Averages & Mixtures — Quant Aptitude Questions & Formulas

Quant Aptitude · choose a topic

Averages & Mixtures

1. Average Basics

\text{Average}=\frac{\text{Sum of observations}}{\text{Number of observations}}

Adding a constant

k

to every term raises the average by

k

; multiplying every term by

k

multiplies it by

k

Replacement: when one member leaves and another joins,

\text{change in average}=\dfrac{\text{new value}-\text{old value}}{n}

2. Weighted Average

\bar{x}=\frac{n_1x_1+n_2x_2+\cdots}{n_1+n_2+\cdots}

The average of two groups always lies between the two group averages, nearer the larger group.

3. Alligation - the Mixture Workhorse

\frac{\text{Quantity of cheaper}}{\text{Quantity of dearer}}=\frac{\text{Dearer price}-\text{Mean price}}{\text{Mean price}-\text{Cheaper price}}

Read it as a cross: the distance from the mean on the far side gives the quantity on this side.

4. Repeated Replacement

x

units are drawn from

V

units of pure liquid and replaced by water,

n

times over:

\text{Liquid left}=V\left(1-\frac{x}{V}\right)^{n}

5. Common Patterns

Average speed for equal distances at

u

and

v

\dfrac{2uv}{u+v}

- never