The ratio of two quantities a and b in the same units, is the fraction a/b 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 represent 5/9 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.
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.
a : b = c : d, then c is called the third proportion to a and b.
Mean proportional between a and b is √(ab).
4. Comparison of Ratios:
We say that (a : b) > (c : d) ⇔ a/b > c/d .
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 (a1/3 : b1/3).
If a/b = c/d , then (a + b)/( a - b) = (c + d)/( c - d). [componendo and dividendo]
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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