Average Formulas

Average is a fundamental concept in mathematics that is crucial for anyone preparing for competitive exams. It represents a central value of a set of numbers and is commonly used to summarize data. Many entrance exams and government job tests include questions that require calculating averages to analyze and compare different data sets effectively.

Read More: Average in Maths

Average Formula

The average in mathematics is calculated using the formula sum of values divided by the number of values. Hence, the average formula is given as

Average = Sum of Values/Number of Values

For given n numbers x₁, x₂, x₃ ,….., x_n the average is given by the formula,

Average = (x₁ + x₂ + … + x_n)/n

Important Formulas on Average

Some important tips and tricks to solve average questions are mentioned below. These formulas will help students and be useful in boards and competitive exams.

Average of first n natural numbers:

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

Average of first n natural number squares,

• Sum of square of first n natural numbers = n(n + 1)(2n + 1)/6
• Average of square of first n natural numbers = (n + 1)(2n + 1)/6

Average of first n natural number cubes:

• Sum of cube of first n natural numbers = [n(n + 1)/2]²
• Average of the cube of first n natural numbers = n(n + 1)²/4

Average of first n natural odd numbers:

• The sum of the first n natural odd numbers = n²
• Average of first n natural odd number = n

Average of first n natural even numbers:

• The sum of first n natural even numbers = n(n + 1)
• Average of first n natural even numbers = n + 1

Solved Examples of Average

Below are some Solved examples to help you understand the concept better.

Also Check: Tricks to solve Average Questions

Problem Statement 1: The average of 12 numbers is 68. If the average of the first 7 numbers is 65 and the average of the next 4 is 72, find the 12th number.

Solution:

Average of 12 observations = 68
Sum of 12 observations = 68 × 12 = 816

Average of first 7 observations = 65
Sum of first 7 observations = 65 × 7 = 455

Sum of 8th, 9th, 10th, and 11th = 72 × 4 = 288

12th number = 816 – 455 – 288 = 73

Problem Statement 2: A car travels at a speed of 80 km/hr on the way to a destination and returns at a speed of 50 km/hr. What is the average speed for the entire trip?

Solution:

Let a be the distance to the destination.

• Total distance traveled in the journey = 2a
• Time to travel to the destination = Distance / speed = a /80
• Time to travel back = Distance / Speed = a / 50
• Total time for the journey = a / 80 + a / 50

Average speed = Total distance / total time
= 2a / (a / 80 + a / 50)
= 4000 × 2a / 130a
= 4000 / 65
= 61.54 km/hr

Hence, the average speed for the entire trip is approximately 61.54 km/hr.

Related Articles:

• Average in Maths
• Difference Between Average And Mean
• Average Aptitude Questions

Summary

Average is an important topic in quantitative aptitude frequently encountered in competitive exams. It involves determining the central value of a set of numbers by calculating the arithmetic mean. Problems related to averages often require summing values and dividing by the total number of items, enabling candidates to analyze data effectively and draw meaningful conclusions.