a.n)  , if and only if,   a   =   c
b (d + b.n) b d

12.   (a + c)   >   a  , if,   a   <   c
(b + d) b b d

13.   (a + c)   <   a  , if,   a   >   c
(b + d) b b d

14.   If   a   >  1,  then,   (a + x)   <   a
b (b + x) b

15.   If   a   <  1,  then,   (a + x)   >   a
b (b + x) b

16.   If   a   >  1,  then,   (a - x)   >   a
b (b - x) b

17.   If   a   <  1,  then,   (a - x)   <   a
b (b - x) b

18.   If      a : b = a1:b1
     and   b : c = b2:c1
     then,  a : b : c = (a1.b2) : (b2.b1) : (b1.c1)

19.   If      a : b = a1:b1
     and   b : c = b2:c1
     and   c : d = c2:d1
     then,  a : b : c : d = (a1.b2.c2) : (b2.b1.c2) : (b1.c1.c2) : (b1.c1.d1)

20.   Proportion is equality of two ratios, i.e. a : b :: c : d. a and d are called extremes and b and c are called means.
   Here, a is 1st proportional, b is 2nd proportional,
             c is 3rd proportional, and d is 4th proportional.

21.   Product of extremes = Product of means, i.e. a × d = c × d

22.   If a : b = b : c then a, b, c are said to be in (continued) proportion.

23.   If a : b = b : c, then b2 = a × c. Here, b is said to be mean proportional between a and cand c is called third proportional.

24.   Invertendo: 
      If,   a   =   c  , then,   b   =   d
b d a c

25.   Alternando: 
      If,   a   =   c  , then,   a   =   b
b d c d
26.   Componendo: 
      If,   a   =   c  , then,   (a + b)   =   (c + d)
b d b d

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

28.   Componendo and Dividendo: 
      If,   a   =   c  , then,   (a + b)   =   (c + d)
b d (a - b) (c - d)

29.   Direct Variation:   If X ∝ Y, then
      X = k.Y,   or,       X   =  k
Y
      Where, k is constant of proportionality
      Also,     X1   =   X2
Y1 Y2

30.   Inverse Variation:
      If    X ∝   1
Y
     then,     X  =   k ,   or,      X.Y  =  k
Y
      Where, k is constant of proportionality
      Also,    X1.Y1  =  X2.Y2,    or,     X1   =   Y2
X2 Y1