Average
~12 min read
- What: Average covers the arithmetic mean, weighted average, replacement effects on the average, and special averages — average of consecutive numbers, average speed in two-leg journeys.
- Why it matters: CDS papers from 2000 to 2023 carry 2–4 questions per sitting on Average — the replacement archetype (one member changes, find new average) recurs constantly.
- Key fact: If a new entry is added to a set of \(n\) numbers with average \(A\), and the average rises by \(d\), the new entry equals \(A + (n+1) d\). One formula solves dozens of CDS variants.
Average is one of those CDS chapters where two or three formulae cover 90% of the question types. Setters love it because it admits clean integer answers, and aspirants love it because once you have the replacement and weighted-average reflexes, the chapter is almost free marks.
This page is built from CDS Previous Year Questions across 2000–2023, plus NCERT Class 8 Tales by Dots and Lines and Class 10 Statistics. Pair with Statistics (which formalises the mean) and Time, Speed and Distance (average speed).
What This Topic Covers
CDS scope: (1) arithmetic mean — sum/count; (2) weighted average — sum of (value × weight) divided by sum of weights; (3) replacement — one member replaced, find new average or the replacement value; (4) addition / removal — entry added or removed, find change in average; (5) average of consecutive numbers — equal to the middle term; and (6) average speed — special case for two-leg journeys.
Why This Topic Matters
- 2–4 CDS questions per paper, with clean integer answers.
- The replacement archetype \(\text{new entry} = A + (n+1)d\) is the highest-ROI shortcut in the chapter.
- Weighted average underpins alligation and many mixture problems.
Exam Pattern & Weightage
| Year / Paper | No. | Subtopics Tested |
|---|---|---|
| 2010-I/II | 3 | Replacement, addition of entry |
| 2011-I/II | 3 | Average speed, weighted average |
| 2012-I/II | 3 | Consecutive average, replacement |
| 2013-I/II | 3 | Average of squares, weighted |
| 2014-I/II | 4 | Replacement, age problems |
| 2015-I/II | 3 | Group merging, weighted |
| 2016-I/II | 3 | Average speed, addition |
| 2017-I/II | 3 | Replacement, consecutive |
| 2018-I/II | 3 | Group merging, weighted |
| 2019-II | 2 | Replacement of multiple members |
| 2020-I/II | 3 | Weighted, replacement |
| 2021-I/II | 3 | Age, average speed |
| 2022-I / 2023-I | 3 | Mixed application |
The average of \(n\) consecutive integers (or any arithmetic progression) equals the middle term. For odd \(n\), this is the literal middle. For even \(n\), it is the average of the two middle terms. So the average of 11, 12, 13, 14, 15 is 13 — no addition needed.
Core Concepts
Arithmetic Mean
Replacement Formula
If one member of a group of \(n\) is replaced and the average changes by \(\Delta A\), then the new member exceeds the old member by exactly \(n \cdot \Delta A\). This is the single most useful identity in CDS average problems.
Addition Formula
If a new entry is added to \(n\) existing members with average \(A_{\text{old}}\), and the new average is \(A_{\text{old}} + \Delta A\), the new entry equals \(A_{\text{old}} + (n+1) \Delta A\). Note the \(+1\) — the entry "lives" in the new larger group.
Weighted Average
If a class has 30 boys averaging 65 marks and 20 girls averaging 75, the class average is \(\tfrac{30 \cdot 65 + 20 \cdot 75}{30 + 20} = \tfrac{1950 + 1500}{50} = 69\) marks.
Average Speed (Two Legs)
Not the arithmetic mean! If you go \(d\) km at speed 40 and return at speed 60, average speed is \(\tfrac{2 \cdot 40 \cdot 60}{40 + 60} = \tfrac{4800}{100} = 48\) km/h — not 50.
For equal distances at different speeds, average speed is the harmonic mean, not the arithmetic mean. For equal times at different speeds, it is the arithmetic mean. CDS exploits this distinction.
Worked Examples
Example 1 — Replacement (2014-I)
Q: The average age of 5 friends is 24 years. One friend leaves; the average becomes 23. What is the age of the friend who left?
- Sum before = \(5 \times 24 = 120\). Sum after = \(4 \times 23 = 92\).
- Friend who left = \(120 - 92 = 28\) years.
Example 2 — Addition with Δ\(A\) (2016-II)
Q: The average weight of 10 students is 45 kg. A new student joins; the average rises by 0.5 kg. What is the new student's weight?
- Apply formula: new entry \(= A_{\text{old}} + (n+1) \Delta A\).
- \(= 45 + 11 \cdot 0.5 = 45 + 5.5 = 50.5\) kg.
Example 3 — Replacement Difference (2017-I)
Q: The average mark of 8 students drops by 2 marks when a student scoring 80 is replaced by another. What is the new student's score?
- Apply formula: new score = old score + \(n \cdot \Delta A = 80 + 8 \cdot (-2) = 80 - 16 = 64\).
Example 4 — Weighted Average (2018-II)
Q: Section A has 40 students with average 60; Section B has 60 students with average 70. Find the combined average.
- Sum A = \(40 \cdot 60 = 2400\). Sum B = \(60 \cdot 70 = 4200\). Total sum = 6600.
- Total count = 100. Combined average = \(6600 / 100 = 66\).
Example 5 — Average Speed (2011-II)
Q: A car travels from A to B at 60 km/h and returns at 40 km/h. Find the average speed.
- Apply harmonic mean: \(\tfrac{2 \cdot 60 \cdot 40}{60 + 40} = \tfrac{4800}{100} = 48\) km/h.
Example 6 — Consecutive Numbers (2012-I)
Q: Find the average of the first 20 natural numbers.
- For first \(n\) natural numbers: sum \(= n(n+1)/2\), average \(= (n+1)/2\).
- Average of first 20 natural numbers \(= 21/2 = 10.5\).
Example 7 — Group Merging with Constraint (2015-II)
Q: Two groups of 6 and 4 have averages 12 and 17 respectively. Find the average of all 10.
- Sum of group 1 = \(6 \cdot 12 = 72\). Sum of group 2 = \(4 \cdot 17 = 68\). Total = 140.
- Average = \(140 / 10 = 14\).
How CDS Tests This Topic
Five archetypes: (1) replacement — one member swaps, find new value or new average; (2) addition / removal — entry added or removed, average shifts; (3) weighted (group merging) — two or more groups combine; (4) average speed two-leg journey — harmonic mean; (5) consecutive or arithmetic progression — middle term.
Exam Shortcuts (Pro-Tips)
Shortcut 1 — The "\(n \cdot \Delta A\)" Identity
Shortcut 2 — Consecutive = Middle Term
Average of \(n\) consecutive integers (or AP) is the middle term (\(n\) odd) or mean of two middle terms (\(n\) even). Skip the sum.
Shortcut 3 — Harmonic Mean for Equal Distance
For equal distances at two speeds: \(\bar{v} = \tfrac{2 v_1 v_2}{v_1 + v_2}\). For three legs of equal distance: \(\bar{v} = \tfrac{3 v_1 v_2 v_3}{v_1 v_2 + v_2 v_3 + v_3 v_1}\).
Shortcut 4 — Sum is What Matters
Translate every average question into a "sum" question. Old sum = \(n_1 A_1\); new sum = old sum ± Δ. Then divide by new count.
Shortcut 5 — Age Problems
"Average age of family of \(n\) members \(t\) years ago was \(A\). What is the present average?" Just add \(t\) to \(A\) (everyone has aged by \(t\) years). Births and deaths add/remove members, so recompute the sum.
Common Question Patterns
Pattern 1 — One Member Replaced
Old average \(A_1\), new average \(A_2\). \(n\) members. Find the new (or old) member. Use \(n \cdot \Delta A\).
Pattern 2 — Entry Added / Removed
New member joins or leaves; average shifts. Apply \(A_{\text{new}} + n \cdot \Delta A\) (for joiner) or sum-based reasoning.
Pattern 3 — Group Merging
Two or more groups with given counts and averages; find combined average. Weighted-mean formula.
Pattern 4 — Two-Leg Speed
Equal distances at different speeds — harmonic mean. Equal times at different speeds — arithmetic mean.
Pattern 5 — Age over Time
Present average versus past or future average. Account for everyone aging by the same Δt; account for any births or deaths.
Preparation Strategy
Week 1. Master the four core formulae: arithmetic mean, replacement (\(n \cdot \Delta A\)), addition into group ((\(n+1) \Delta A\)), and weighted mean. Drill 20 problems mixing the four.
Week 2. Average speed (harmonic vs arithmetic), age problems, and group-merging variants. Layer in Time, Speed and Distance for average-speed cross-training.
Mock testing. Take timed papers. Track whether you get the +1 right in addition problems and whether you remember harmonic mean for equal-distance two-leg speeds. Both are common slip points. Use CDS mock tests for pace.
Drill Average in Real Time
CDS mocks with replacement, group-merging, and average-speed problems. Five archetypes — five formulae — reflex.
Start Free Mock TestFrequently Asked Questions
What is the replacement formula for averages?
If one member of a group of \(n\) is replaced and the average changes by \(\Delta A\), then the new member exceeds the old by exactly \(n \cdot \Delta A\). Example: average of 8 students drops by 2 when one scoring 80 is replaced. New score = \(80 + 8 \cdot (-2) = 64\).
What if a new entry is added to the group?
The formula becomes \(\text{new entry} = A_{\text{old}} + (n+1) \cdot \Delta A\), where \(n\) was the old count and \(\Delta A\) is the change in average. The \(+1\) matters because the entry is part of the new (larger) group.
Why is average speed for equal distances not the arithmetic mean?
Average speed is total distance / total time. For equal distances at speeds \(v_1\) and \(v_2\), times are \(d/v_1\) and \(d/v_2\). Total time = \(d(1/v_1 + 1/v_2)\); total distance = \(2d\). So \(\bar{v} = \tfrac{2d}{d(1/v_1 + 1/v_2)} = \tfrac{2 v_1 v_2}{v_1 + v_2}\) — the harmonic mean.
What is the average of \(n\) consecutive integers?
It equals the middle term. For \(n\) odd, this is the literal middle integer. For \(n\) even, it is the mean of the two middle integers. Example: average of 11, 12, 13, 14, 15 is 13; average of 11, 12, 13, 14 is 12.5.
How do I find the combined average of two groups?
Weighted average: \(\bar{x} = \tfrac{n_1 \bar{x}_1 + n_2 \bar{x}_2}{n_1 + n_2}\). Compute total sum (counts × averages added), divide by total count. Example: 40 students average 60, 60 students average 70 ⇒ combined = \(\tfrac{2400 + 4200}{100} = 66\).
How do I handle age problems where everyone ages by the same amount?
Each member's age increases by \(\Delta t\), so the sum increases by \(n \Delta t\) and the average also increases by \(\Delta t\). For "present vs past average", just add (or subtract) \(\Delta t\). If a member is born or dies in between, recompute the sum carefully.
Which CDS Maths topics connect to Average?
Statistics formalises the arithmetic mean alongside median and mode. Time, Speed and Distance uses average speed (harmonic mean for two-leg journeys). Mixture problems use weighted average through alligation.