entrayn_trial

22IIMIIM
2IIM - An IIM Alumni Venture - Classes for CAT
E Mail: prep@2iim.com Web: www.2IIM.com
Averages
Teaching Notes – Ascent Education

22IIMIIM
Theory
Averages – Basic concepts
2
Average is a measure that gives a general idea about a sample space.
The formula for computing average is straight-forward :
Average =
One basic example
What is the average score in 3 subjects if you score 44, 50, and 56 in
them?
objectsofnumberTotal
Total

22IIMIIM
Theory
Averages – Basic concepts
3
Average is a measure that gives a general idea about a sample space.
The formula for computing average is straight-forward :
Average =
One basic example
What is the average score in 3 subjects if you score 44, 50, and 56 in
them? Average is 50
objectsofnumberTotal
Total

22IIMIIM
Question
4
The marks scored by Ram in 5 subjects in an examination are 44, 56, 72,
60 and 28. What is the average marks scored by Ram in the 5 subjects?
Que:

22IIMIIM
Solution
5
The marks scored by Ram in 5 subjects in an examination are 44, 56, 72,
60 and 28. What is the average marks scored by Ram in the 5 subjects?
Ram – 44, 56, 72, 60 and 28
Average =
=
= 52
Soln:
5
2860725644 ++++
5
260

22IIMIIM
Theory
6
 Total score in 5 subjects is 420, average is equal to?
 Average weight of 6 boys is 72. Their total weight is equal to?
 Total rainfall for a few days is 108 cms, average rainfall per day is 12
cms. For how many days was rainfall measured?

22IIMIIM
Theory
7
 Total score in 5 subjects is 420, average is equal to? 84
 Average weight of 6 boys is 72. Their total weight is equal to? 288
 Total rainfall for a few days is 108 cms, average rainfall per day is 12
cms. For how many days was rainfall measured? 9 days

22IIMIIM
Theory
Averages – Standard framework
8
Example question: The average age of the students of a class A comprising
30 students is 16 years and that of class B comprising 20 students is 11
years. What is the average age of the students of the two classes taken
together?
Number * Average = Total
30 16 480
20 11 220

22IIMIIM
Theory
Averages – Standard framework
9
Example question: The average age of the students of a class A comprising
30 students is 16 years and that of class B comprising 20 students is 11
years. What is the average age of the students of the two classes taken
together?
30 16 480
20 11 220
50 = 14 700
50
700

22IIMIIM
Question
10
The average expense of a company during a year was Rs.250,000 a
month. During the first 5 months, the company spent an average of
Rs.350,000 per month and the during the last 6 months the company
spent an average of Rs.180,000 per month. How much did the company
spend in the 6th
month?
Que:

22IIMIIM
Solution
11
spend in the 6th
month?
Soln:
5 350k 1750 k

22IIMIIM
Solution
12
spend in the 6th
month?
Soln:
5 350k 1750 k
1 X x
Sixth
month

22IIMIIM
Solution
13
spend in the 6th
month?
Soln:
5 350k 1750 k
1 x x
6 180k 1080 k
Last six
months

22IIMIIM
Solution
14
spend in the 6th
month?
Soln:
5 350k 1750 k
1 x x
6 180k 1080 k
12 250k 3000 kFull year

22IIMIIM
Solution
15
1750 + x + 1080 k = 3000k
x = Rs. 170,000
Company spent Rs. 170k in the sixth month.
Soln:

22IIMIIM
Question
16
The average monthly rainfall during the months of April, May and June in
a year was 6 cms. The average monthly rainfall during the months of May,
June and July during the same year was 7.2 cms. If the rainfall in July was
12 cms, what was the rainfall in the month of April?
Que:

22IIMIIM
Solution
17
Soln:
April May June July
Avg = 6
Total = ? Avg = 7.2
Total = ?

22IIMIIM
Solution
18
July -> 12 cms
May + June = (May + June + July ) – July = 21.6 – 12 = 9.6 cms
April = (April + May + June) – ( May + June) = 18 – 9.6 = 8.4 cms
Soln:
April May June July
Avg = 6
Total = 18 Avg = 7.2
Total = 21.6

22IIMIIM
Question
19
The average weight of A and B is 40 kgs. The average weight of B and C is
50 kgs and that of C and A is 45 kgs. What are the weights of A, B and C?
Que:

22IIMIIM
Solution
20
Soln:
Average Total
A + B 40 80
B + C 50 100
C + A 45 90

22IIMIIM
Solution
21
A + B + C = 135
C = 55 kgs [(A + B + C) – (A + B)]
A = 35 kgs
B = 45 kgs
Soln:
Average Total
A + B 40 80
B + C 50 100
C + A 45 90
2(A + B + C) = 270Add all three
equations

22IIMIIM
Question
22
There are four boxes of different weights. These boxes were weighed in
sets of 3 boxes at a time and the following readings were noted - 96, 98,
100 and 102 kgs. What is the weight of the heaviest of the four boxes?
Que:

22IIMIIM
Solution
23
There are four boxes of different weights. These boxes were weighed in
sets of 3 boxes at a time and the following readings were noted - 96, 98,
100 and 102 kgs. What is the weight of the heaviest of the four boxes?
a + b + c = 96
a + c + d = 98
a + b + d = 100
b + c + d = 102
Whenever, we have symmetric equations, adding them up gives
something interesting
3(a + b + c + d) = 396
=> a + b + c + d = 132
a = 132 – 102 = 30 kgs a = a + b + c + d – ( b + c + d)
b = 132 – 98 = 34 kgs, c = 132 – 100 = 32 kgs, d = 132 – 96 = 36 kgs
Soln:

22IIMIIM
Question
24
The GPA of a student at IIM Calcutta in the 6th
term was 1.5 times the
average in the previous 5 terms. By what percentage has his CGPA gone
up after the 6th
term as compared to that at the end of the 5th
term?
Que:

22IIMIIM
Solution
25
up after the 6th
term??
Soln:
5 a 5a
1 1.5a 1.5a

22IIMIIM
Solution
26
up after the 6th
term??
↑ By 8.33%
Soln:
5 a 5a
1 1.5a 1.5a
6
= 1.0833a
6.5a
6
6.5

22IIMIIM
Question
27
The captain of a football team of 11 players is 30 years old. The goal
keeper is 34 years old. If these two players are replaced by two new
players, the average age of the team drops by 1 year. What is the average
age of the two new players?
Que:

22IIMIIM
Solution
28
Soln:
11 a 11a

22IIMIIM
Solution
29
Soln:
11 a 11a
2 - 64Two people
drop out

22IIMIIM
Solution
30
Soln:
11 a 11a
2 - 64
9 - 11a – 649 remain

22IIMIIM
Solution
31
Soln:
11 a 11a
2 - 64
9 - 11a – 64
2 x 2x
Two
replacements

22IIMIIM
Solution
32
Soln:
11 a 11a
2 - 64
9 - 11a – 64
2 x 2x
11 a – 1
11a – 64 + 2x
Or 11(a - 1)
Average
drops by
1
Back to 11
players

22IIMIIM
Solution
33
11(a – 1) = 11a - 64 + 2x
11a – 11 = 11a – 64 + 2x
53 = 2x
x = 26.5 years
Soln:
There is another very simple way of thinking about this. Two people have
been replaced and the average falls by 1. So, the total should fall by 1 *
11 = 11. The sum of the two ages leaving = 30 + 34 = 64. => the sum of the
two ages entering the team = 64 - 11 = 53

22IIMIIM
Theory
Averages – Weighted average
 Normal average assumes that all measurements have equal weightages.
But we can attach different weights to each measurement as well.
Weighted average formula is as follows.
 Average A = , where w1 and w2 are weightages and a1 and
a2 are averages are the respective averages
One basic example
 Average weight of 20 boys in a class is 50 kgs. Average weight of 30 girls
in the same class is 40 kgs. What is the overall average weight?
Overall average = = 44 kgs.
34
3020
40*3050*20
+
+
21
2211
ww
awaw
+
+

22IIMIIM
Question
35
The average age of the students of a class A comprising 30 students is 16
years and that of class B comprising 20 students is 11 years. What is the
average age of the students of the two classes taken together?
Que:

22IIMIIM
Solution
36
The average age of the students of a class A comprising 30 students is 16
years and that of class B comprising 20 students is 11 years. What is the
average age of the students of the two classes taken together?
Average = = 14 years
Soln:
No of student * Average = Total
30 16 480
20 11 220
50 700
50
700

22IIMIIM
Question
37
The average weight of a class comprising 30 students is 30 kgs. If the
weight of the teacher is also included, the average weight increases by 2
kgs. What is the weight of the teacher?
Que:

22IIMIIM
Solution
38
Soln:
30 30 900

22IIMIIM
Solution
39
Soln:
30 30 900
1 x x
Teacher
added

22IIMIIM
Solution
40
X = 92 kgs
Soln:
30 30 900
1 X X
31 32 992
Avg
increases
by 2

22IIMIIM
Solution
41
Alternate method:
If the average has to remain constant even after the addition of the
teacher, teacher’s weight should be 30 kgs. But we see that the average
has increased by 2 kgs. This means that the total has increased by 31*2 =
62 kgs due to the increase in average. Hence, on the whole, the total has
increased by 30kgs (the weight of the teacher for the average to remain at
30) + 62 kgs (the increase in total weight due to increase in average) = 92
kgs.
Hence weight of the teacher = 92 kgs.
Soln:

22IIMIIM
Question
42
The average marks scored by students of a class of 20 students is 50. If
the student who scored the highest marks and the one who scored the
lowest marks are removed, then the average drops by 1.5 marks. If the
difference between the highest and the lowest marks is 50, what is the
lowest mark in the class?
Que:

22IIMIIM
Solution
43
Soln:
20 50 1000

22IIMIIM
Solution
44
Soln:
20 50 1000
18 48.5 8732 students
removed
Avg
falls by
1.5

22IIMIIM
Solution
45
H + L = 127 H – L = 50 (given in the question)
Solving, we get
=> H = 88.5, L = 38.5
Soln:
20 50 1000
18 48.5 873
2 127
Highest +
Lowest
Avg
falls by
1.5

22IIMIIM
Solution
46
Alternate method:
If the average has to remain constant even after the removal of 2
students, the two students should have “taken out” 100 marks.
However, the average falls => they take out more than 100
How much more than 100?
The average falls by 1.5. Average for 18 students falls by 1.5 => Total falls
by 1.5 * 18 = 27
=> They take out 27 more than 100. Or they take out 127
Soln:

22IIMIIM
Question
47
In a group of 20 students, one student weighing 30 kgs leaves the group
and the average weight of the group drops by 0.2 kgs. What is the
average weight of the remaining 19 students?
Que:

22IIMIIM
Solution
48
Soln:
20 a 20a

22IIMIIM
Solution
49
Soln:
20 a 20a
1 30 30One student
leaves

22IIMIIM
Solution
50
19(a – 0.2) = 20a – 30
a = 30 – 3.8 = 26.2
A – 0.2 = 26
Average weight of 19 students = 26 kgs
Soln:
20 a 20a
1 30 30
19 (a – 0.2)
20a – 30 or
19(a – 0.2)
Avg
falls by
0.2

22IIMIIM
Question
51
A batsman has a certain average in 24 innings. After scoring 400 runs in
the next 6 innings, his average increases by 5 runs. What was his average
in the first 24 innings?
Que:

22IIMIIM
Solution
52
A batsman has a certain average in 24 innings. After scoring 400 runs in
the next 6 innings, his average increases by 5 runs. What was his average
in the first 24 innings?
30(x + 5) = 24x + 400 => 30x + 150 = 24x + 400
6x = 250 => x =
Soln:
24 X 24x
6 - 400
30 x + 5
30(x + 5) or
24x + 400
41.67==
3
125
6
250

22IIMIIM
Question
53
In Hotel Pallav, rooms are numbered 101 - 130 on the first floor, 231 - 270
on the second floor, 371 - 420 in the third floor. In the month of June, the
occupancy was 50%, 75% and 70% on the first, second and third floors
respectively. If the rent charged per room per day is Rs. 200, Rs. 150 and
Rs. 100 in the first, second and third floor respectively, find the average
income per day per occupied room for the month of June 2007.
Que:

22IIMIIM
Solution
54
Soln:
Number
No of
rooms
Occupancy
101 – 130 30 50%
231 – 270 40 75%
371 – 420 50 70%

22IIMIIM
Solution
55
Soln:
Number
No of
rooms
Occupancy
No of room
occupied
Rent
101 – 130 30 50% 15 200
231 – 270 40 75% 30 150
371 – 420 50 70% 35 100

22IIMIIM
Solution
56
Soln:
Number
No of
rooms
Occupancy
No of room
occupied
Rent Amount
101 – 130 30 50% 15 200 3000
231 – 270 40 75% 30 150 4500
371 – 420 50 70% 35 100 3500
11000

22IIMIIM
Solution
57
Average rent per occupied room =
= = =
= Rs. 137.5
Soln:
80
11000
8
1100
4
550
2
275

22IIMIIM
Question
58
20 friends go out for dinner. 8 of them spend Rs.32.5 each for the dinner
while each of the rest spend Rs.5 more than the average expense of all
20. What was the total money spent?
Que:

22IIMIIM
Solution
59
Soln:
8 32.5 260

22IIMIIM
Solution
60
Soln:
8 32.5 260
12 x + 5 12(x + 5)
20 x 20x
Avg for the rest is 5 more than
overall average

22IIMIIM
Solution
61
260 + 12(x + 5) = 20x => 260 + 12x + 60 = 20x
320 = 8x => x = 40
=> Total = 40 x 20 = Rs. 800
Soln:
8 32.5 260
12 x + 5 12(x + 5)
20 x 20x

22IIMIIM
Question
62
A school has 5 classes that have 20, 30, 40, 50 and 60 students. If the pass
percentages are 30%, 40%, 50% and 60% and 70% in each of the classes
respectively, find the pass % of the school.
Que:

22IIMIIM
Solution
63
A school has 5 classes that have 20, 30, 40, 50 and 60 students. If the pass
percentages are 30%, 40%, 50% and 60% and 70% in each of the classes
respectively, find the pass % of the school.
Overall Pass % = x 100 = 55%
Soln:
Number of student Pass %
No of student who
passed
20 30 % 6
30 40 % 12
40 50 % 20
50 60% 30
60 70 % 42
200
110

22IIMIIM
Question
64
Typist A can type a sheet in 7 minutes, typist B in 9 minutes and typist C in
6 Minutes. What is the average number of sheets typed per hour, per
typist?
Que:

22IIMIIM
Solution
65
Typist A can type a sheet in 7 minutes, typist B in 9 minutes and typist C in
6 Minutes. What is the average number of sheets typed per hour, per
typist?
Typist A sheets per hour =
Typist B sheets per hour =
Typist C sheets per hour =
Average per hour=
= =
Soln:
7
60
3
20
9
60
=
10
60
60
=





 ++
21
210140180
3
1






++ 10
3
20
7
60
3
1
63
530

22IIMIIM
Question
66
The average weight of a body calculated through eight different
experiments is 60 units. The average weight from the first three
experiments is 58 units. The difference between the weight recorded in
the fourth and fifth experiment is 5 units, with the fourth one being
heavier. The average of the last three experiments is 62 units. Find the
weight in the fourth experiment.
Que:

22IIMIIM
Solution
67
Soln:
a1 a2 a3 a4 a5 a6 a7 a8
Average = 60, Total = 480

22IIMIIM
Solution
68
Soln:
a1 a2 a3 a4 a5 a6 a7 a8
Average = 58
Total = 174

22IIMIIM
Solution
69
a5 = a4 – 5
Soln:
a1 a2 a3 a4 a5 a6 a7 a8
Average = 58
Total = 174
Average = 62
Total = 186

22IIMIIM
Solution
70
480 = 174 + a4 + a5 + 186 [∴a5 = a4 – 5]
120 = 2a4 – 5
a5 = 57.5, a4 = 62.5 units
Soln:
a1 a2 a3 a4 a5 a6 a7 a8
Average = 58
Total = 174
Average = 62
Total = 186

22IIMIIM
Solution
71
Alternate method:
The first three average 58, last three average 62, so these six must
average 60. All 8 put together average 60. So, a4 and a5 should average
60.
⇒ a4 + a5 = 120
⇒ a4 - a5 = 5
⇒ a4 = 62.5
The intuitive method is usually better 
Soln:

22IIMIIM
Prep Edge

22IIMIIM
Question
73
Consider a class of ‘n’ people with an average weight of ‘a’ kgs. When a
person weighing 72 kgs joins the class, the average weight increases by 1
kg. When another person weighing 48 kgs join the class, the average
weight is ‘a’. How many students were originally there in the class?
A.11
B.12
C.13
D.60
Que:

22IIMIIM
Solution
74
Consider a class of ‘n’ people with an average weight of ‘a’ kgs. When a
person weighing 72 kgs joins the class, the average weight increases by 1
kg. When another person weighing 48 kgs join the class, the average
weight is ‘a’. How many students were originally there in the class?
Person 1’s weight = 72 kgs, avg ↑ +1 kg
Person 2 weight = 48 kgs, avg back to same level.
2 people weighing 72 + 48 = 120 kgs. Have no impact on average
⇒Avg. to begin with was 60 kgs
Addition of person weighing 72 kgs increases avg by 1kg. => 12 kgs more
than average increases average by 1. => No of students should now be
equal to 12. => No of students originally = 11
⇒ Choice (A)
Soln:

22IIMIIM
Question
75
A class of 40 students has an average weight of 40 kgs. When student A
weighing ‘a’ kgs leaves this class, the average increases. When a second
student B weighing ‘b’ kgs subsequently leaves the class the average
decreases but stays above 40.
A.b > a > 40
B.a < 40 < b < 80 – a
C.a < 40 < 80 – a < b
D.a < b < 40
Que:

22IIMIIM
Solution
76
A class of 40 students has an average weight of 40 kgs. When student A
weighing ‘a’ kgs leaves this class, the average increases. When a second
student B weighing ‘b’ kgs subsequently leaves the class the average
decreases but stays above 40.
Student A leaves the class and average increases => a < 40
Once student A leaves, let us assume new average is k.
We know k > 40. When second student leaves, average decreases
=> b > k
⇒ a < 40 < k < b
When both students leave, average > 40
⇒ a + b < 80, => b < 80 - a
⇒ a < 40 < b < 80 - a => Choice (B)
Soln:

22IIMIIM
Question
77
The first 25 natural numbers are placed in 5 sets of 5 elements in each.
What is the minimum average of the lowest elements in each of the 5
sets?
A.10
B.9
C.3
D.11
Que:

22IIMIIM
Solution
78
The first 25 natural numbers are placed in 5 sets of 5 elements in each.
What is the minimum average of the lowest elements in each of the 5
sets?
One of the lowest elements has to be 1. Now, if we take the 5–element
set that has 1 as its lowest element, the best-case scenario would be to
place large numbers in this group. So as not to “waste” small numbers. So
the five groups can each have 1, 2, 3, 4 and 5 as one of their elements.
⇒ Minimum average = = 3
Choice (C)
Soln:
5
54321 ++++

22IIMIIM
Question
79
Set S contains {1, 7, 3, 5, 8}. The average of all 3-element subsets of S are
computed and placed in set P. What is the average of all the elements in
set P?
A.3
B.4
C.4.8
D.3.6
Que:

22IIMIIM
Solution
80
Set S contains {1, 7, 3, 5, 8}. The average of all 3-element subsets of S are
computed and placed in set P. What is the average of all the elements in
set P?
The average of all average of 3 element subsets will be equal to the
average of the 5 numbers. All selections are symmetric. So each number
will occur an identical number of times. 
Average = = 4.8
Choice (C)
Soln:
5
85731 ++++

22IIMIIM
Question
81
Consider a class of 15 students. Average weight of the 8 lightest students
is 37 kgs. The average weight of the 8 heaviest students is 46 kgs. The
median weight of the class = 43 kgs. What is the average weight of the
entire class?
A.39.8
B.39.6
C.43
D.41.4
Que:

22IIMIIM
Solution
82
Consider a class of 15 students. Average weight of the 8 lightest students
is 37 kgs. The average weight of the 8 heaviest students is 46 kgs. The
median weight of the class = 43 kgs. What is the average weight of the
entire class?
Let weights a1, a2, a3 …………………… a15 such that
a1 ≤ a2 ≤ a3 ………………………< a15
= 37, = 46
a8 = 43
(1)a1 + a2 …….. a8 = 37 x 8 = 296
(2)a8 + a9 ……… a15 = 46 x 8 = 368
(3)a8 = 43
Soln:
8
a...............aa 821 +
8
a..............aa 1598 +

22IIMIIM
Solution
83
(1) + (2) – (3) => a1 + a2 + a3 ………. a15
⇒ 296 + 368 – 43 = 621
Average = = 41.4
Choice (D)
Soln:
15
621

22IIMIIM
Question
84
Average score obtained by girls in a school is 82 marks. Average of the
overall school is 80 marks. If there are twice as many boys as girls in the
school, what is the average marks scored by the boys?
A.78
B.76
C.79.5
D.79
Que:

22IIMIIM
Solution
85
Average score obtained by girls in a school is 82 marks. Average of the
overall school is 80 marks. If there are twice as many boys as girls in the
school, what is the average marks scored by the boys?
Number of girls = n
No of boys = 2n
= 80
⇒ 82n + 2na = 240n
⇒ 2a = 240 – 82
⇒ 2a = 158
⇒ a = 79
⇒ Choice (D)
Soln:
3n
ax2n82xn +

22IIMIIM
Question
86
Average score of n students in a class in m. Average of the remaining
(m + 1) students is (n – 1). What is the overall average?
A.
B.
C.
D.None of these
Que:
1nm
1n)(m2mn
++
++−
12n
1nm2mn
+
−−−
n1m
1nm2mn
++
−−−

22IIMIIM
Solution
87
Average score of n students in a class in m. Average of the remaining
(m + 1) students is (n – 1). What is the overall average?
Weighted average
=
=
=
None of these
Choice (D)
Soln:
1mn
1)-1)(n(mmxn
++
++
1mn
1-nmnmnm
++
+−+
1mn
1-n)(m2mn
++
−−

entrayn_trial

Recommended

Recommended

More Related Content

Viewers also liked

Viewers also liked (8)

Similar to entrayn_trial

Similar to entrayn_trial (15)

More from rajeshb1980

More from rajeshb1980 (8)

Recently uploaded

Recently uploaded (20)

entrayn_trial

Editor's Notes