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.
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 ,….., xn the average is given by the formula,
Average = (x1 + x2 + … + xn)/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 numbers = 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
Average - Questions and Answers
Below are some Solved examples to help you understand the concept better.
Question 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 = 816Average of first 7 observations = 65
Sum of first 7 observations = 65 × 7 = 455Sum of 8th, 9th, 10th, and 11th = 72 × 4 = 288
12th number = 816 – 455 – 288 = 73
Question 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/hrHence, the average speed for the entire trip is approximately 61.54 km/hr.
Question 3: Average Fuel Efficiency Suppose you want to calculate the average fuel efficiency of a car for a road trip. You recorded the following distances covered on different days: 250 miles, 200 miles, 180 miles, and 300 miles. The car's fuel tank capacity is 10 gallons.
Solution:
Sum of all values = 250 + 200 + 180 + 300 = 930 miles
Total number of values = 4
Average = 930 miles / 4 = 232.5 miles per day
Question 4: Average Test Scores (Weighted Average) Consider a class with two tests: Test 1 and Test 2. The weightage of Test 1 is 40% and Test 2 is 60%. The scores obtained by a student in each test are as follows: Test 1 score = 80% and Test 2 score = 90%.
Solution:
Weighted average = (Weightage of Test 1 * Test 1 score) + (Weightage of Test 2 * Test 2 score)
Weighted average = (0.40 * 80) + (0.60 * 90)
Weighted average = 32 + 54 = 86
Therefore, the weighted average test score for the student is 86%.
Question 5: Average Daily Temperature Consider the daily temperatures recorded over a week: 25°C, 28°C, 30°C, 27°C, 26°C, 23°C, and 24°C. Let's calculate the average daily temperature.
Solution:
Sum of all values = 25 + 28 + 30 + 27 + 26 + 23 + 24 = 183
Total number of values = 7
Average = 183 / 7 = 26.14 (rounded to two decimal places)
Therefore, the average daily temperature over the week is approximately 26.14°C.