Average — Study Notes for MH Police Bharti
Overview
Average is one of the most reliable scoring topics in the Maharashtra Police Bharti mathematics section. Questions are straightforward once you master the basic formula and its variations. Examiners typically test your ability to find the average of a given set, work backward to find a missing number, or handle scenarios where items are added, removed, or replaced.
This topic connects directly with other quantitative areas like ratio, percentage, and data interpretation. A solid grip on average problems saves time in the exam because most can be solved in under 30 seconds with the right approach.
To score full marks here, you must understand three things: the basic average formula, the concept of weighted average, and how changes in a group affect the overall average.
Key Concepts
- Average (Mean) is the sum of all observations divided by the number of observations. It represents the "central" value of a data set.
- Weighted Average applies when different items have different importance (weights). Multiply each value by its weight, sum them, then divide by total weight.
- Effect of Adding a Number: If a new number is added to a group, the new average shifts toward that number. If the new number equals the old average, average remains unchanged.
- Effect of Removing a Number: Removing a number above the average pulls the average down; removing one below pulls it up.
- Replacement Rule: When one item is replaced by another, change in sum = (new value – old value). This change divided by count gives shift in average.
- Average of Consecutive Numbers: For any arithmetic progression, average = (first term + last term) ÷ 2.
- Age-based Problems: Average age of a group changes uniformly with time—after T years, average increases by T (assuming no one leaves or joins).
Formulas / Key Facts
| Concept | Formula |
|---|---|
| Basic Average | Average = Sum of observations ÷ Number of observations |
| Sum from Average | Sum = Average × Number of observations |
| Weighted Average | Weighted Avg = (w₁x₁ + w₂x₂ + ... + wₙxₙ) ÷ (w₁ + w₂ + ... + wₙ) |
| Average of first n natural numbers | (n + 1) ÷ 2 |
| Average of first n even numbers | n + 1 |
| Average of first n odd numbers | n |
| Average of consecutive numbers from a to b | (a + b) ÷ 2 |
| New average after adding one number | (Old Sum + New Number) ÷ (n + 1) |
| Change in average on replacement | Change = (New value – Old value) ÷ n |
Worked Examples
Example 1: Basic Average Calculation
Problem: The marks of 5 students are 72, 85, 68, 90, and 75. Find the average marks.
Solution:
- Sum = 72 + 85 + 68 + 90 + 75 = 390
- Number of students = 5
- Average = 390 ÷ 5 = 78
Example 2: Finding a Missing Number
Problem: The average of 6 numbers is 24. If five of them are 20, 25, 30, 18, and 27, find the sixth number.
Solution:
- Total sum = Average × Count = 24 × 6 = 144
- Sum of five known numbers = 20 + 25 + 30 + 18 + 27 = 120
- Sixth number = 144 – 120 = 24
Example 3: Weighted Average
Problem: In a class, 30 boys have an average score of 60 and 20 girls have an average score of 70. Find the average score of the entire class.
Solution:
- Total marks of boys = 30 × 60 = 1800
- Total marks of girls = 20 × 70 = 1400
- Total students = 30 + 20 = 50
- Combined average = (1800 + 1400) ÷ 50 = 3200 ÷ 50 = 64
Example 4: Replacement Problem
Problem: The average weight of 10 persons is 65 kg. One person weighing 70 kg is replaced by a new person, and the average becomes 64.5 kg. Find the weight of the new person.
Solution:
- Change in average = 64.5 – 65 = –0.5 kg
- Total change in sum = –0.5 × 10 = –5 kg
- Weight of new person = 70 – 5 = 65 kg
Example 5: Average of Consecutive Numbers
Problem: Find the average of all integers from 15 to 45.
Solution:
- Average = (First + Last) ÷ 2 = (15 + 45) ÷ 2 = 60 ÷ 2 = 30
Common Mistakes
| Wrong Thinking | Correct Approach |
|---|---|
| Adding a number always increases the average | Only true if the new number is greater than the current average. If it equals the average, no change occurs. |
| Using simple average when weights are given | When groups have different sizes, you must use weighted average, not just (Avg₁ + Avg₂) ÷ 2. |
| Forgetting to update count after addition/removal | When a person joins, divide by (n + 1); when one leaves, divide by (n – 1). |
| Confusing sum with average in backward problems | Always calculate total sum first using Sum = Average × Count, then find the unknown. |
| Assuming average of 1 to n is n ÷ 2 | Correct formula is (n + 1) ÷ 2. For example, average of 1 to 10 is 5.5, not 5. |
Quick Reference
- Average = Sum ÷ Count — memorize and apply instantly.
- Sum = Average × Count — the key to all backward problems.
- Weighted Average: Multiply each average by its group size, sum, then divide by total group size.
- Consecutive integers a to b: Average = (a + b) ÷ 2.
- First n natural numbers: Average = (n + 1) ÷ 2.
- Replacement shortcut: Change in sum = Change in average × Count.