# average Topic-wise Practice Test, Examples With Solutions & More Shortcuts

#### average & IT'S TYPES

Useful for Management (CAT, XAT, MAT, CMAT, IIFT, SNAP & other), Bank (PO & Clerk) SSC (CGL, 10+2, Steno, FCI, CPO, Multitasking), LIC (AAO & ADO) CLAT, RRB, UPSC and Other State PSC Exams

#### model 1 basic average questions

Practice test for basic average with questions asked in various ...

#### model 2 average of consecutive numbers

Latest quantitative aptitude average multiple choice questions, fully solved answers ...

#### model 3 twice, thrice, one third etc. of numbers

Average on twice, thrice, one-third of Numbers based multiple choice ...

#### model 4 find nth average from 1st & last number

New average quantitative aptitude multiple choice questions with fully solved ...

#### model 5 find new average from error

Average aptitude MCQs based on finding the new average from ...

#### model 6 find average of excluded number

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#### model 7 average on ages/weight

Practice test for Finding average using total Ages, Weights based ...

#### model 8 find monthly income

Important quantitative aptitude average multiple choice questions, fully solved answers ...

#### model 9 average on cricket/exam

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## Quantitative Aptitude: Average Concepts, Tricks, Formulas with Examples

The concept of average is a basic of arithmetic operations and is important to solve many problems. Especially ‘average’ questions are regularly asked in all competitive exams here.

We covered various types of questions useful for all examinations. The examples which will represent the questions which include finding Basic arithmetic means of numbers/series, Average on age, Average on income, Average on distance, Increase/decrease in group average, minimum/maximum of quantity/number for a specific average. Tabulation based/rate of occurrence-based means will give you a good detailed idea about averages.

Smart working, Full concentration, Accuracy on finding answers, speed of answering, and Regular cross-checking is must for ‘Average’ formulae. Do calculations with care will make your success easier.

### What is an Average?

Average can be defined as a single value that is meant to type a dataset. It gives a measure of the middle or expected value of the dataset. Though there are many measures of central tendency, average typically refers to the arithmetic mean and is defined as the ratio of the sum of items to the number of items in a dataset,

Therefore, Average$=(\text"Total value of all the items"/\text"Number of items")$

On other words:

An average or an arithmetic mean of given data is the sum of the given observations divided by number of observations.

For example, if we have to find out the average of 15, 25, 35 and 45, then the required average will be

$(15+25+35+45)/4=120/4=30$

Similarly, finding the average of 30, 50 and 40

$(30 + 50 + 40)/ 3= 120 / 3 = 40$

For example, let us take heights of the students in a class. If the number of students is, say for example, 5, and their heights are 168, 170, 169, 174 and 166, then the average can be calculated using the above formula as:

$\text"Average"=(168+170+169+174+166)/5=847/5=169.4$

Therefore, we found the formula of Average is,

$\text"Average"=(\text"Sum of observations"/\text"Number of observations")$

### Properties of Average:

(i) Average of a given data is less than the greatest observation and greater than the smallest observation of the given data

For example, Average of 3, 7, 9 and 13

$\text"Average"=\text"3+7+9+13"/4 = 32/4=8$

Clearly, 8 is less than 13 and greater than 3.

(ii) If the observations of given data are equal, then the average will also be the same as observations

For example, Average of 6, 6, 6 and 6 will be 6 because

$(\text"6+6+6+6"/4)=24/4=6$

(iii) If 0 (zero) is one of the observations of a given data, then that 0 (zero) will also be included while calculating average.

For example, Average of 3, 6 and 0 is 3 because

$\text"3+6+0"/3=9/3=3$

• If all the Numbers get increased by a, then their average is also increased by a.
• If all the numbers get decreased by a, then their average is also decreased by a.
• If all the numbers are multiplied by a, then their average is also multiplied by a.
• If all the numbers are divided by a, then their average is also divided by a.

EX 1: Find out the average of 308, 125, 45, 120 and 102

Sol: As we known that,

Average =$\text"Sum of given observations"/\text"Number of observations"$

$=\text"308+125+45+120+102"/5=700/5=140$

Ex 2: If the weight of A is 60 kg, weight of B is 45 kg and weight of C is 54 kg, then what is the average weight of three persons?

$=\text"60+45+54"/3=159/3=53kg$

## Quantitative Aptitude: Average Rule's, Formula's & Shortcut's

Important "27 Rules" for solving all different kind of Average based problems asked in all competitive exams

Rule 1:
Average of two or more numbers/quantities is called the mean of these numbers, which is given by

Average(A) = $\text"Sumof observation/quantities"/\text"No of observation/quantities"$
∴ S = A × n
(or)
Average of numbers = ${x_1 + x_2 + ... + x_n}/n$
or, Average = ${∑ ↙{i = 1} ↖{n}{x_i}}/n$

Rule 2:
If the given observations (x) are occuring with certain frequency (A) then,

Average = ${A_1x_1 + A_2x_2 + A_3x_3 +...+ A_nx_n}/{x_1 + x_2 +...+ x_n}$
where, $A_1, A_2, A_3, ... A_n$ are frequencies

Rule 3:

The average of 'n' consecutive natural numbers starting from 1

i.e. Average of 1,2,3, .....n = ${n +1}/2$

Rule 4:

The average of squares of 'n' consecutive natural numbers starting from 1

i.e. Average of $1^2, 2^2, 3^2, 4^2 ... x^2 = {(n + 1)(2n + 1)}/6$

Rule 5:

The average of cubes of first 'n' consecutive natural numbers

i.e. Average of $1^3, 2^3, 3^3 ... n^3 = {n(n + 1)^2}/4$

Rule 6:

The average of first 'n' consecutive even natural numbers

i.e. Average of 2, 4, 6, ..... 2n = (n + 1)

Rule 7:
The average of first 'n' consecutive odd natural numbers

i.e. Average of 1, 3, 5, ..... (2n – 1) = n

Rule 8:
The average of certain consecutive numbers a, b, c, ......... n is

$a+b+c+d+....+n= {a + n}/2$

Rule 9:
The average of 1st 'n' multiples of certain numbers

x = ${x(l + n)}/2$

Rule 10:
If the average of '$n_1$' numbers is $a_1$ and the verage of '$n_2$' numbers is $a_2$, then average of total numbers $n_1$ and $n_2$ is

Average = ${n_1a_1 + n_2a_2}/{n_1 + n_2}$

Rule 11:
If A goes from P to Q with speed x km/h and returns from Q to P with speed y km/h, then the average speed of total journey is

Average speed = ${2xy}/{x + y} = \text"total distance"/\text"total time taken"$

Rule 12:
If a distance is travelled with three different speeds a km/h, b km/h and c km/h, then

Average speed of total journey = ${3abc}/{ab + bc + ca}$km/h

Rule 13:
If the average of m numbers is x and out of these 'm' numbers the average of n numbers is y. (or vice versa) then the average of remaining numbers will be

1. Average of remaining numbers = ${mx - ny}/{m - n}$ (if m > n)
2. Average of remaining numbers = ${ny - mx}/{n - m}$ (if n > m)

Rule 14:
In three numbers, if 1st number is 'a' times of 2nd number and 'b' times of 3rd number and the average of all three numbers is x, then 1st number

Average = ${3ab}/{a + b + ab}$x

Rule 15:
From three numbers, first number is 'a' times of 2nd number, 2nd number is 'b' times of 3rd number and the average of all three numbers is x, then,

First number = ${3ab}/{1 + b + ab}$x
Second number = ${3b}/{1 + b + ab}$x
Third number = ${3b}/{1 + b + ab}$x

Rule 16:
If from (n + 1) numbers, the average of first n numbers is 'F' and the average of last n numbers is 'L', and the first number is 'f' and the last number is 'l'

then, f – l = n(F – L)

Rule 17:
't' years before, the average age of N members of a family was 'T' years. If during this period 'n' children increased in the family but average age (present) remains same,

then, Present age of n children = n.T – N.t

Rule 18:
If in the group of N persons, a new person comes at the place of a person of 'T' years, so that average age, increases by 't' years

∴ Then, the age of the new person = T + N.t

If the average age decreases by 't' years after entry of new person,

∴ Then, the age of the new person = T – N.t

Rule 19:
The average age of a group of N students is 'T' years.
1. If 'n' students join, the average age of the group increases by 't' years,

then, Average age of new students
$= T + ({N}/n + 1)t$

2. If the average age of the group decreases by 't' years,

then, Average age of new students
= T - $({N}/n + 1)$t

Rule 20:
If the average of 'n' observations is 'x' and from these the average of 1st 'm' observations is 'y' and the average of last 'm' observations is 'z' then,

mth observation = m(y + z) – nx
(m + 1)th observation = nx – m(y + z)

Rule 21:
If the average age (height) of 'n' persons is x year (cms) and from them 'm' persons went out whose average age (height) is 'y' years (cms) and same number of persons joined whose average age (height) is 'z' years (cms) then what is the average age (height) of n persons ?

Average age = $[x - {m(y - z)}/n]$ years (cms).

Rule 22:

Average of bowler = $\text"Total runs"/\text"No of wickets"$
Total runs = Average (A) x y,
where y = Number of wickets.

Rule 23:
If in a group, one member is replaced by a new member, then,

Age of new member = ( age of replaced member) ± xn

where, x = increase (+) or decrease (–) in average n = Number of members.

Rule 24:
If a new member is added in a group then.

Age (or income) of added member = Average (or income) ± x (n + 1)

where, x = increase (+) or decrease (–) in average age (or income) n = Number of members.

Rule 25:
If a member leaves the group, then

income (or age) of left member = Average income (or age) ± x (n – 1)

where, x = increase (+) or decrease (–) in average income (or age) n = Number of members.

Rule 26:
If average of n numbers is m later on it was found that a number 'a' was misread as 'b'.

The correct average will be = m + ${(a - b)/n}$

Rule 27:
If the average of n numbers is m later on it was found that two numbers a and b misread as p and q.

The correct average = m + ${(a + b - p - q)}/n$

## Different types of questions or problems based on Average of quantitative aptitude which are asked in various examinations:

• Model 1: Simple average questions,
• Model 2: Average of consecutive numbers,
• Model 3: Twice, thrice, one-third etc. of numbers,
• Model 4: Find nth number from 1st & last number,
• Model 5: Find new average from error or replacing one,
• Model 6: Find average of excluded number,
• Model 7: Average Vs ages/weight,
• Model 8: Find monthly income,
• Model 9: Average Vs Cricket/Exam, and
• Model 10: Find group average by combining/splitting. Refer: Average Practice & Mock Test with online live Quiz

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