Average — Study Notes for HSSC CET
Overview
Average is one of the most frequently tested topics in the Quantitative Ability section of HSSC CET. It appears both as direct calculation questions and as a component within other topics like Data Interpretation, Time and Distance, and Profit-Loss problems.
The concept is straightforward — average represents the central value of a dataset. However, exam questions often add layers of complexity through missing values, addition/removal of items, or weighted distributions. Mastering averages gives you quick wins in the exam while also building skills needed for more advanced problems.
Most can be solved within 30–60 seconds if you know the shortcuts. Focus on understanding the relationship between sum, count, and average rather than memorising formulas blindly.
Key Concepts
- Simple Average = Sum of all observations ÷ Number of observations. All items carry equal importance.
- Weighted Average applies when different items have different weights or frequencies. Each value is multiplied by its weight before summing.
- The Sum Principle: Average × Number of items = Total Sum. This relationship is the foundation of almost every average problem.
- Effect of adding/removing an item: When a new value is added, the change in average depends on how far the new value is from the old average.
- Average of consecutive numbers: For any arithmetic progression, the average equals the middle term (or average of two middle terms if count is even).
- Combining two groups: When two groups merge, the combined average lies between the individual averages, closer to the group with more members.
- Age-based problems: Average age of a group increases by 1 year after 1 year passes, assuming no additions or removals.
Formulas / Key Facts
| Concept | Formula |
|---|---|
| Simple Average | Average = Sum ÷ n |
| Sum from Average | Sum = Average × n |
| Weighted Average | (w₁×x₁ + w₂×x₂ + ... + wₙ×xₙ) ÷ (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 when one item is replaced | New Avg = Old Avg + (New value − Old value) ÷ n |
| Combined average of two groups | (n₁ × A₁ + n₂ × A₂) ÷ (n₁ + n₂) |
Quick fact: Sum of first n natural numbers = n(n+1)/2, so average = (n+1)/2.
Worked Examples
Example 1: Basic Average Calculation
Problem: The marks obtained by 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 marks
Example 2: Finding Missing Value
Problem: The average of 6 numbers is 42. If five of the numbers are 38, 45, 40, 50, and 35, find the sixth number.
Solution:
- Total sum = 42 × 6 = 252
- Sum of five numbers = 38 + 45 + 40 + 50 + 35 = 208
- Sixth number = 252 − 208 = 44
Example 3: Replacement Problem
Problem: The average weight of 10 persons is 65 kg. If one person weighing 70 kg is replaced by another person, the average becomes 64.5 kg. Find the weight of the new person.
Solution:
- Old total = 65 × 10 = 650 kg
- New total = 64.5 × 10 = 645 kg
- Decrease in total = 650 − 645 = 5 kg
- Weight of new person = 70 − 5 = 65 kg
Shortcut: Change in average × n = Difference caused by replacement → (−0.5) × 10 = −5, so new person is 5 kg less than old = 65 kg.
Example 4: Weighted Average
Problem: In a class, 30 boys scored an average of 60 marks and 20 girls scored an average of 70 marks. Find the average marks of the entire class.
Solution:
- Total marks of boys = 30 × 60 = 1800
- Total marks of girls = 20 × 70 = 1400
- Combined total = 1800 + 1400 = 3200
- Total students = 30 + 20 = 50
- Class average = 3200 ÷ 50 = 64 marks
Note: The answer (64) is closer to 60 than to 70 because boys (with lower average) outnumber girls.
Common Mistakes
- Mistake: Adding averages directly when combining groups. Fix: Always calculate total sums first, then divide by total count. Averages cannot be added directly unless group sizes are equal.
- Mistake: Forgetting that average changes when the group size changes. Fix: If a person joins or leaves, recalculate using new count. Average = New Sum ÷ New Count.
- Mistake: Confusing "average of first n even numbers" with "average of even numbers from 1 to n." Fix: First 5 even numbers are 2,4,6,8,10 (avg = 6). Even numbers from 1 to 5 are 2,4 (avg = 3). Read carefully.
- Mistake: In age problems, assuming average remains constant over time. Fix: If 1 year passes, add 1 to the average age (assuming same group). If someone new joins, calculate separately.
- Mistake: Using weighted average formula when simple average is needed, or vice versa. Fix: Use weighted average only when items have different frequencies or importance explicitly stated.
Quick Reference
- Average = Sum ÷ Count — the only formula you truly need.
- To find sum: Multiply average by count.
- Consecutive numbers a to b: Average = (a + b) ÷ 2.
- First n natural numbers: Average = (n + 1) ÷ 2.
- When one value changes by x: Average changes by x ÷ n.
- Combined average always lies between individual averages, weighted toward the larger group.