34 days to CAT

Lesson 15 min· Section 1 of 4

The straight-up lesson, built for the exam.

Averages

Formula sheet

Everything for this chapter, in one place. $n$ is the count and $A$ the average.

The only definition


Average	$A = \dfrac{\text{sum}}{n}$
The form you will actually use	$\text{sum} = \text{average} \times \text{count}$
Deviations	the gaps above $A$ and the gaps below $A$ cancel exactly

Adding or removing one value


Add a value $v$ to $n$ items	new average $= \dfrac{nA+v}{n+1}$
The shift	$\dfrac{v-A}{n+1}$ — so the average rises if and only if $v > A$
Remove a value $v$	new average $= \dfrac{nA-v}{n-1}$
Replace $v$ with $w$	the total moves by $w-v$ , the average by $\dfrac{w-v}{n}$

Two groups combined


Weighted average	$A = \dfrac{n_1A_1 + n_2A_2}{n_1+n_2}$
Alligation	$\dfrac{n_1}{n_2} = \dfrac{A_2 - A}{A - A_1}$
In words	the combined average sits closer to the bigger group

Shifting and scaling


Add $k$ to every value	the average goes up by $k$
Multiply every value by $k$	the average is multiplied by $k$
Assumed mean	$A = \text{guess} + \text{average of the deviations from the guess}$

Evenly spaced values


Average of an AP	$\dfrac{\text{first}+\text{last}}{2}$ , which is the middle term
First $n$ natural numbers	$\dfrac{n+1}{2}$
$n$ consecutive integers	the middle one when $n$ is odd

Averages that are not plain averages


Average speed	$\dfrac{\text{total distance}}{\text{total time}}$ — never the average of the speeds
Equal distances at two speeds	the harmonic mean $\dfrac{2xy}{x+y}$
Equal money spent at different rates	the average rate is the harmonic mean, not the arithmetic one
Equal quantity bought at different rates	then it is the arithmetic mean

Why this chapter matters

1.3

questions in a typical Quant section

5th

of 25 chapters

≈ 4 of 66 marks a section ride on it.

Asked in23 of 24 papers

20172025

Year by yearper section

2017: 12017: 1 per section
2018: 1.32018: 1.3 per section
2019: 0.72019: 0.7 per section
2020: 1.12020: 1.1 per section
2021: 1.32021: 1.3 per section
2022: 2.32022: 2.3 per section
2023: 12023: 1 per section
2024: 12024: 1 per section
2025: 0.72025: 0.7 per section

Question type32 asked

MCQ 62%Type-in 38%

Type-in answers carry no negative marking.

From every Averages question in CAT papers since 2017.