Average
⚡ Unlock time-saving calculation tricks and mental math techniques. Solve complex problems in seconds with proven shortcut methods used by top performers.
Shortcut Techniques
Study MaterialShortcut Techniques – Average
Shortcut techniques in Average help candidates solve aptitude questions quickly and accurately in SSC, Banking, Railway, CDS, NDA, CAT, UPSC, and placement examinations.
Instead of lengthy calculations, these methods focus on:
- Fast average calculations
- Consecutive number tricks
- Average speed shortcuts
- Replacement concepts
- Addition and removal techniques
- Arithmetic progression shortcuts
- Mental calculation methods
Why Learn Shortcut Techniques?
- Improves calculation speed significantly.
- Reduces lengthy arithmetic calculations.
- Useful in average age and average speed problems.
- Helps solve data interpretation questions faster.
- Improves overall quantitative aptitude performance.
Shortcut #1: Basic Average Formula
Always remember the fundamental relationship:
Average = Sum of Observations / Number of Observations
From this:
- Sum = Average × Number of Terms
- Number of Terms = Sum / Average
Example:
Average of 8 numbers is 15.
Total sum:
= 15 × 8
= 120
Shortcut #2: Consecutive Numbers Trick
For consecutive numbers:
Average = (First Number + Last Number) / 2
✔ No need to add all numbers individually.
Example:
Average of numbers from 21 to 31:
= (21 + 31) / 2
= 52 / 2
= 26
Shortcut #3: Arithmetic Progression Trick
If numbers are in arithmetic progression:
- Odd number of terms → Average = Middle term
- Even number of terms → Average = Average of two middle terms
Example:
Average of:
5, 10, 15, 20, 25
Middle term = 15
Shortcut #4: Average of First n Natural Numbers
Formula:
Average = (n + 1) / 2
Example:
Average of first 50 natural numbers:
= (50 + 1) / 2
= 25.5
Shortcut #5: Average of First n Odd Numbers
Formula:
Average = n
Example:
Average of first 11 odd numbers:
= 11
Shortcut #6: Average of First n Even Numbers
Formula:
Average = n + 1
Example:
Average of first 20 even numbers:
= 20 + 1
= 21
Shortcut #7: Addition and Removal Technique
When a person or value is added:
- Increase in average × New total members = Added value difference
When a person or value is removed:
- Decrease in average × Remaining members = Removed value difference
Example:
Average of 20 students increases by 2 after adding one student.
Extra total weight:
= 21 × 2
= 42
Shortcut #8: Replacement Technique
If one value is replaced:
Change in Average = Difference / Total Number of Terms
Example:
A number 40 is replaced by 60 in a set of 10 numbers.
Increase in average:
= (60 − 40) / 10
= 20 / 10
= 2
Shortcut #9: Average Speed Trick (Equal Distances)
For equal distances covered at speeds A and B:
Average Speed = 2AB / (A + B)
✔ This is harmonic mean, not arithmetic mean.
Example:
A person travels equal distances at 40 km/h and 60 km/h.
Average Speed:
= (2 × 40 × 60) / (40 + 60)
= 4800 / 100
= 48 km/h
Shortcut #10: Average Speed for Three Equal Distances
Formula:
Average Speed = 3ABC / (AB + BC + CA)
Example:
Speeds are 30 km/h, 15 km/h, and 10 km/h.
Average Speed:
= (3 × 30 × 15 × 10) / (450 + 150 + 300)
= 13500 / 900
= 15 km/h
Shortcut #11: Weighted Average Technique
Weighted average is used when quantities have different frequencies.
Formula:
Weighted Average = Total Weighted Sum / Total Frequency
Example:
Average marks of 30 students = 60
Average marks of 20 students = 80
Combined average:
= [(30 × 60) + (20 × 80)] / 50
= (1800 + 1600) / 50
= 68
Shortcut #12: Deviation Method
Choose a base value close to all observations and calculate deviations.
✔ Useful when numbers are large and closely spaced.
Example:
Find average of:
48, 49, 50, 51, 52
Take base = 50
Deviations:
−2, −1, 0, +1, +2
Sum of deviations = 0
Average = 50
Shortcut #13: Symmetry Trick
If numbers are equally distributed around a central value, the average is the central value.
Example:
40, 45, 50, 55, 60
Average = 50
Shortcut #14: Zero Inclusion Trick
Zero is also counted as an observation.
Example:
Average of 0, 10, 20:
= (0 + 10 + 20) / 3
= 30 / 3
= 10
Shortcut #15: Fast Combined Average Formula
When two groups are combined:
Combined Average = (Total Sum of Both Groups) / (Total Number of Terms)
Important Exam Tips
- Memorize all standard average formulae.
- Use middle-term shortcut for consecutive numbers.
- Apply harmonic mean formula in equal-distance speed problems.
- Use deviation method for large calculations.
- Be careful while adding or removing observations.
- Always verify total number of observations.
- Use approximation and elimination techniques in MCQs.
Shortcut techniques in Average help candidates improve calculation speed, reduce errors, and solve aptitude questions efficiently in competitive examinations.