# CAT Quant : Averages and Alligations – Important Concepts and Formulas

## Averages

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 Q
_{1}, Q_{2}, Q_{3}, … Q_{n}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, (Q_{i}– P) is the deviation. - For an AP series x
_{1}, x_{2}, x_{3}, … x_{n-2}, x_{n-1}, x_{n}; 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, f
_{1}, f_{2}, f_{3}, …f_{n}are the corresponding weights of each value x_{1}, x_{2}, x_{3}, …x_{n}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.

## Mixtures

- Mixing of two or more qualities of things produces a mixture.
- Mixtures is basically application of averages and weighted averages.
- If x
_{1}is the quantity (number of items) of one particular item of quality A_{1}, and x_{2}is the quantity (number of items) of the second item of quality A_{2}, 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

- 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 –**- A
_{1}, A_{2}and A_{w}are always rate units, while n_{1}and n_{2}are quantity units.

- A

**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.