Chapter 03 : Ratio and Proportion > Ratio

Ratio and proportion is the continuation of the basic concepts for the entrance tests. The application of the topics is far and wide, a direct application which will be covered here is “Allegation and Mixtures”.

Ratio:

The comparison of two quantities of the same unit is the ratio of one quantity to another. The ratio of A and B is written as A: B or A / B, where A is called the antecedent and B the consequent.

Example: The ratio of 10 kg to 20 kg is 10:20 or 10/20, which is 1:2 or ½, where 1 is called the antecedent and 2 the consequent.

Properties of Ratio

1. a : b = ma : mb, where m is a constant

2. a : b : c = A : B : C is equivalent to a / A = b /B = c /C, this is an important property and has to be used in ratio of three things.

3. If a / b = c / d , then

(a+b) / b=(c+d) / d – Componendo
Example: ½ = 2/4, so (1+2)/2 = (2+4)/4 => 3/2 = 6/4 => 3/2 = 3/2

(a-b) / b = (c-d) / d – Dividendo
Example: 10/4 = 20/8, so (10-4)/4 = (20-8)/8 => 6/4 = 12/8 => 3/2 = 3/2

(a+b) / (a-b) = (c+d) / (c-d) – Componendo and Dividendo
Example: 10/4 = 20/8, so (10+4)/(10-4) = (20+8)/(20-8) => 14/6 = 28/12 => 7/3 = 7/3

Application: These properties have to be used with quick mental calculations; one has to see a ratio and quickly get to results with mental calculations.
Example: 10/4 = 20/8, should quickly tell us that 14/4(7/2) = 28/8(7/2), 6/4(3/2) = 12/8(3/2) and 14/6(7/3) = 28/12(7/3)

4. If a / b = c / d =e / f = ……, then (a+c+e+…..)/(b+d+f+….) = each of the individual ratio
Example: ½ = 2/4 = 4/8, there fore (1+2+4)/(2+4+8) = 7/14 = ½

5. If A > B then (A+C)/ (B+C) < A/B Where A, B and C are natural numbers
Example: 3>2, then (3+4)/ (2+4) = 7/6 (1.16)

6. If A < B then (A+C)/ (B+C) > A/B Where A, B and C are natural numbers.