Ratio and Proportion - Important Formulae

1. Ratio:
   The ratio of two quantities a and b in the same units, is the fraction  and we write it as a : b.
   In the ratio a : b, we call a as the first term or antecedent and b, the second term orconsequent.

   Eg. The ratio 5 : 9 represents	5	with antecedent = 5, consequent = 9.
   	9

   Rule: The multiplication or division of each term of a ratio by the same non-zero number does not affect the ratio.
   Eg. 4 : 5 = 8 : 10 = 12 : 15. Also, 4 : 6 = 2 : 3.
2. Proportion:
   The equality of two ratios is called proportion.
   If a : b = c : d, we write a : b :: c : d and we say that a, b, c, d are in proportion.
   Here a and d are called extremes, while b and c are called mean terms.
   Product of means = Product of extremes.
   Thus, a : b :: c : d  (b x c) = (a x d).
3. Fourth Proportional:
   If a : b = c : d, then d is called the fourth proportional to a, b, c.
   Third Proportional:
   a : b = c : d, then c is called the third proportion to a and b.
   Mean Proportional:
   Mean proportional between a and b is ab.
4. Comparison of Ratios:

   We say that (a : b) > (c : d)	a	>	c	.
   	b		d

   Compounded Ratio:
   The compounded ratio of the ratios: (a : b), (c : d), (e : f) is (ace : bdf).
5. Duplicate Ratios:
   Duplicate ratio of (a : b) is (a² : b²).
   Sub-duplicate ratio of (a : b) is (a : b).
   Triplicate ratio of (a : b) is (a³ : b³).
   Sub-triplicate ratio of (a : b) is (a^1/3 : b^1/3).

   If	a	=	c	, then	a + b	=	c + d	.     [componendo and dividendo]
   	b		d		a - b		c - d

6. Variations:
   We say that x is directly proportional to y, if x = ky for some constant k and we write,x  y.
   We say that x is inversely proportional to y, if xy = k for some constant k and

   we write, x	1	.
   	y

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