Episode 8 — Aptitude and Reasoning / 8.8 — Average
8.8 Quick Revision -- Average
Use this sheet for last-minute revision before exams. It covers every formula, shortcut, and common trap in one place.
1. Core Formulas
Average = Sum / Count
Sum = Average x Count
Count = Sum / Average
2. Weighted Average
Weighted Avg = (w1*x1 + w2*x2 + ... + wn*xn) / (w1 + w2 + ... + wn)
Key property: Combined average ALWAYS lies between the individual averages and is closer to the average of the LARGER group.
3. Average of Standard Sequences
First n natural numbers (1, 2, ..., n):
Sum = n(n+1)/2 Average = (n+1)/2
First n even natural numbers (2, 4, ..., 2n):
Sum = n(n+1) Average = n + 1
First n odd natural numbers (1, 3, ..., 2n-1):
Sum = n^2 Average = n
Consecutive integers from a to b:
Average = (a + b) / 2 Count = (b - a) + 1
Squares of first n natural numbers:
Sum = n(n+1)(2n+1)/6 Average = (n+1)(2n+1)/6
Cubes of first n natural numbers:
Sum = [n(n+1)/2]^2 Average = n(n+1)^2/4
4. Average Speed
Same distance, two speeds:
Avg Speed = 2*S1*S2 / (S1 + S2) [Harmonic Mean]
Same time, two speeds:
Avg Speed = (S1 + S2) / 2 [Arithmetic Mean]
Same distance, three speeds:
Avg Speed = 3*S1*S2*S3 / (S1*S2 + S2*S3 + S1*S3)
General:
Avg Speed = Total Distance / Total Time
5. Adding / Removing / Replacing Elements
Adding element x to n numbers with average A:
New Average = (nA + x) / (n + 1)
Removing element x from n numbers with average A:
New Average = (nA - x) / (n - 1)
Replacing element x with y in n numbers:
Change in average = (y - x) / n
New average = A + (y - x) / n
6. Quick Shortcut: Finding the New/Removed Element
New element (when avg changes from A to A'):
x = A' + n(A' - A) where n = original count
Removed element (when avg changes from A to A'):
x = A - (n-1)(A' - A) where n = original count
7. Uniform Change to All Elements
Each element + k => New Avg = Old Avg + k
Each element - k => New Avg = Old Avg - k
Each element x k => New Avg = Old Avg x k
Each element / k => New Avg = Old Avg / k
8. Combined Average of Groups
Two groups: Avg = (n1*A1 + n2*A2) / (n1 + n2)
Three groups: Avg = (n1*A1 + n2*A2 + n3*A3) / (n1 + n2 + n3)
9. Alligation (Finding Ratios)
n1 : n2 = (A2 - Ac) : (Ac - A1)
Where:
A1, A2 = individual group averages (A1 < A2)
Ac = combined average
n1, n2 = group sizes
10. Age Problems
After t years: New average = Current average + t
Before t years: Old average = Current average - t
New baby born (age 0): Sum stays same, count increases by 1.
Person leaves/dies: Sum decreases, count decreases by 1.
11. Batting/Bowling Average
Batting Avg = Total runs / Innings played
Bowling Avg = Total runs conceded / Wickets taken
Runs in (n+1)th innings to increase avg by x:
= Old Avg + (n+1) x (increase)
12. Wrong Entry Correction
Correct Sum = Wrong Sum - Wrong Value + Correct Value
Correct Avg = Correct Sum / n
13. Grouped Data (Frequency Distribution)
Average = Sum(fi * xi) / Sum(fi)
For class intervals, use mid-value:
xi = (Lower limit + Upper limit) / 2
14. Deviation Method (Assumed Mean)
Step 1: Pick assumed mean A (close to middle value)
Step 2: Compute deviations di = xi - A
Step 3: Average = A + (Sum of di) / n
Best when numbers are large and clustered together.
15. Mean vs Median vs Mode
Mean: Sum of values / Count of values
Median: Middle value when sorted (or avg of two middle for even count)
Mode: Most frequently occurring value
Empirical relation: Mode ≈ 3 x Median - 2 x Mean
16. Common Traps -- DO NOT Fall For These
TRAP 1: Average speed for equal distances is NOT (S1+S2)/2.
Use 2*S1*S2/(S1+S2).
TRAP 2: Weighted average is NOT (A1+A2)/2 unless groups are equal size.
TRAP 3: When adding/removing elements, the COUNT changes.
New count = n+1 (adding) or n-1 (removing).
TRAP 4: "Average increases by x" is NOT the same as "new average is x."
TRAP 5: In age problems, after t years ALL members age by t,
so total sum increases by n*t.
TRAP 6: Counting from a to b: count = b - a + 1 (NOT b - a).
TRAP 7: For consecutive even/odd numbers, the average may not be
an integer. Check: 4 consecutive even numbers with avg 27 means
n + 3 = 27, so n = 24, numbers are 24, 26, 28, 30.
17. Exam Strategy
1. Read whether it says "average" or "sum" -- know what is given and what is asked.
2. For large numbers, use the deviation method.
3. For consecutive / AP numbers, use (First + Last) / 2 -- no need to add.
4. For speed problems, always ask: "Same distance or same time?"
5. For age problems, set up equations carefully with time shifts.
6. For combined averages, use alligation for ratio-based questions.
7. Verify: combined average must lie between individual averages.
8. Double-check: did the count change? (Adding/removing elements)
18. Key Numbers to Remember
Average of 1 to 100 = 50.5
Average of 1 to 50 = 25.5
Average of 1 to 200 = 100.5
Average of 1^2 to 10^2 = 38.5
Average of first 10 even = 11
Average of first 10 odd = 10
19. Formula Quick-Reference Card
+--------------------------------------------------+-----------------------------------+
| Situation | Formula |
+--------------------------------------------------+-----------------------------------+
| Basic average | Sum / n |
| Find sum | Avg x n |
| Weighted average | Sum(wi*xi) / Sum(wi) |
| Consecutive a to b | (a + b) / 2 |
| First n natural | (n + 1) / 2 |
| First n even | n + 1 |
| First n odd | n |
| Avg speed (equal dist) | 2ab / (a + b) |
| Avg speed (equal time) | (a + b) / 2 |
| Add element x | (nA + x) / (n + 1) |
| Remove element x | (nA - x) / (n - 1) |
| Replace x by y | A + (y - x)/n |
| All elements + k | A + k |
| All elements x k | A x k |
| Combined average | (n1*A1 + n2*A2) / (n1+n2) |
| Alligation ratio | n1:n2 = (A2-Ac) : (Ac-A1) |
| After t years | A + t |
| Wrong entry fix | (nA - wrong + correct) / n |
| Batting avg change | Runs = A + (n+1)*increase |
+--------------------------------------------------+-----------------------------------+
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