Average, also known as the arithmetic mean, is a measure that summarizes a set of numbers by dividing the sum of these numbers by the count of values in the set. This simple yet powerful tool is widely used in various fields such as statistics, economics, and everyday life to determine central tendencies.

For students preparing for competitive exams like SSC, IBPS, and GRE, mastering average-related problems is essential. These exams often include questions that test your ability to calculate averages, work with consecutive numbers, and solve real-world problems involving averages. Practice questions help reinforce these concepts, ensuring that you can quickly and accurately handle them under exam conditions.

This article includes definitions of different types of averages along with their formula and practice problems on Average with solution.

What is Average?

An average is a measure that represents the central or typical value in a set of data. It is a statistical concept used to summarize and describe a set of values.

Average = Sum of all given observations/ Total number of observation

There are several types of averages, including:

• Arithmetic Mean (Arithmetic Average)
• Weighted Mean (Weighted Average)
• Geometric Mean (Geometric Average)
• Harmonic Mean (Harmonic Average)

Arithmetic Mean

The arithmetic mean is the most common type of average. It is calculated by adding all the values in a dataset and dividing the sum by the number of values.

Arithmetic mean= [Tex]\frac{\sum_{i=1}^n x_i}{n}[/Tex]

Where x_i represents each value in the dataset, and n is the number of values.

Weighted Average

The formula for calculating the weighted mean is used to calculate the average of a set of values that have different weights is:

Weighted Average = (w₁x₁ + w₂x₂ + . . . + w_nx_n)/(w₁ + w₂ + ……… w_n)

Geometric Mean

The geometric mean is calculated by multiplying all the values together and then taking the n-th root (where n is the number of values).

Geometric Mean= [Tex]\left( \prod_{i=1}^n x_i \right)^{\frac{1}{n}}[/Tex]

Harmonic Mean

The harmonic mean is the reciprocal of the arithmetic mean of the reciprocals of the values. It is often used in situations where the values are rates or ratios.

Harmonic Mean= [Tex] \frac{n}{\sum_{i=1}^n \frac{1}{x_i}}[/Tex]

Problems on Average with Solutions

Problem 1: Calculate the average of the following numbers: 6, 8, 2, 3, 12, 14.

Solution:

Average = Sum of all given observations/ Total number of observations

Total number of observations = 6

Mean = 6 + 8 + 2 + 3 + 12 + 14 / 6 = 45/6 = 7.5

Thus, the average of 6, 8, 2, 3, 12, 14 is 7.5

Problem 2: Find the average of the first 15 natural numbers.

Solution:

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

Sum of first 15 natural numbers = 15 × ( 15 + 1) / 2 = 15 × 8 = 120

Average = 120 / 15 = 8

Thus, the average of first 15 natural numbers is 8.

Problem 3: Calculate the weighted average of the following data: (3, 1), (4 2), (7, 3), where the first number is the value and the second number is the weight.

Solution:

Weighted Mean = 3 × 1 + 4 × 2 + 7 × 3 / 1 + 2 + 3 = 3 + 8 + 21 / 6 = 32 / 6 = 5.33

Therefore, the weighted average of the following data is 5.33

Problem 4: The average of five numbers is 16. Four of the numbers are 12, 15, 18, and 20. Find the fifth number.

Solution:

Total sum = 5 × 16 = 80

Sum of four numbers = 12+ 15 + 18 + 20 = 65

Now, Fifth number = 80 − 65 = 15

Thus, the fifth number is 15.

Problem 5: A student scored 90, 85, 90, 78, and 92 in five subjects. What is his average score?

Solution:

Average score = Sum of all scores/ Total number of scores

= 90 + 85 + 90 + 78 + 92 / 5

= 435 /5 = 87

Thus, the average score of student is 87.

Problem 6: Find the average of the first five even numbers.

Solution:

The first five even numbers are = 2, 4, 6, 8, 10

Average = (2 + 4 + 6 + 8 + 10)/ 5 = 30 / 5 = 6

The average of first five even numbers is 6.

Problem 7: The average of 8, 12, and 16 is added to the average of 20, 24, and 28. What is the sum of these two averages?

Solution:

Average of first set of numbers = 8 + 12 + 16 / 3 = 36 / 3 = 12

Average of second set of numbers = 20 + 24 + 28 / 3 = 72 / 3 = 24

Sum of averages = 12 + 24 = 36.

Problem 8: The average of 12 numbers is 50. If each number is increased by 14, then what will be the new average?

Solution:

Given the average of 12 numbers is 50

If each number is increased by 14

Average = Sum of all given observations/ Total number of observation,

then, new average = 50 + 14 = 64

Note: If each number is increased by any number x , then new average becomes -> “old average” + x

Problem 9: The average score of a student in 6 tests is 75. If the scores in five tests are 70, 80, 85, 65, and 75, find the score in the sixth test.

Solution:

Average Score = Sum of scores in all tests/ Number of tests

⇒ 75 = Sum of scores in all tests / 6

⇒ Total Score = 6 × 75 = 450

Score in sixth test = Total score – Sum of five scores

Sum of five scores = 70 + 80 + 85 + 65 + 75 = 375

Thus, Score in sixth test = 450 – 375 = 75.

Problem 10: The average of two classes of students is 70 and 80 respectively. If the first class has 30 students and the second class has 20 students, find the combined average.

Solution:

Combined Average = [( 70 × 30) + (80 × 20)]/(30 + 20) = (2100 + 1600)/ 50 = 3700 / 50 = 74

The combined average of given two classes is 74.

Problem 11: The average weight of a group of 10 people is 70 kg. If the weights of nine of them are 65, 75, 70, 68, 72, 74, 69, 71, and 73 kg, find the weight of the tenth person.

Solution:

Average weight = Sum of weight of all people/ Number of people

⇒ 70 = Sum of weight of all people / 10

⇒ Total weight = 70 × 10 = 700

Sum of nine weights = 65 + 75 + 70 + 68 + 72 + 74 + 69 + 71 + 73 = 637

Weight of tenth person = Total weight – sum of weight of nine people

Weight of tenth person = 700 – 637 = 6

Problem 12: The average of six numbers is 15. If the sum of five of these numbers is 65, find the sixth number.

Solution:

Given: Average = 15

⇒ Sum of All Observations/6 = 15

⇒ Sum of All Observations = 6 × 15 = 90

Sum of five numbers = 65

Thus, Sixth number = 90 − 65 = 25.

Problem 13: Calculate the geometric mean of the numbers 2, 8, and 32.

Solution:

Multiply all the values together

2 × 8 × 32 = 512

Take the cube root (since there are three values):

Geometric Mean = ∛(512) = 8

Thus, the geometric mean of 2, 8, and 32 is 8.

Problem 14: Calculate the harmonic mean of the numbers 2, 3, and 6.

Solution:

Calculate the reciprocals of each value:

1/2, 1/3, and 1/6

Sum the reciprocals:

1/2 + 1/3 + 1/6 = (3 + 2 + 1)/6 = 6/6 = 1

Divide the number of values by the sum of the reciprocals (there are three values):

Harmonic Mean = 3/1

Thus, the harmonic mean of 2, 3, and 6 is 3.

Average Practice Problems: Worksheet

Problem 1: Calculate the average of the following numbers: 13 , 14, 17, 23, 12.

Problem 2: Find the average of the first 30 natural numbers.

Problem 3: The average of five numbers is 30. Four of the numbers are 25, 35, 40, and 20. Find the fifth number.

Problem 4: The average of 15 numbers is 72. If each number is increased by 10, then what will be the new average?

Problem 5: The average of two classes of students is 50 and 80 respectively. If the first class has 40 students and the second class has 15 students, find the combined average.

Problem 6: The average score of a student in 7 tests is 70. If the scores in five tests are 70, 70, 80, 85, 65, and 75, find the score in the seventh test.

Problem 7: The average of 14, 16, and 20 is added to the average of 18, 25, and 27. What is the sum of these two averages?

Problem 8: Calculate the average of first eight odd numbers.

Problem 9: A student scored 91, 88, 93, 88, and 92 in five subjects. What is his average score?

Problem 10: If the average of four numbers is 35, and the first three numbers are 30, 40, and 45, find the fourth number.

Read More,

• Average
• Weighted Average Formula
• Arithmetic Mean
• Geometric Mean
• Harmonic Mean

Practice Problems on Average with Solution- FAQs

What is average or mean?

Average is a measure that represents the central value in a set of data. It is a statistical concept used to summarize and describe a set of values.

Average = Sum of all given observations/ Total number of observation

What are different types of averages/ means?

There are several types of averages, including:

• Arithmetic Mean (Arithmetic Average)
• Weighted Mean (Weighted Average)
• Geometric Mean (Geometric Average)
• Harmonic Mean (Harmonic Average)

What is Geometric mean used for ?

The geometric mean is used to calculate average growth rates, average ratios, and average multiplicative rates. It provides a more accurate measure than the arithmetic mean when dealing with proportional growth or with numbers whose values are exponential in nature.

How is Harmonic Mean calculated?

The harmonic mean is calculated as the reciprocal of the arithmetic mean of the reciprocals of a set of numbers.

Harmonic Mean = [Tex]n/∑_{i=1}^n1/n_i[/Tex]