averages level 2

8/10/2019 Averages Level 2

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AVERAGES LEVEL 2C.S.VEERARAGAVAN


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DEFINITION A measure of Central TendencySimple Average (or) Mean is defined as the ratio of sum ofthe quantities to the number of quantities.

Average = Sum of the quantitiesNo. of the quantities

=


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The average of 20 numbers is zero. Of them, at the most,

how many may be greater than zero?

Average = 0.

Average = sum of 20 numbers20 = 0.

Hence sum of 20 numbers = 0.

It is possible that 19 numbers may be greater than zero

and if their sum is a ,

then the 20 th number should be a.

Hence maximum = 19.


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Weighted Average or MeanIf some body asks you to calculate the combined averagemarks of both the sections of class X- A and X- B, when both

sections have 60% and 70% average marks respectively?

Then your answer will be 65% but this is wrong as you do not know

the total number of students in each sections.

So to calculate weighted average, we must have to know the

number of students in both the sections.

Let N 1 , N 2 , N 3 , . N n be the weights attached tovariable values X 1 , X 2 , X 3 , .. X n respectively.

Then the weighted arithmetic mean is given by

X =

n321

nn332211

N.....NNN

XN.....XNXNXN


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ExampleThe average marks of 30 students in a section of class X are 20while that of 20 students of second section is 30. Find the averagemarks for the entire class X.

Solution

We can do the question by using both the Simple average &weighted average method.

Simple average = =

= 24

By the weighted mean method, Average = 20 + 30

= 12 + 12 = 24.

studentsof number Total

studentsallof marksof Sum

2030

20303020

50

30

50

20


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Example

The average of 11 results is 50.

If the average of first 6 results is 49 and that of the last 6 is 52,

find the 6th result.

(1) 48 (2) 50 (3) 60 (4) 56

Solu t ion :

The sum of 11 results = 11 50 = 550

The sum of the first 6 results = 49 6 = 294

The sum of the last 6 results = 52 6 = 312

So the 6 th result is = (294 + 312) 550 = 56

Answer (4)


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Average age of a family of four members is 20. what is theaverage age (in years) of the same family of four membersafter four years?1)21 2)24 3) 28 4) 36

If each number is increased / decreased by a certainquantity n, then the mean also increases or decreasesby the same quantity.

Answer is 24.


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If each number is multiplied/ divided by acertain quantity n, then the mean also getsmultiplied or divided by the same quantity.

Example :

If the average of x, y, z is k, then average of3x, 3y, 3z will be 3k

Real Facts A bout Average


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If the same value is added to half of thequantities and same value is subtractedfrom other half quantities then there will not

be any change in the final value of theaverage.

Example :

If the average of a, b, c, d is k, then averageof a + x, b + x, c x, d x is also k.

Real Facts A bout Average


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The average of the numbers which are in arithmetic progression is

the middle number or the average of the first and last numbers.

Example:

The average of 10 consecutive numbers starting from 21 is the

average of 5th & 6th number, i.e. average of 25 and 26 which is 25.5

Example:

The average of 4, 7, 10, ., 34 = 4+342 = 19


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The average weight of a class of 40 students is 40 kg. If anew student, whose weight is 40 kg, joins the class, thenwhat is the new average weight of the class? 1)40 kg 2)41kg 3)44 kg 4)80kg.

Total weight of 40 students = 1600 kg.

New average =+

=

=40 kg.

Shortcut: If age of new entrant is the old average then therewill be no change in average.


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In a company the average height of female employees is160 cm and the average height of male employee is170cm.If the total number of employees is 330, what is theaverage height of all the employees of the company?

Since the number of female or male is not given, wecannot determine the average height of all.


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The average weight of girls in a class of 50 students is 40kg. which of the following could be the average weight of30 of the boys is 50 kg? 1)42 kg 2) 43 kg 3) 45 kg 4) 47kg?If there are exactly 30 boys,

GIRLS CLASS BOYSNUMBER 20 30

AVG WEIGHT 40 50

X

50 X X 40

= 1500 30X = 20X 800

X = = 46

As there are possibly more boys the average willbe greater than 46 but not less


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The average marks scored by girls in a class of 40students in an examination is 80.If the number of boys in the class and their average marksin the same examination are 25 and 72 respectively, whatare the average marks of all the students in the class?1) 74 2) 75 3) 76 4) 78

Sum of marks of 25 boys = 72 x 25 = 1800

Sum of marks of 15 girls = 80 x 15 = 1200

Average marks of total students = 300040 = 75.


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In a company, the average salary for female employees is`.20,000 per month and that of male employees is `.20,000per month. What is the average salary per month for allthe employees of the company?1) 19000 2) 20000 3) 210004) cannot be determined.Let number of female be f and male be m.

Average = + ( )

+ = 20000


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The average marks scored by Suresh in 5 out of six

subjects is 54. to get an average of 60 in 6 subjects

together, what is the score that Suresh must get in the

sixth subject?

Total marks in 5 subjects = 5x 54 = 270.

Total marks in 6 subjects = 6 x 60 = 360.

Marks in 6 th subject should be

= 360 270 = 90.


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The ratio of the number of girls to that of boys is 2:3. If the

average weight of girls is 35 kg and that of boys is 40 kg,

what is the average weight of the entire class? (in kg)

1) 36 2) 37 3) 38 4) 39

Number of girls = 2x and boys = 3x.

Sum of age of girls = 70x and boys = 120x

Average age = 190x5x = 38.


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The average number of chocolates with four is 4.

Another child joins them.

Now, the average number of chocolates with the five children is

5.

What is the number of chocolates with the fifth child?

1) 1 2) 9 3) 5 4) 6

No of chocolates with four = 4x 4 = 16.

Total no after fifth child joined = 5 x 5 = 25.

Number of chocolates with fifth child = 9.


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A bag contains 2 kg of potatoes and 3 kg of tomatoes.

If the cost of potatoes is Rs.5 per kg and that of tomatoes

is Rs.8 per kg, then what is the average price per kilogram

of the vegetables in the bag?

1) Rs.6 2) Rs.6.60 3)Rs.6.80 4)Rs.7

Total cost of potatoes = 10 and tomatoes = 24

Average cost = 10 245 = Rs.6.80


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A total of 44 chocolates were distributed among 10 children such that

each girl gets 5 chocolates and each boy gets 4 chocolates. What is the

number of girls?

1) 6 2) 5 3) 4 4) 3

Average chocolates for girls = 5 and boys = 4.

Total average chocolates for the group = 4.4

Using rule of allegation

ab

= 5 4.44.4 4

= 0.60.4

= 32

As a+b = 10, a = 6 and b = 4

Hence there are 4 girls.

4 5

4.4 0.6 0.4


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The average income of a man for first 9 months is Rs.10,000

per month.

Last 3 months is Rs. 20,000.

Then which of the following is necessarily true regarding the

average income per month for the man?

1) It is between Rs.10,000 and Rs. 15,000.

2) It is exactly Rs.15,000

3) It is between Rs.15,000

Mostly the income was 10000.

Hence average also should be closer to 10000


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Age of new entrant =New average + No of old members X change in average.Example

The average age of 30 boys of a class is equal to 14 years.When the age of the class teacher is included the averagebecomes 15 years.Find the age of the class teacher.

= 15+30(15-14) = 45 years.


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Age of one who left =New average No. of old members X change inaverage.Example

There are 50 boys in a class. Their average weight is45 kg. When one boy leaves the class, the averagereduces by 100 gms. Find the weight of the boy wholeft the class.

= 44.9-50x(-0.1) = 44.9+5 = 49.9kg.


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Age of new person = Age of the removed person + No. of members x change inaverage.

ExampleThe average weight of 8 men is increased by 1.5 g. whenone of the men who weighs 65 kg is replaced by a newman. The weight of the new man is:

= 65 + (8x1.5) = 65+12 = 77 kg.


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The average monthly salary of 12 workers and 3 managers in a

factory was Rs. 600.When one of the manager whose salary was Rs. 720, was replaced

with a new manager, then the average salary of the team went down

to 580.

What is the salary of the new manager? A. 570 B.420 C.690 D.640

Salary of new person =

Salary of the removed person + No. of members x change in average.

Salary of new person = 720 + (15)(-20)

= 720 300

= 420


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Average Speed

If d1 & d 2 are the distances covered at speeds v 1 & v2 respectively

and the time taken are t 1 & t2 respectively, then the average speedover the entire distance (d 1 + d 2) is given by

takentimeTotal

eredcovcetandisTotal Average Speed =

takentimeTotal

eredcovcetandi sTotal

21

21

tt

dd

2

2

1

1

21

v

d

v

d

dd

TIP:

Average speed can never be double or more than double ofany of the two speeds.


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# If both the time taken are equal i.e. t 1 = t 2 = t then

Average Speed

Average speed2

vv21

# If both the distances are equal i.e. d 1 = d 2 = d then,

Average speed21

21

vv

vv2


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Example: A man travels at a speed of 60 kmph on a journey from A to Band returns at 100 kmph. Find his average speed for the

journey (in kmph).

(1) 80 (2) 72

(3) 75 (4) None of these

Solu t ion :

Average speed = 2 X 60 X 10060+100 = 75 Answer (3)


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The average of marks obtained by n candidates in acertain examination is T.If the average marks of passed candidates is P and that offailed candidates is F.

Then the number of candidates who PASSED the exam isgiven by: n T FP F

The average marks obtained by 90 candidates in a certain

examination is 38, if the average marks of passedcandidates is 40 and that of the failed candidates is 30,what is the number of candidates who passed the exam?

Here: n= 90, T=38, F= 30 and P= 40. 90 38 3040 30 = 72 .


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The average of marks obtained by n candidates in acertain examination is T.If the average marks of passed candidates is P and that offailed candidates is F.

Then the number of candidates who passed the exam isgiven by: n P TP F

The average marks obtained by 90 candidates in a certain

examination is 38, if the average marks of passedcandidates is 40 and that of the failed candidates is 30,what is the number of candidates who FAILED the exam?

Here: n= 90, T=38, F= 30 and P= 40. 90 40 38

40 30 =

18 .


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The average age of N persons is x years.If one new person joins them.Then the average age is increased by y years.

Then the age of new comer is x + (1 + N) y years


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The average age of N persons is x years.If M persons joins them, the average age isincreased by y years then the average age of new

comers is 1 years

The average age of N persons is x years.

If M persons joins them, the average age isdecreased by y years then the average age of new

comers is x 1

years


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The average age of N persons is x years.If M persons left, then the average age is increased by yyears, then the average age of outgoing persons is

1

y years.

The average age of N persons is x years.If M persons left, then the average age is decreased by y

years. Then the average age of outgoing persons is 1

y years


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In a group of N persons whose average age isincreased by y years when a person of x years isreplaced by a new man.Then the age of new comer is x + Ny years.


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The captain of a cricket team of 11 members is 26 years old and the wicket keeperis 3 years older. If the ages of these two are excluded, the average age of theremaining players is one year less than the average age of the whole team. What isthe average age of the team?

A. 23 years B. 24 years C. 25 years D. None of these

Let the average age of the whole team by x years.

11x - (26 + 29) = 9(x -1)

11x - 9x = 46

2x = 46

x = 23.

So, average age of the team is 23 years.


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The average monthly income of P and Q is Rs. 5050. The averagemonthly income of Q and R is Rs. 6250 and the average monthly incomeof P and R is Rs. 5200. The monthly income of P is:

A. 3500 B. 4000 C. 4050 D. 5000

Let P, Q and R represent their respective monthly incomes. Then, wehave:P + Q = (5050 x 2) = 10100 .... (i)Q + R = (6250 x 2) = 12500 .... (ii)P + R = (5200 x 2) = 10400 .... (iii)

Adding (i), (ii) and (iii), we get: 2(P + Q + R) = 33000 or P + Q + R =16500 .... (iv)Subtracting (ii) from (iv), we get P = 4000.P's monthly income = Rs. 4000.


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The average age of husband, wife and their child 3 years agowas 27 years and that of wife and the child 5 years ago was20 years. The present age of the husband is:

A. 35 years B. 40 years C. 50 years D. None of these

Sum of the present ages of husband, wife and child= (27 x 3 + 3 x 3) years = 90 years.

Sum of the present ages of wife and child= (20 x 2 + 5 x 2) years = 50 years.

Husband's present age = (90 - 50) years = 40 years.


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A car owner buys petrol at Rs.7.50, Rs. 8 and Rs. 8.50 per litre forthree successive years. What approximately is the average cost perlitre of petrol if he spends Rs. 4000 each year?

A. Rs. 7.98 B. Rs. 8C. Rs. 8.50 D. Rs. 9

Total quantity of petrol consumed in 3 years = 40007.50 40008.50 litres

= 4000 215 217 litres

=76700

51 litresTotal amount spent = Rs. (3 x 4000) = Rs. 12000.

Average cost = Rs. 12000X5176700 = Rs.6120767 = Rs. 7.98


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In Arun's opinion, his weight is greater than 65 kg but less than 72 kg.His brother does not agree with Arun and he thinks that Arun's weight isgreater than 60 kg but less than 70 kg. His mother's view is that hisweight cannot be greater than 68 kg. If all are them are correct in theirestimation, what is the average of different probable weights of Arun?

A. 67 kg. B. 68 kg. C. 69 kg.D. Data inadequate E. None of theseLet Arun's weight by X kg.

According to Arun, 65 < X < 72 According to Arun's brother, 60 < X < 70.

According to Arun's mother, X


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The average weight of A, B and C is 45 kg. If the average weight of A and B be40 kg and that of B and C be 43 kg, then the weight of B is:

A. 17 kg B. 20 kgC. 26 kg D. 31 kg

Let A, B, C represent their respective weights. Then, we have:

A + B + C = (45 x 3) = 135 .... (i)

A + B = (40 x 2) = 80 .... (ii)

B + C = (43 x 2) = 86 ....(iii)

Adding (ii) and (iii), we get: A + 2B + C = 166 .... (iv)

Subtracting (i) from (iv), we get : B = 31.

B's weight = 31 kg.


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The average weight of 16 boys in a class is 50.25 kgand that of the remaining 8 boys is 45.15 kg. Find theaverage weights of all the boys in the class.

A. 47.55 kg B. 48 kgC. 48.55 kg D. 49.25 kg

Required average =50.25X16 + 45.15X8

16+8

=804+361.20

24 =

1165.2024

=48.55


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A library has an average of 510 visitors on Sundaysand 240 on other days. The average number ofvisitors per day in a month of 30 days beginning

with a Sunday is: A. 250 B. 276 C. 280 D. 285

Since the month begins with a Sunday, to there willbe five Sundays in the month.

Required average = 510X5 + 240X255+25

=855030 =285


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If the average marks of three batches of 55, 60 and45 students respectively is 50, 55, 60, then theaverage marks of all the students is:

A. 53.33 B. 54.68 C. 55 D. None of these

Required Average =55X50 60X55 45X60

55 60 45

2750 3300 2700160

=54.68


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A pupil's marks were wrongly entered as 83 insteadof 63. Due to that the average marks for the classgot increased by half (1/2). The number of pupils inthe class is:

A. 10 B. 20C. 40 D. 73

Let there be x pupils in the class.

Total increase in marks = X x = = (83 63)

x= 40.


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Average of first n natural numbers = (n+1)/2

Average of the squares of the first n natural numbers =

[(n+1)(2n+1)]/6

Average of the cubes of the first n natural numbers =[n2(n+1) 2]/4

Average of the first n even natural numbers = n

Average of the first n odd natural numbers = (n+1 )


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A man travels a certain distance at x kmph andan equal distance at y kmph.Then the average speed during whole journey =(2xy/(x+y))kmph.

Average of n average is not equal to the { (sum ofn averages )/ n }.


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If all values are increased or decreased by a certainquantity, the average also increases or decreases by

the same quantity.

If all values are multiplied or divided by a certainquantity, the average also gets multiplied or divided

by the same quantity.


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When a new value X is added to a set of B valueswith average A, if the average increases by Y, then

the added value X = A + (B + 1)Y.When a new value X is added to a set of B valueswith average A, if the average decreases by Y, then

the added value X = A (B + 1)Y.


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When a value X is removed from a set of B valueswith average A, if the average increases by Y, then

the deleted value X = A (B 1)Y.

When a value X is removed from a set of B valueswith average A, if the average decreases by Y, then

the deleted value X = A + (B 1)Y.


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There are twice the number of two wheelers as there are three wheelers and thenumber of 4 wheelers are equal to the number of two wheelers.The average number of wheel per vehicle is1) 3 2) 4 3) 5 4) None of these.

No. of 2wheelers

No. of 3wheelers

No. of 4wheelers

2x x 2x

No. of wheels 2 * 2x = 4x 3 * x = 3x 4 * 2x = 8x

Average wheels = 15x3x = 5


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The average of earnings of a man forthe first four months,the fifth to the eighth months andthe last four months in a particular year areRs.5000 per month, Rs.6000 per month andRs.10000 per month respectively.What is the average salary per month in that year?1) Rs.6000 2) Rs.7000 3) Rs.8000 4) Rs.9000Since the period for each of averages is same

Average = 5000+6000+100003 =21000

3 = 7000


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Thank you


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From 10 litre of milk, one litre is taken out and replaced byone litre of water, then one

averages level 2

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