RATIO AND PROPORTION
The ratio of two quantities of the same units is a fraction that one quantity is of the other.
The ratio a : b represents a fraction a/b . The first term of a ratio is called antecedent while the second term is known as consequent. Thus, the ratio 5 : 7 represents 5/7 with antecedent 5 änd consequent 7.
The multiplication or division of each term of a ratio by a same non-zero number does not affect the ratio.
Thus 4:5 = 8:10 = 12:15 = 16:20 etc.
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.
In a proportion, the first and fourth terms are known as extremes, while second and third terms are known as means.
We have, Product of Means • Product of Extremes
Fourth Proportional: If a: b = c: d, then d is called the fourth proportional to a, b, c.
Third Proportional: The third proportional to a, b is the fourth proportional to a, b, b.
Mean Proportional : Mean proportional between a and b is √ab.
Comparison of Ratios : We say that (a: b) > (c : d) if a / b > c / d
Compounded Ratio : The compounded ratio of the ratios (a : b), (c : d), (e : f) is (ace : bdf )
Some More Definitions:
(i) a2 : b2 is called the duplicate ratio of a: b.
(ii) √a :√b is called sub-duplicate ratio of a:b.
(iii) a³ : b³ is called triplicate ratio of a:b.
(iv) If a/b=c/d then a+b/a-b=c+d/c-d (Componendo & dividendo)
Variation : We say that x is directly proportional to y
if x = ky for some constant k and we write, x y,
Also, we say that x is inversely proportional to y,
if x=k/y for some constant k and we write x is inversely propotional to y.