Average : Important Terms, Formulas and Shortcuts

Average is an important topic under Quantitative Aptitude. Almost all competitive exams consists of questions on Average.

Now lets start the lesson !

What Is Average ?

The result obtained by addition of several quantities together and then dividing this result obtained by the number of quantities is called Average.

In other words, Average is the sum of all quantities divided by the total number of quantities.

"Equal Distribution" is the term used to define Average.Average can sometimes be also called as "Arithmetic Mean".

We obtain the average of a number using formula that is sum of observations divided by Number of observations.

Average=(Sum of observations / Number of observations)

Suppose we have to determine average of 10,20,25,30 and 35 then 
Average=(10+20+25+30+35)/5=120/5=24

Some Basic Formulas on Average of Numbers are as follows :

• Average of N natural number = ( N + 1 ) / 2

• Average of N even number = ( N + 1 )

• Average of N odd number = N

• Average of N consecutive natural number = ( First Number + Last Number ) / 2

• Average of sum of squares of first N natural number = ( N + 1 ) ( 2 N + 1 ) / 6

Some Important Points You Should Know :

• If each term in an Average is increased by N, then average of all terms will also increase by N

• If each term in an Average is decreased by N, then average of all terms will also decrease by N

• If each term in an Average is multiplied by N, then the average of all terms will also get multiplied by N

• If each term in an Average is divided by N, then the average of all terms will also get divided by N

Shortcuts to solve questions asked on Average :

• If a person travels a distance at a speed of x km/hr and the same distance at a speed of y km/hr then the average speed during the whole journey is given by 2xy/(x+y)

Suppose Ram travels a distance of 10 km at 4km/hr and the next 10 km at 5 km/hr then the average speed of Ram=(2x4x5)/(4+5)=40/9=4.44 km/hr

• If a person covers A km at x km/hr and B km at y km/hr and C km at z km/hr, then the average speed in covering the whole distance is (A+B+C)/[(A/x)+(B/y)+(C/z)]

Suppose a person covers 10 km at 4 km/hr and 5 km at 5 km/hr and 10 km at 3 km/hr, then the average speed in covering the whole distance = (10+5+10)/[(10/4)+(5/5)+(10/3)]=25/(2.5+1+3.33)=25/6.83=3.66 km/hr

When a person leaves the group and another person joins the group in place of that person then 

• If the average age is increased,

Age of new person = Age of separated person + (Increase in average × total number of persons)

• If the average age is decreased, 

Age of new person = Age of separated person - (Decrease in average × total number of persons)

When a person joins the group then

• In case of increase in average,

Age of new member = Previous average + (Increase in average × Number of members including new member)

• In case of decrease in average,

Age of new member = Previous average - (Decrease in average × Number of members including new member)

In the Arithmetic Progression there are two cases :

• when the number of terms is odd,the average will be the middle term

Suppose we have to determine  the average of 13, 14, 15, 16, 17 then Average is the middle term when the number of terms is odd, but before that let’s checks whether it is in A.P or not, since the common difference is same so the series is in A.P. So the middle term is 15 which is our average of the series

• when number of terms is even,then the average will be the average of two middle terms

Suppose we have to determine the average of 13, 14, 15, 16, 17, 18 then We have discussed that when the number of terms are even then the average will be the average of two middle terms.Now the two middle terms are 15 and 16, but before that the average we must check that the series should be A.P. Since the common difference is same for each of the term we can say that the series is in A.P. and the average is (16+15)/2 = 15.5

• If the average of N Numbers is X and the average of M Numbers is Y then, 

Average of ( N + M ) Numbers = ( NX + MY ) / ( N + M )

Suppose Average of 20 Numbers is 15 and Average of 30 Numbers is 20 then, Average of 50 Numbers = ( 20 x 15 ) + ( 30 x 20 ) / ( 20 + 30 ) = ( 300 + 600 ) / 50 = 900 / 50 = 18

• If the average marks obtained by N students is M. If the average marks of passed student is X and average of failed student is Y then,

Number of students failed = N ( X - M ) / ( X - Y )

Suppose average marks obtained by 50 students in an exam is 45. If average marks obtained by passed student is 55 and average marks obtained by failed students is 30 then, number of failed student = 50 ( 55 - 45 ) / ( 55 - 30 ) = 50 ( 10 ) / 25 = 500 / 25 = 20

• If the average of N quantities is M and when a quantity is removed the average it becomes P then,

Value of removed quantity = [ N ( M - P ) + P ]

Suppose average of a set of 25 numbers is 35 and when a number is removed from the set the average becomes 34, then the removed number = [ 25 ( 35 - 34 ) + 34 ] = 25 + 34 = 59

• If a batsman score N runs in his Xth innings and thus lead to increase in his average by M then,

Average after Xth inning is [ N - M ( X - 1 ) ]

Suppose a batsman has scored 90 run in his 40th inning and the average is increase by 2 then the average after 40th inning = 90 - 2 ( 40 - 1 ) = 90 - 78 = 12

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