Average — Study Notes for SSC CGL Tier 1

Overview

Average is a foundational topic in SSC CGL Quantitative Aptitude, appearing in 2–3 direct questions per exam. Understanding averages is crucial not just for standalone problems but also for topics like Time and Work, Speed and Distance, and Data Interpretation. The exam tests three core areas: basic average calculation of numbers, weighted average scenarios (mixing groups with different averages), and age-related puzzles that require careful equation setup.

Mastery requires fluency with the fundamental formula, the ability to quickly adjust averages when elements are added or removed, and skill in translating word problems into algebraic expressions. Most questions are moderate difficulty, solvable within 60–90 seconds if you recognize the pattern. Strong performance here builds confidence and saves time for harder topics.

The topic rewards methodical practice: understand the concept once, then solve 20–30 varied problems to internalize the patterns. Focus on speed—knowing when to use shortcuts versus full calculation makes the difference between attempting 23 versus 25 questions in the exam.

Key Concepts

• **Average definition**: The average (arithmetic mean) of n numbers equals their sum divided by n. It represents a central value where all numbers could be "leveled" to be equal.

• **Balancing principle**: The average lies between the smallest and largest values. Deviations above the average exactly balance deviations below it—this principle enables quick mental checks.

• **Effect of adding/removing**: When you add a number greater than the current average, the new average increases. Remove a number below average, and the average rises. The magnitude of change depends on how far the new/removed value sits from the existing average.

• **Weighted average**: When combining groups with different averages, the overall average depends on both the group averages and their sizes. Larger groups pull the combined average toward their value more strongly.

• **Age problems structure**: These typically involve relationships between current ages and ages from x years ago or y years in the future. Set up equations carefully: if average age increases by 5 years in 5 years, that's expected; look for the puzzle element.

• **Sum-based approach**: Often easier to work with total sums than individual values. If average of 5 numbers is 20, their sum is 100—work from there rather than finding each number.

Formulas / Key Facts

**Basic Average Formula** Average = Sum of all values / Number of values OR Sum = Average × Number of values

**Change in Average When One Value is Replaced** New Average = Old Average + (New Value - Old Value) / Total Count

**Weighted Average of Two Groups** Combined Average = (n₁ × A₁ + n₂ × A₂) / (n₁ + n₂) where n₁, n₂ are group sizes and A₁, A₂ are their averages

**Average of Consecutive Numbers** For consecutive integers from a to b: Average = (a + b) / 2 For first n natural numbers: Average = (n + 1) / 2

**Average of First n Natural Numbers** Average = (n + 1) / 2; Sum = n(n + 1) / 2

**Average of First n Even Numbers** Average = n + 1; Sum = n(n + 1)

**Average of First n Odd Numbers** Average = n; Sum = n²

**Deviation Method for Average** Choose any assumed average A. Find deviations. Actual Average = A + (Sum of deviations / n)

Worked Examples

**Example 1: Basic Average with Replacement** The average of 5 numbers is 27. If one number 25 is replaced by 40, what is the new average?

*Solution:* Original sum = 27 × 5 = 135 Change in sum = 40 - 25 = 15 New sum = 135 + 15 = 150 New average = 150 / 5 = 30

*Quick method:* Change in average = Change in value / Count = 15 / 5 = 3 New average = 27 + 3 = 30

**Example 2: Weighted Average** The average weight of 30 students in class A is 40 kg. The average weight of 20 students in class B is 35 kg. Find the average weight of all 50 students.

*Solution:* Total weight of class A = 30 × 40 = 1200 kg Total weight of class B = 20 × 35 = 700 kg Combined total weight = 1200 + 700 = 1900 kg Combined average = 1900 / 50 = 38 kg

**Example 3: Age-Related Problem** The average age of a family of 5 members is 24 years. If the youngest member is 8 years old and was born when the family had 4 members, what was the average age of the family at that time?

*Solution:* Current total age = 24 × 5 = 120 years 8 years ago, the 4 existing members' total age = 120 - 8 - (8 × 4) = 120 - 8 - 32 = 80 years Average age 8 years ago = 80 / 4 = 20 years

**Example 4: Average Income Problem** The average monthly income of A and B is Rs. 5050. The average monthly income of B and C is Rs. 6250. The average monthly income of A and C is Rs. 5200. What is the monthly income of A?

*Solution:* A + B = 5050 × 2 = 10,100 ... (i) B + C = 6250 × 2 = 12,500 ... (ii) A + C = 5200 × 2 = 10,400 ... (iii)

Adding all three: 2(A + B + C) = 33,000 Therefore: A + B + C = 16,500 ... (iv)

From (iv) - (ii): A = 16,500 - 12,500 = Rs. 4,000

Common Mistakes

**Mistake: Forgetting to adjust for time in age problems** Wrong thinking: "Average age increased by 4 years means total age increased by 4." Correct fix: If n people age by t years, total age increases by n × t. An increase beyond this indicates additions/replacements.

**Mistake: Directly averaging the averages** Wrong thinking: "Average of 40 and 35 is 37.5, so combined average is 37.5." Correct fix: You cannot average averages unless groups are equal in size. Always use weighted average formula considering group sizes.

**Mistake: Confusing sum and average in multi-step problems** Wrong thinking: Working with average values when the calculation requires sum. Correct fix: Convert to sum immediately (Sum = Average × Count), perform all operations, then convert back to average if needed.

**Mistake: Not checking if answer is in reasonable range** Wrong thinking: Accepting an average that lies outside the range of given values. Correct fix: Average must lie between minimum and maximum values. If your answer violates this, recheck calculations.

**Mistake: Mishandling negative deviations in deviation method** Wrong thinking: Taking absolute values of all deviations before summing. Correct fix: Keep signs intact—negative deviations below assumed average, positive above. The algebraic sum matters.

Quick Reference

• **Core formula**: Average = Sum / Count. Work backward: Sum = Average × Count.
• **Replacement shortcut**: New Avg = Old Avg + (Difference / Total numbers).
• **Weighted average**: Larger group pulls the combined average toward its value.
• **Consecutive numbers**: Average = (First + Last) / 2.
• **Age problems**: Set up equations for "total age t years ago/hence" carefully.
• **Speed check**: Average must lie between min and max values—use to verify answers.