Ratio and Proportion

     I. 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, 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 the same non-zero number does not affect the ratio.

Ex.  4: 5 = 8: 10 = 12: 15 etc. 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. (i) Fourth Proportional: If a : b = c: d, then d is called the fourth proportional  to a, b, c.

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

   (iii) Mean Proportional: Mean proportional between a and b is square root of ab

4. (i) COMPARISON OF RATIOS:

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

        (ii) COMPOUNDED RATIO:

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

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

      (ii) Sub-duplicate ratio of (a : b) is (√a : √b).

     (iii)Triplicate ratio of (a : b) is (^a3 : ^b3).

     (iv) Sub-triplicate ratio of (a : b) is (a ⅓ : b ⅓ ).

     (v) If (a/b)=(c/d), then  ((a+b)/(a-b))=((c+d)/(c-d))    (Componendo and dividendo)

6. VARIATION:

(i) We say that x is directly proportional to y, if x = ky  for some constant k and

     we write, x µ y.

(ii) We say that x is inversely proportional to y, if xy = k for some constant k and

       we write, x∞(1/y)