In mathematics, the average, or arithmetic mean, is a central value that represents the typical value of a set of numbers. It is calculated by adding up all the numbers and dividing the sum by the count. The average provides a useful measure to analyze and compare data, offering insights into trends and patterns.
Assignment-1
Que 1: A student was given a task to calculate the average of a set of 12 numbers: 3, 11, 7, 9, 15, 13, 8, 19, 17, 21, 14, and x. When the student calculated the average, he found it to be 12. Now find the value of x.
(a) 3
(b) 7
(c) 17
(d) 31
Solution: -9-1-5-3+3+1-4+7+5+9+2+(x-12)=0
5+x-12=0
x=7
Option (B) is correct.
Que 2: If we have three batches of students with 55, 60, and 45 students respectively with the average marks of 50, 55, and 60 respectively, then find the average marks of all the students combined?
(a) 54.68
(b) 53.33
(c) 55
(d) None
Solution: let average= 55
-5×55+0×60+5×45=-50
Average= 55-(50/160)=54.68
Option (a) is correct.
Que 3: The average of n numbers x_1, x_2,…. x_n is \overline{x} . Then the value of \sum\limits_{i=1}^{n}(x_i-\overline{x}) is equal to:
(a) n
(b) 0
(c) n\overline{x}
(d) \overline{x}
Solution: Option (b) is correct.
Que 4: The average monthly expenditure of a family was 2,200 during the first three months, 2,550 during the next four months, and 3,120 during the last five months of the year. If the total savings during the year was 1,260. Find the average monthly income of the family?
(a) 1260 rs
(b) 1280 rs
(c) 2805 rs
(d) 2850 rs
Solution: let the average expenditure= 2550
-350×3+0×4+570×5= 1800
Average=2550+1800/12=2700
Monthly income= 2700+1260/12=2805
Option (c) is correct.
Que 5: The average (mean) of 9 observations is 16. An additional observation is included, and as a result, the new average becomes 17. Find the value of the 10th observation?
(a) 9
(b) 16
(c) 26
(d) 30
Solution: Total increment= 10 observations ×10
The value of 10th observation= 16+10=26
Option (c) is correct.
Que 6: The average of a set of natural numbers is 15. If we add 30 to the first number and subtract 5 from the last number, the new average becomes 17.5. Find the number of natural numbers in the set?
(a) 15
(b) 30
(c) 20
(d) 10
Solution: Increment in average= 17.5-15=2.5
30-5=25
25/n=2.5
n=10
Option (d) is correct.
Que 7: The average weight of 15 oarsmen in a boat increased by 1.6 kg when one of the crew members, who weighed 42 kg, was replaced by a new man. Find the weight of the new man in kilograms?
(a) 67
(b) 65
(c) 66
(d) 43
Solution: Total increment= 1.6×15=24
The weight of the new man= 42+24=66
Option (c) is correct.
Que 8: A librarian bought 50 story books for his library. However, he realised that by spending Rs. 76 more, he could purchase an additional 14 books. As a result, the average price per book would be reduced by Re. 1. Find the average price (in Rs.) of each book he initially bought?
(a) 15
(b) 10
(c) 25
(d) 20
Solution: (x-1)×64 -50x=76
x=10
Option (b) is correct.
Que 9: Three science classes, A, B, and C, take a Life Science test. The average score of class A is 83. The average score of class B is 76. The average score of class C is 85. Additionally, the average score of classes A and B combined is 79, and the average score of classes B and C combined is 81. Find the average score of classes A, B, and C combined?
(a) 81.5
(b) 81
(c) 80.5
(d) 80
Solution: A⇒ 83, B⇒76, C⇒85
A+B⇒79 so A:B=3:4
B+C⇒81 so B:C=4:5
A:B:C= 3:4:5
Let Average score= 83
0×3 -7×4+2×5=-18
Average= 83-18/12=81.5
Option (a) is correct.
Que 10: The average high temperature of the first four days of a week is 25°C, while the average of the last four days is 25.5°C. If the overall average temperature of the entire week is 25.2°C, Find the temperature on the fourth day.
(a) 25.5°C
(b) 25°C
(c) 25.2°C
(d) 25.6°C
Solution: -0.2×4+0.3×4=0.4
The temperature on the fourth day=25.2+0.4=25.6°C
Option (d) is correct.
Que 11: The average of the first 7 integers in a series of 13 consecutive odd integers is 37. Find the average of the entire series?
(a) 37
(b) 39
(c) 41
(d) 43
Solution: Odd series⇒ a, a+2, a+4, ……a+24
Average of first seven⇒ 4th term=a+6=37
Average of the entire series⇒ 7th term=a+12= 37+6=43
Option (d) is correct.
Que 12: The average of the marks of 17 students in an examination was initially calculated as 71. However, it was later discovered that the mark of one student was incorrectly recorded as 65 instead of 56, and another student’s mark was recorded as 24 instead of 50. Find the correct average?
(a) 70
(b) 71
(c) 72
(d) 73
Solution: Correction= -9+26=17
New average= 71+17/17=72
Option (c) is correct.
Que 13: Find the average of the squares of the first ten natural numbers?
(a) 35.75
(b) 36
(c) 37.25
(d) 38.50
Solution: The sum of the series= 1/6×n(n+1)(2n+1)
Average= Sum/n= 1/6×(n+1)(2n+1)=1/6×11×21=38.5
Option (d) is correct.
Que 14: A man spends in 8 months an amount equal to what he earns in 6 months. Over the course of a year, he manages to save Rs. 6000. Find his average monthly income?
(a) 2400 rs
(b) 2000 rs
(c) 2150 rs
(d) 1800 rs
Solution: 6×I=8×E
I:E=4:3
I:E:S=4:3:1
Monthly saving= 6000/12=500
Monthly income=500×4=2000
Option (b) is correct.
Que 15: The average weight of three men, A, B, and C, is 84 kg. Another man, D, joins them, and the average weight of the four becomes 80 kg. If E, whose weight is 3 kg more than D, replaces A, the average weight of B, C, D, and E becomes 79 kg. Find the weight of A?
(a) 65 kg
(b) 70 kg
(c) 75 kg
(d) 80 kg
Solution: A+B+C⇒ 84
A+B+C+D⇒ 80
D=84 -4×4=68 kg
E=68+3=71 kg
A=71+4×1=75 kg (Replacement)
Option (c) is correct.
Assignment-2
Que 1: The average of a group of 20 items is 55. If we take out two items with values of 45 and 30, and look at the remaining items, the new average of those items will be..
(a) 57.1
(b) 57.3
(c) 56.9
(d) 56
Solution: 55-45=10
55-30=25
25+10=35 will be distributed among the remaining 18 people
New average= 55+35/18= 56.9
Option (c) is correct.
Que 2: A cricketer has a bowling average of 24.85 runs per wicket. After taking 5 wickets for 52 runs in a match, his average decreases by 0.85. How many wickets he had taken till the last match.
(a) 64
(b) 72
(c) 80
(d) 96
Solution: Let number of wickets taken till the last match=n
First average= 24.85 runs per wicket
Second average= 52/5=10.4 runs per wicket
Final average= 24
0.85×n-13.6×5=0
n=80
Option (c) is correct.
Que 3: A cricket player has an average of 32 runs over 10 innings. To increase his average by 4 runs, Find out how many runs he must score in his next inning.
(a) 76
(b) 70
(c) 81
(d) 68
Solution: Let the score in his last inning= k
-10×4+(k-36)×1=0
k=76
Option (a) is correct.
Que 4: In a hostel, there were initially 35 students. When the number of students is increased by 7, the expenditure on food per day increases by 42 rs. However, this increase in students causes the average expenditure per student to decrease by 1 rs. What was the initial expenditure on food per day?
(a) 400 rs
(b) 432 rs
(c) 442 rs
(d) 420 rs
Solution: Let initial average= x
42(x-1)-35x=42
x=12
Initial expenditure= 12×35=420 rs
Option (d) is correct.
Que 5: Two years ago, the average age of a family consisting of 8 members was 18 years. Recently, a baby was added to the family, and the average age of the family remains the same as it was two years ago. Find out the age of the baby?
(a) 2 year
(b) 1.5 years
(c) 1 year
(d) 2.5 years
Solution: Let the age of the baby=k
New average of 8 members= 20
Average of 9 members= 18
2×8-1×(18-k)=0
k=2
Option (a) is correct.
Que 6: In a family of 5 members, the average age of the family members currently is 33 years. The youngest member in the family is 9 years old. Find the average age of the family just before the birth of the youngest member?
(a) 30 years
(b) 29 years
(c) 25 years
(d) 24 years
Solution: Average of 5 members 9 years ago= 33-9=24
The average of 4 members= (5×24)/4=30 years
Option (a) is correct.
Que 7: In a class, there are initially 30 boys with an average age of 15 years. One boy, who is 20 years old, leaves the class, but two new boys join in his place. The ages of the two new boys differ by 5 years. After these changes, the average age of all the boys in the class remains 15 years. Find out the age of the younger newcomer?
(a) 20 years
(b) 15 years
(c) 10 years
(d) 8 years
Solution: There will be no change in the replacement of two persons of age 20-20 years.
Now with the arrival of a new person of 15 years, the average of 31 boys will become equal to 15.
Total age of new persons= 20+15=35
x+(x+5)=35
x=15 years
Option (b) is correct.
Que 8: When the average age of a husband, wife, and their son was 42 years, the son got married. One year after the marriage, a child was born. When the child turned five years old, the average age of the family became 36 years. Find the age of the daughter-in-law at the time of marriage?
(a) 26 years
(b) 25 years
(c) 24 years
(d) 23 years
Solution: H+W+S⇒ Average=42
After 6 years, H+W+S⇒ Average=48
Age of Child=5 years
Age of Daughter in law=k
12×3+(k-36)×1-31×1=0
k=31
Required age= 31-6=25
Option (b) is correct.
Que 9: A team consists of 5 members who are weighed one after another. After each member is weighed, the average weight of the team is calculated. It is observed that the average weight increases by one kilogram each time. How much heavier is the last player compared to the first player?
(a) 4 kg
(b) 20 kg
(c) 8 kg
(d) 5 kg
Solution: The average increases by 1 after each new member is added, so their weights are in A.P. with a common difference of 2.
a, a+2, a+4, a+6, a+8
Difference in weights= 8 kg
Option (c) is correct.
Que 10: Ten years ago, the ages of the members of a joint family, consisting of eight people, added up to 231 years. Three years later, one member died at the age of 60, and a child was born during the same year. After another three years, one more member died at the age of 60, and another child was born during the same year. What is the approximate current average age of this eight-member joint family?
(a) 23 years
(b) 22 years
(c) 20 years
(d) 24 years
Solution: Initial sum=231
After 3 years⇒231+8×3-60+0=195
After 3 more years⇒195+8×3-60+0=159
After 4 more years⇒159+8×4=191
Average= 191/8≈24 years
Option (d) is correct.
Que 11: In a class, there are 20 boys and 30 girls. In the mid-semester examination, the average score of the girls was 5 higher than that of the boys. In the final exam, the average score of the girls dropped by 3, but the average score of the entire class increased by 2. By how much did the average score of the boys increase in the final exam?
(a) 9.5
(b) 10
(c) 4.5
(d) 6
Solution: 20 Boys⇒ x
30 Girls⇒ x+5
Let average=x
20×0+5×30=150
Actual average= x+150/50=x+3
New average of entire class= x+5
New average of 30 Girls= x+2
Let the new average of 20 Boys=k
{k-(x+5)}×20-3×30=0
k=x+9.5
Option (a) is correct.
Que 12: The average of 30 integers is 5. Out of these 30 integers, exactly 20 of them do not exceed 5. What is the highest possible value for the average of these 20 integers?
(a) 3.5
(b) 5
(c) 4.5
(d) 4
Solution: Minimum average of 10 integers=6
Let the maximum average of 20 integers=k
-(5-k)×20+1×10=0
k=4.5
Option (c) is correct.
Que 13: The average weight of students in a class increases by 600 grams when some new students join the class. The average weight of the new students is 3 kilograms more than the average weight of the original students. What is the ratio of the number of original students to the number of new students?
(a) 3:5
(b) 1:2
(c) 4:1
(d) 3:1
Solution: The number of old students=n
The number of new students=k
Average weight of n students= x
So, the average weight of k students= x+3
The average weight of the entire class= x+0.6
-0.6×n+2.4×k=0
n/k=4/1
Option (c) is correct.
Que 14: If a and b are non-negative real numbers such that a + 2b = 6, What is the average of the maximum and minimum possible values of (a + b)?
(a) 3
(b) 4
(c) 3.5
(d) 4.5
Solution: a+b+b=6
So the minimum value of b can be 0 and maximum value can be 3
a+b=6-b
(a+b)_{max}=6-0=6
(a+b)_{min}=6-3=3
Average= (6+3)/2=4.5
Option (d) is correct.
Que 15: Five students, including Amit, appear for an examination with possible marks between 0 and 50 (inclusive). The average marks for all the students is 38, and exactly three students scored more than 32. It is known that no two students received the same marks, and Amit scored the lowest among the five students. What is the difference between the highest and lowest possible marks that Amit could have obtained?
(a) 22
(b) 21
(c) 24
(d) 20
Solution: Maximum possible score of Amit=31
Let, the minimum score of Amit=k
k+50+49+48+32=38×5
k=190-179=11
Difference= 31-11=20
Option (d) is correct.
Que 16: In an examination, the average marks of students in sections A and B are 32 and 60, respectively. The number of students in section A is 10 less than that in section B. If the average marks of all the students across both sections combined is an integer, what is the difference between the maximum and minimum possible number of students in section A?
(a) 50
(b) 63
(c) 48
(d) 72
Solution: Let, the number of students in section A=x
The number of students in section A=x+10
Average of the entire class=k
32x+60.(x+10)=k.(2x+10)
k=\frac{46x+300}{x+5}.
k=46+\frac{70}{x+5}.
x=2, …., 65
Difference= 65-2=63
Option (b) is correct.
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