Averages – Concept and Basics
The average value is the ratio of the sum of the number of values in a given set to the total number of values in the set. We use averages in a variety of reallife situations. So, the average is defined as the sum of the observations divided by the number of observations.
Average is calculated as the sum of the observations divided by the number of observations.
Difference Between Mean and Average
Average is defined as the sum of the values divided by the total number of words.
Mean is defined as the sum of the list’s largest and lowest numbers divided by two.
Averages Formula
Students can use the following important average formulas to answer about averages:
Average of first n natural numbers = (n+1) / 2
Average of squares of first n natural numbers = (n+1) (2n+1) / 6
Average of cubes of first n natural numbers = n (n+1)^{2} / 4
Average of first n even numbers = n + 1
Average of squares of first n even numbers = 2 (n+1) (2n+1) / 3
Average of cube of first n even numbers = 2n (n+1)2
Average of first n odd numbers = n
Average of squares of first n odd numbers = (2n+1) (2n1) / 3
Average of cube of first n odd numbers = n(2n^{2} – 1)
Practice Quizzes
Averages Quiz 1 – Coming Soon  Averages Quiz 2 – Coming Soon  Averages Quiz 3 – Coming Soon 
Sample Questions: Average – Related Problems
Q1. 20 boys go for dinner. 16 of them spent Rs. 64 each on their dinner and the rest spent Rs 4 more than the average expenditure of all 20. What was the total money spent by them?
a) 1200
b) 1500
c) 1800
d) 1300
e) None of these
Solution:
Q2. If the square root of the average ‘A’ of 11 numbers is increased by A, the result is another value B which is the average of first 10 of the 11 numbers used before. Then the 11th number (in terms of A is) is –
a. A + 10√A
b. A – 10√A
c. 10A + √A
d. 10A – √A
e. None of the above
Solution:
Let the sum of first ten numbers be x & let the 11th no. be y
So, A = (x + y) / 11 …(1) Also, B = x / 10 …(2)
ATQ, B = A + √A
or, x/10 = A+ √A {using (2)}
or, x =10(A+ √A) putting in (1)
y = 11A – x = 11A – (10 A+ 10√A) = (A – 10√A)
Q3. In a town a standard tax of Rs. 2000 has to be paid every month on an annual salary up to Rs. 45000 and an additional amount which is 15% of the salary exceeding Rs. 45000 has to be paid. If Ramesh pays an average of Rs. 2750 per month, then what is his average monthly salary?
a. Rs. 8,400
b. Rs. 8,750
c. Rs. 12,000
d. Rs. 9,000
e. Rs. 12,500
Solution:
Standard tax for 12 months = 2000 × 12 = Rs. 24,000
Amount paid by Ramesh = 2750 × 12 = Rs. 33,000
Difference = Rs. 9,000
Let exceeding salary = Rs. x. Then
15x/100 = 9000 ⇒ x =60,000
So, total annual salary of Ramesh = 45,000 + 60,000 = Rs. 1,05,000
Required monthly Salary = Rs. 8750
Q4. Average marks of boys of class is 78 while average marks of girls of the same class is 86. If average marks of all students is M and ratio of number of boys to number of girls in the class is N : 5 then which of the following can be the value of (M – N).
a. 90
b. 96
c. 88
d. 87
e. 80
Solution:
78 < M < 86
0 < N
M – N < 86
Only option possible is option E.
Q5. Average of 11 numbers is 71 and the average of first 5 numbers is 73. If average of last 4 numbers is 84 and the ratio of 6th and 7th number is 9 : 7, then find the 6th number.
a) 27
b) 36
c) 45
d) 54
e) None of these
Solution:
Let the 6th number = 9x, then 7th number = 7x
According to question
11 × 71 = 5 × 73 + 9x + 7x + 4 × 84
⇒ 781 = 365 + 16x + 336
⇒ 16x = 781 – (365 + 336) = 80
⇒ x = = 5
Hence, the 6th number = 9x = 9 × 5 = 45
Q6. In a class of seven students, one student of weight 140 kgs is replaced by a student of weight 119 kgs, then the average weight of the class is increased or decreased by how much kg?
a) 21
b) 14
c) 7
d) 5
e) 3
Solution:
Required decrease in the average weight of the class = (140119)/ 7 = 21/7 = 3 kgs.
Q7. The average marks obtained by 80 students in Math is 42. If the average marks of the passed students is 48 while the average marks of the failed students is 32, then find the difference of number of students who passed and failed in the math exam.
a) 30
b) 40
c) 50
d) 60
e) 20
Solution: E
Let passed students and failed students be P and F
42 × 80 = 48 × P + 32 × F ……(i)
and P + F = 80 …….(ii)
Solving (i) and (ii) , we get
P = 50 and F =30
Required difference = 50 – 30 = 20
Q8. There are two sections in a company. Section A has 30 employees and section B has 40 employees. The average salary of employees of section A and employees of section B is Rs. 5000 and Rs. 6000 respectively. Find the average salary of all the employees in the company.
a) Rs. 5571.43
b) Rs. 5000.02
c) Rs. 4872.12
d) Rs. 5000
e) None of these
Solution: A
Required average salary of all the employees in the company =(5000*30 + 6000 * 40)/(30+40)= Rs. 5571.428
Averages FAQs
Q. What is average?
The average value is the ratio of the sum of the number of values in a given set to the total number of values in the set.
Q. What is the sign for averages?
Averages are the mean value expressed as x bar (x̄), also referred to as the Average Symbol. The average sign can alternatively be represented as μ.
Q. What is the average of negative numbers?
If a negative number appears in a given list of numbers, candidates can determine the average using the following formula: Average = Sum of observations / Number of observations.
Q. How can I determine the average of a list?
Students can use the procedures outlined below to successfully compute the average of a given list of numbers of values.
Step 1: First, add all of the numbers from the list.
Step 2: Divide the estimated sum by the number of terms specified in the list.
Step 3: Use the Average formula to calculate and summarize the results. The average of numbers can be stated as follows: Average = the sum of the observations divided by the number of observations.
Q. What is the difference between the mean and the average?
The average is the sum of values divided by the total number of terms, whereas the mean is the sum of the list’s highest and smallest numbers divided by two.
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