Ratio and Proportion

Ratio :

a/b or a : b

a → antecedent

b → consequent

• If the ratio between P, Q and R is a : b : c then P = ak, Q = bk and R = ck where, k = constant
• If x is to be divided in a : b : c then, 1st part = ax/(a+b+c), 2nd part = bx/(a+b+c) and 3rd part   = cx/(a+b+c)

Proportion : 

Equality of two ratios is called proportion and a : b = c : d is written as a : b :: c : d.

• If a : b :: b : c then, b² = ac and quantity c = b²/a is called 3rd proportion of a and b.
• If a : b :: c : d then, ad = bc and d is called 4th proportion of a, b and c.
• If a : b :: b : c then, b² = ac and b is called the mean of a and c.

Important points :

1. If a : b = c : d or a/b = c/d, then

Componendo Rule :  (a + b)/b = (c + d)/d

Dividendo Rule :       (a - b)/b = (c - d)/d

Componendo and Dividendo Rule :

                                   (a + b)/(a - b) = (c + d)/(c - d)

2. Income of two persons are in ratio → a : b

    Their expenditure ratio                  →  c : d

If each of them saves $x, 

then their incomes are given by x(d - c)a/(ad - bc) and x(d - c)b/(ad - bc) respectively

and expenditures are x(b - a)c/(ad - bc) and x(b - a)d/(ad - bc) respectively.