CAT Quant : Averages and Alligations – Important Concepts and Formulas


The average of a number is a measure of central tendency of a set of numbers. $$Average = \frac{Sum \ of \ value \ of \ all \ items}{Number \ of \ items \ in \ group}$$ $$A_n = \frac{x_1 + x_2 + x_3 + ….. + x_n}{n}$$ $$ A_w \ [weighted] = \frac{x_1n_1 + x-2n_2 + x_3n_3 + ….. +x_kn_k}{n_1 + n_2 + n_3 + …. + n_k}$$

Arithmetic Mean

  • Average is also called “Mean” or “Mean Value”
  • Arithmetic Mean is defined as the number by which we can replace each and every number of the set without changing the total of the set of numbers.
  • The average will always be greater than the smallest value and less than the largest value in the group
  • If the value of each item is increased or decreases by the same value p, then average of the group or items will also increase or decrease by the same value, p respectively.
  • If value of each item is multiplied or divided by the same value p, then the average of the group will also get multiplied or divided by the same value p respectively.
  • The net deficit due to the numbers below the average always equals the net surplus due to the numbers above the average.
  • If the average age of a group of persons is x-years today then after n years their age will be (x+n) years. Also n years ago, their average age would have been (x-n).
  • If there are n items and they are denoted by Q1, Q2, Q3, … Qn then the average of these items is given by – $$A = P + \frac{1}{n}\sum_{i=1}^{n} (Q_i – P)$$ Note : In the given equation, (Qi – P) is the deviation.
  • For an AP series x1, x2, x3, … xn-2, xn-1, xn ; the average is the average of equidistant terms from end. $$A(\bar{x}) = \frac{x_1 + x_2}{2} = \frac{x_2 \ – \ x_{n-1}}{2}$$

Weighted Average

  • The average where a weight is assigned to each of the values that are needed to be averaged is termed as Weighted Average.
  • The formula for calculating weighted average is as follows – $$\bar{x} = \frac{f_1x_1 + f_2x_2 + f_3x_3 + …..f_nx_n}{f_1 + f_2 + f_3 + ….f_n}$$ where, f1, f2, f3, …fn are the corresponding weights of each value x1, x2, x3, …xn respectively.
  • Note – In the equation for weighted averages $$x_1 \leq \bar{x} \leq x_2$$ $$ if \ f_1 > f_2 \longrightarrow weighted \ average \ will \ be \ closer \ to \ x_1 \\ if \ f_2 > f_1 \longrightarrow weighted \ average \ will \ be \ closer \ to \ x_2$$
  • Instead of taking actual values we can also take ratio of the weightages.
  • Methods of deviations can also be used.


  • Mixing of two or more qualities of things produces a mixture.
  • Mixtures is basically application of averages and weighted averages.
  • If x1 is the quantity (number of items) of one particular item of quality A1, and x2 is the quantity (number of items) of the second item of quality A2, and then both are mixed together to give a new mixture. The Weighted average value (p) of the quality of mixture is given by – $$p = \frac{x_1A_1 + x_2A_2}{A_1 + A_2}$$
  • If there are more than two groups of items mixed, the weighted average formula can be applied in a similar manner.
  • A mixture can also be a solution (a liquid mixed with another liquid). The concentration of the solution is expressed as the proportion (or percentage) of the liquid in the total solution.


  • Alligations is nothing but a faster technique of solving questions based on the weighted average situation. $$\frac{n_1}{n_2} = \frac{A_2 – A_w}{A_w – A_1} \\ A_w = \frac{n_1A_1 + n_2A_2}{n_1 + n_2}$$
  • This is called the rule of Alligation, This rule connects quantities and prices in mixtures. Also written as $$\frac{Quantity \ of \ Cheaper}{Quantity \ of \ Dearer} =\frac{Rate \ of \ Dearer – Average \ Rate}{Average \ Rate – Rate \ of \ Cheaper}$$
  • Graphical Representation –
    • A1, A2 and Aw are always rate units, while n1 and n2 are quantity units.
here, A1 < Aw
  • Special Case –
    • If there is P volume of pure liquid initially and in each operation, Q volume is taken out and replaced by Q volume of water, then at the end of n such operations, concentration (k) of the liquid in the solution is $$k = {\LARGE[} \frac{P – Q}{P} {\LARGE]}^n$$
    • This gives the concentration (k) of the liquid as a “proportion” of the total volume of solution. $$P \ left = Capacity * [1 – fraction \ of \ P \ withdrawn]^n \ for \ n \ operation$$
    • If the volume of the liquid is to be found out at the end of n operations, it is given by “kP” i.e. the concentration (k) multiplied by total volume of solution P.

Scroll to Top