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

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).
 
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.
 
Comparison of Ratios:
 
We say that (a : b) > (c : d) .
 
Compounded Ratio:
 
The compounded ratio of the ratios: (a : b), (c : d), (e : f) is (ace : bdf).
 
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 (a1/3 : b1/3).
 
If , then . [componendo and dividendo]
 
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,




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