Ratio and Proportion

  1. Proportion: The equality of two ratios id called proportion.

If a : b : : c : d we read it as a, b, c, d are in proportion.

Here a and b are called extremes while b and c are called mean terms.

Product of means = Product of extremes.

Thus,  a : b : : c : d => (b * c) = (a * d)

  1. Duplicate ratio of (a : b ) is (a2 : b2)

Sub-duplicate ratio of (a : b ) is (a1/2 : b1/2).

  1. 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)

  1. Suppose a container contains x units of liquid from which y units are taken out and replaced by water. After n  operations, the quantity of pure liquid in the final mixture = [x{1-y/x}n] units.
  2. Ratio: 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 as the second term or consequent.

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

Rule: The multiplication or division of each term of a ratio by same non-zero number does not affect the ratio.

Ex. 4 : 5 = 8 : 10 = 12 : 15 etc. also, 4 : 6 = 2 : 3.

  1. Comparison of Ratios: We say that a : b > c: d = a /b > c/d.

Compounded Ratio: The compounded ratio of the ratios (a : b), (c : d), (e : f) = (ace :bdf)

  1. a. Fourth Proportional :  If a : b = c: d then d is called the fourth proportional to a, b, c.

b. Third Proportional: If a : b = b : c then c is called the third proportional to  a and b.

c. Mean Proportional: Mean proportional between a and b is (ab)1/2

Ratio and Proportion Topic

Formulas

Examples

Exercise 1