Study Notes: Average (SSC GD Mathematics)
Overview
Average is one of the most straightforward yet frequently tested topics in SSC GD Elementary Mathematics. The concept is simple: average represents the central value when a quantity is equally distributed among all members. Mastering average problems requires understanding the basic formula, knowing how to handle entry/exit of members, and quickly computing weighted averages. Students often lose marks on age-related average problems due to confusion about time shifts (years ago/years hence). This topic overlaps with ratio, percentage, and time-speed problems, so a strong grasp here builds confidence across multiple question types.
The exam tests three main variations: simple mean of numbers, weighted averages where different groups have different values, and age-based problems where you track changes over time. All three follow the same core principle but require slightly different setup techniques. Spend time understanding how adding or removing a member changes the total, and practice converting word problems into the average equation quickly.
Key Concepts
- Average (Mean) is the sum of all observations divided by the number of observations. Formula: Average = Total Sum ÷ Number of Items. This is the foundation of every average problem.
- Total Sum is always the product of average and count: Total = Average × Number of Items. This reverse calculation is crucial for solving most problems where the total is unknown.
- When a new member joins a group, calculate the new total by adding the new member's value to the old total, then divide by the new count. When a member leaves, subtract that member's value from the total.
- Weighted Average applies when different groups contribute differently. If group A has average x with n₁ members and group B has average y with n₂ members, the combined average = (n₁x + n₂y) ÷ (n₁ + n₂). Never just average the two averages unless group sizes are equal.
- In age problems, remember that when time passes, every person's age increases by the same amount, so the total age of a group increases by (number of people × years passed). The average age increases by exactly the same number of years when no one joins or leaves.
- Replacement problems occur when one member is replaced by another. The change in total equals the difference between the new and old member's value. Change in Average = Change in Total ÷ Number of Items.
- Many problems give you the average and ask for one unknown value. Set up the equation: Sum of all known values + Unknown = Average × Count. Solve for the unknown.
Formulas / Key Facts
- Basic Average Formula: Average = Sum of all observations ÷ Number of observations
- Total Sum Formula: Total = Average × Number of observations
- Average after adding one item: New Average = (Old Total + New Item) ÷ (Old Count + 1)
- Average after removing one item: New Average = (Old Total − Removed Item) ÷ (Old Count − 1)
- Weighted Average of two groups: Combined Average = (n₁ × Avg₁ + n₂ × Avg₂) ÷ (n₁ + n₂)
- Age increase over time: If a group of n people ages by t years, Total Age increases by n × t, and Average Age increases by t
- Effect of replacement: Change in Total = New Member's Value − Old Member's Value; Change in Average = Change in Total ÷ Count
- Finding unknown value: Unknown = (Required Average × Count) − Sum of Known Values
Worked Examples
Example 1: Basic Average The average of five numbers is 28. If one number is excluded, the average becomes 25. Find the excluded number.
Step 1: Total of 5 numbers = 28 × 5 = 140 Step 2: Total of 4 numbers = 25 × 4 = 100 Step 3: Excluded number = 140 − 100 = 40 Answer: 40
Example 2: Weighted Average The average weight of 30 students in class A is 45 kg and the average weight of 20 students in class B is 50 kg. Find the average weight of all 50 students.
Step 1: Total weight of class A = 30 × 45 = 1350 kg Step 2: Total weight of class B = 20 × 50 = 1000 kg Step 3: Combined total = 1350 + 1000 = 2350 kg Step 4: Combined average = 2350 ÷ 50 = 47 kg Answer: 47 kg
Example 3: Age Problem The average age of a family of 5 members is 24 years. If the youngest member is 4 years old, what was the average age of the family at the time of birth of the youngest member?
Step 1: Present total age = 24 × 5 = 120 years Step 2: Four years ago (at birth of youngest), the youngest contributed 0 years Step 3: Four years ago, the other 4 members were each 4 years younger: reduction = 4 × 4 = 16 years Step 4: Total age 4 years ago = 120 − 16 − 4 = 100 years Step 5: Average 4 years ago = 100 ÷ 4 = 25 years Answer: 25 years
Common Mistakes
Mistake: Averaging the averages directly when group sizes differ. A student calculates (45 + 50) ÷ 2 = 47.5 when combining class averages. Fix: Always multiply each average by its count first, add the totals, then divide by the total count. Use the weighted average formula.
Mistake: Forgetting that in age problems, everyone ages together. When calculating "5 years ago," students subtract 5 only from the total instead of (5 × number of people). Fix: Remember the rule — total age changes by (number of people × years). If 4 people age 5 years, total increases by 20 years, not 5.
Mistake: Confusing the change in total with the value being replaced. When one student scoring 60 is replaced by another scoring 70, thinking the new average depends only on 70. Fix: The change in total is 70 − 60 = 10. This changes the average by 10 ÷ count. The replacement creates a difference, not an absolute value.
Mistake: Not converting word problems into the basic formula. Students try to solve age problems by guessing rather than writing Total = Average × Count first. Fix: Always start by writing what you know in formula form. Convert all "average" statements into totals immediately. This reveals what's missing.
Mistake: Calculation errors when working with large numbers or fractions, especially in weighted averages. Fix: Break calculations into steps. Use approximation to check if your answer makes sense — the combined average must lie between the two group averages.
Quick Reference
- Average = Total ÷ Count; Total = Average × Count — memorize both directions
- To find one unknown value: Unknown = (Average × Count) − Sum of Known Values
- Weighted average always lies between the individual averages, closer to the larger group
- In age problems: when n people age t years, total age increases by n × t
- Replacement effect: Change in average = (New Value − Old Value) ÷ Total Count
- When joining/leaving: recalculate total first, then divide by new count