CAT Quant: Ratio, Proportion and Variation – Important Formulas and Concepts

Ratio

  • Ratio is the relation one quantity bears to another of the same kind. The ratio of two quantities “a” and “b” is represented as a:b and read as “a is to b”. Here “a” is called “antecedent” and “b” is the “consequent”.
  • If we multiply or divide the numerator and denominator of a ratio by the same number, the ratio remains unchanged. $$\frac{a}{b} =\frac{ma}{mb} \ {\Large or } \ \frac{a}{b} =\frac{a/m}{b/m}$$
  • The value of a given ratio has –
    • direct relationship with value of numerator
    • inverse relationship with value of denominator
  • The ratio of two fractions can be expressed as a ratio of two integers. $$\frac{a}{b} : \frac{c}{d} = \frac{a/b}{c/d} =\frac{ad}{bc}$$
  • If either or both the terms of a ratio are a surd quantity , then the ratio will never evolve integral numbers unless the surd quantities are equal.
  • If a/b = c/d = e/f = g/f = k then, $$k = \frac{a+c+e+g}{b+d+f+h}$$
  • If a1/b1 , a2/b2 , …….. an/bn are unequal fractions then, the given value below lies between the lowest and the highest of these fractions. $$\frac{a_1+a_2+a_3+…..a_n}{b_1+b_2+b_3+…..b_n}$$
  • If we have two equations containing 3 unknowns as – $$a_1x + b_1y + c_1z = 0 \ , \ a_2x + b_2y + c_2z = 0$$ The value of x, y, z cannot be resolved without 3rd equation. However, in absence of a 3rd equation, we can find the proportion x:y:z by $$b_1c_2 – b_2c_1 : c_1a_2 – c_2a_1 : a_1b_2 – a_2b_1$$
  • A ratio is said to be a ratio of greater or less inequation or equality in accordance to whether the antecedent is greater than, less than or equal to consequent.
    • ratio A : B where A>B – Ratio of greater inequality.
    • ratio A : B where A<B – Ratio of less inequality.
    • ratio A : B where A=B – Ratio of equality.
  • In a given ratio –
    • a/b > 1 (ratio of greater inequality) and k is a positive number, then –
      • (a + b)/(b + k) < a/b
      • (a – k)/(b – k) > a/b
    • a/b < 1 (ratio of lesser inequality) and k is a positive number
      • (a +k)/(b +k) > a/b
      • (a – k)/(b – k) < a/b

Proportion

  • When two ratios are equal, the four quantities composing them are said to be proportional. The below proportional values are read as “a is to b as c is to d” $$\frac{a}{b} = \frac{c}{d} \ \rightarrow \ a:b :: c:d$$
  • Thus a, b, c, d are proportional here. The terms “a and d” are called the Extremes and terms “c and b” are called the Means
  • If four quantities are in proportion, then- $$Product \ of \ Extremes = Product \ of \ Means \\ ( “ad” = “bc” )$$
  • Continued Proportion – If three quantities a, b and c are in continued proportion, then a : b = b : c $$ac = b^2$$
    • “b” is said to be a Mean Proportion between a and c
    • “c” is said to be a Third Proportion between a and
  • Componendo – Dividendo – If four quantities are proportional, i.e $$\frac{a}{b} = \frac{c}{d} \ \Longrightarrow \ \frac{a+b}{a-b} =\frac{c+d}{c-d}$$
    • The converse is also true, i.e $$\frac{a+b}{a-b} =\frac{c+d}{c-d} \ \Longrightarrow \ \frac{a}{b} = \frac{c}{d}$$
    • Also another application includes – $$\frac{a}{b} = \frac{c}{d} = \frac{e}{f} …… \ \Longrightarrow \ \frac{a+c+e+….}{b+d+f+….}$$
  • Duplicate Ratio – If three quantities are proportional the first is to the third is the duplicate ratio of the first to the second.
    • a : b = a2 : b2

Variation

  • When two quantities are such that as one quality changes in value, the other quantity also changes in value bearing certain relationship to the changes in value of the first quantity.
  • Direct Variation (Constant Ratio) – When “A varies directly as B” [A Symbol B) it means –
    • Logical Implication – When A increases, B increases. When A decreases, B decreases.
    • Calculation Implication – If A increases by 10%, B will also increase by 10%.
    • Equation Implication – The ratio A/B is constant. Conversely, when ratio of two quantities is constant, we conclude they vary directly.
    • If A (symbol) B, then A = kB, where K is a constant, it is called the constant of proprtionality.
  • Inverse Variation (Constant Product) – When “A varies inversely as B” [A symbol B] the following implications arise –
    • Logical Implication – When A increases, B decreases. When A decreases, B increases.
    • Calculation Implication – If A decreases by 9.09%, then B will increases by 10%
    • Equation Implication – The product A*B is constant. Conversely, when product AB is constant, we conclude they vary inversely.
    • If A (symbol) B, then A = k/B where K is a constant and is called the constant of proportionality.
  • Joint Variation – If there are 3 quantities A, B and C such that –
    • “A” varies with “B” when “C” is constant.
    • “A” varies with “C” when “B” is constant.
    • Then A is said to vary jointly with B and C when both B and C are varying.
    • Examples – $$1) \ A \propto B \ , \ A \propto C \\ \Rightarrow \ A \propto BC \\ \Rightarrow \ A = kBC $$ $$1) \ A \propto B \ , \ A \propto \frac{1}{C} \\ \Rightarrow \ A \propto \frac{B}{C} \\ \Rightarrow \ A = \frac{kB}{C} $$

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