Study Notes: Average (SSC GD Mathematics)

Overview

Average is one of the most straightforward yet frequently tested topics in SSC GD Elementary Mathematics. Every year, 2–3 direct questions on average appear, often combined with age problems, group comparisons, or weighted averages. 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