Average

1. Definition of Average:

• The average of a set of numbers is the sum of all the numbers divided by the total count of numbers.
• Formula: Average=

2. Key Concepts:

• Simple Average: Calculating the average of a set of values.
• Weighted Average: When different values have different levels of importance (weights), the average is calculated considering these weights.

Weighted Average =

• Average of Groups: Finding the average of combined groups.
• Change in Average: Calculating the new average when a number is added or removed.

3. Types of Problems:

• Basic Average Problems: Direct application of the average formula.
• Problems involving multiple groups: Questions where you need to calculate the average of combined groups.
• Finding Missing Numbers: Given the average and some numbers, find the missing number(s).
• Adjustment of Average: How the average changes when a new value is added or removed.
• Mixed Average Problems: Problems that involve different groups with different averages and require finding the combined average.

4. Common Scenarios:

• Daily Life Applications: Calculating averages in scenarios like average marks, average speed, average age, etc.
• Problems on Ages: Determining average age of a group, the impact on average age when a person joins or leaves.
• Problems on Runs/Marks: Finding the average marks or runs scored by a player/team.

5. Techniques and Shortcuts:

• Sum Formula: Directly use the sum of the observations to find the average.
• Balancing Method: For simpler and quicker calculations by balancing deviations from an assumed average.
• Use of Arithmetic Mean: Simplifying complex problems by breaking them down into smaller, manageable parts.

6. Importance in Exams:

• Banking Exams: Problems on averages frequently appear in the Quantitative Aptitude section, often involving mixed data sets and requiring quick and accurate calculations.
• SSC Exams: Averages are tested in both the Quantitative Aptitude and Data Interpretation sections. Problems can range from basic calculations to more complex scenarios involving grouped data.

7. Practice Tips:

• Focus on Speed and Accuracy: Regular practice to improve calculation speed.
• Use of Approximation: In competitive exams, sometimes approximation helps in saving time.
• Understand Patterns: Recognize common patterns in problems to apply the correct approach quickly.

8. Example Problems:

• Find the average of five numbers.
• Calculate the new average when a number is added to the group.
• Determine the missing number if the average of a set is known.

Here are some example problems related to averages that are commonly encountered in banking and SSC exams:

Example 1: Basic Average Calculation

Problem: Find the average of the following numbers: 12, 15, 18, 22, and 25.

Solution:

• Sum of numbers = 12 + 15 + 18 + 22 + 25 = 92
• Number of observations = 5
• Average =92/5=18.4

Answer: The average is 18.4.

 

Example 2: Weighted Average

Problem: In a class, there are 30 boys and 20 girls. The average marks of boys are 75, and the average marks of girls are 80. Find the overall average marks of the class.

Solution:

• Total marks of boys = 75×30=225075 \times 30 = 225075×30=2250
• Total marks of girls = 80×20=160080 \times 20 = 160080×20=1600
• Total marks = 2250 + 1600 = 3850
• Total number of students = 30 + 20 = 50
• Overall average = 3850/50=77

Answer: The overall average marks are 77.

 

Example 3: Adjusting the Average

Problem: The average of 5 numbers is 20. If one more number, 30, is added to the set, what is the new average?

Solution:

• Original sum of numbers = 20×5=10020 \times 5 = 10020×5=100
• New sum of numbers = 100 + 30 = 130
• New number of observations = 6
• New average = 130/6≈21.67

Answer: The new average is approximately 21.67.

 

Example 4: Finding a Missing Number

Problem: The average of 4 numbers is 25. If one of the numbers is 30, what is the average of the remaining three numbers?

Solution:

• Total sum of 4 numbers = 25×4=10025 \times 4 = 10025×4=100
• Sum of the remaining 3 numbers = 100 – 30 = 70
• Average of the remaining 3 numbers = 70/3​≈23.33

Answer: The average of the remaining three numbers is approximately 23.33.

 

Example 5: Combined Groups

Problem: The average marks of 30 students in section A is 60, and the average marks of 20 students in section B is 70. What is the average marks of all 50 students?

Solution:

• Total marks of section A = 60×30=180060 \times 30 = 180060×30=1800
• Total marks of section B = 70×20=140070 \times 20 = 140070×20=1400
• Total marks = 1800 + 1400 = 3200
• Total number of students = 30 + 20 = 50
• Combined average = 3200/50​=64

Answer: The combined average marks of all 50 students are 64.

 

Example 6: Impact of Adding a New Value

Problem: The average of 6 numbers is 10. If a new number 16 is added, what will be the new average?

Solution:

• Original sum of numbers = 10×6=6010 \times 6 = 6010×6=60
• New sum of numbers = 60 + 16 = 76
• New number of observations = 7
• New average = 76/7​≈10.86

Answer: The new average is approximately 10.86.