Table of contents | |
Introduction | |
What is an Average | |
Weighted Mean | |
Real Facts About Average | |
Average Speed | |
Examples |
Example 1: Find the average of first five natural numbers.
- Sol: First five natural numbers:
1, 2, 3, 4, 5
Step 1: Calculate the sum
Sum = 1 + 2 + 3 + 4 + 5
Sum = 1 + 2 + 3 + 4 + 5= 15
Step 2: Count the total number of values
Total numbers = 5
Step 3: Calculate the average
Average = 155 = 3
Therefore, The average of the first five natural numbers is 3.
Example 2: If a person with age 45 joins a group of 5 persons with an average age of 39 years. What will be the new average age of the group?
- Sol: Total age will be 45 + 5× 39 = 240. And there will be 6 persons now.
So, the average will be 240/6 = 40.
(or)
Since, 45 is 6 more than 39, by joining the new person, the total will increase by 6 and so the average will increase by 1.
So, the average is 39 + 1 = 40.
Example 3: Two students with marks 50 and 54 leave class VIII A and move to class VIII B. As a result, the average marks of class VIII A fall from 48 to 46. How many students were there initially in class VIII A?
- Sol: The average of all the students of class VIII A is 46, excluding these two students.
They have 4 and 8 marks more than 46. So, with the addition of these two students, 12 marks are adding more, and hence the average is increasing 2.
There should be 6 students in that class, including these two. This is the initial number of students.
Example: The average marks of 30 students in a section of class X are 20 while that of 20 students of the second section is 30. Find the average marks for the entire class X.
Sol: Given:
- Section A: Number of students = 30, Average marks = 20
- Section B: Number of students = 20, Average marks = 30
Step 1: Calculate Total Marks for Each Section
Total marks for Section A:
Total MarksA = 301 × 20 = 600
Total marks for Section B:
Total MarksB = 201 × 30 = 600
Step 2: Calculate Total Marks for the Entire Class
Total Marksclass = Total MarksA + Total MarksB = 600 + 600 = 1200
Step 3: Calculate Total Number of Students
Total Students = 30 + 20 = 50
Step 4: Calculate the Average Marks for the Entire Class
Average Marksclass = 120050 = 24
Therefore , average marks of the entire class is 24.
Fact 1: If each number is increased/decreased by a certain quantity n, then the mean also increases or decreases by the same quantity.
Fact 2: If each number is multiplied/ divided by a certain quantity n, then the mean also gets multiplied or divided by the same quantity.
Fact 3: If the same value is added to half of the quantities and the same value is subtracted from the other half quantities, then there will not be any change in the final value of the average.
Example: The average of 10 consecutive numbers starting from 21 is:
Sol: The average is simply the middle number, which is the average of 5th & 6th no. i.e, 25 & 26 i.e. 25.5
Example 1: A man travels 120 km at 60 km per hour and returns the same distance at 40 km per hour. What is his average speed for the entire journey?
- Sol: The formula for average speed in a round trip is:
- 2 × Speed1 × Speed2Speed1 + Speed2
- Here, Speed1 = 60 km per hour and Speed2 = 40 km per hour.
- Using the formula:
- 2 × 60 × 4060 + 40 = 4800100 = 48 km per hour
- Therefore , the answer to this question is 48 km per hour
Example 2: A grocer mixes two types of sugar. Type A costs ₹30 per kg, and Type B costs ₹50 per kg. How many kilograms of Type B sugar must he mix with 10 kg of Type A to get a mixture worth ₹40 per kg?
- Sol: Let x be the quantity of Type B sugar in kg.
- Using the weighted average formula:
- 30 × 10 + 50 × x10 + x = 40
- Now, simplify the equation:
- 300 + 50x = 40 × (10 + x)
- 300 + 50x = 400 + 40x
- Rearrange terms:
- 50x – 40x = 400 – 300
- 10x = 100
- x = 10
Example 3: A company has three departments:
Sol: First, calculate the total salary for each department:
- Department A: 10 × 60,000 = 600,000
- Department B: 15 × 80,000 = 1,200,000
- Department C: 5 × 50,000 = 250,000
Now, sum all the salaries and divide by the total number of employees:
600,000 + 1,200,000 + 250,00010 + 15 + 5 = 2,050,00030 = 68,333.33
Answer: ₹68,333.33
Example 4: A jar contains 10 liters of a solution with 30% salt. Two liters are removed and replaced with pure water. What is the new concentration of salt?
- Sol: Step 1: Initial amount of salt in the solution = 30 × 10100 = 3 liters of salt
- Step 2: Salt removed = 30 × 2100 = 0.6 liters of salt
- Step 3: Remaining salt after removal: 3 − 0.6 = 2.4 liters.
- The total volume of the solution is still 10 liters after replacing the removed 2 liters with pure water.
- Step 4: New concentration of salt = 2.410 = 0.24 or 24%
Example 5: The average marks of 3 classes are as follows:
Sol: First, calculate the total marks for each class:
- Class A: 50 × 75 = 3,750
- Class B: 30 × 80 = 2,400
- Class C: 20 × 85 = 1,700
Now, sum the total marks and the total number of students:
Total marks:
3,750 + 2,400 + 1,700 = 7,850
Total number of students:
50 + 30 + 20 = 100
The overall average is:
7,850100 = 78.5
110 videos|110 docs|120 tests
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1. What is the difference between average and weighted mean? |
2. How is the average speed calculated? |
3. Why is it important to understand averages in the GMAT exam? |
4. Can the average be affected by outliers? |
5. What are some real-life applications of averages? |
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