Averages are among the most scoring and predictable topics in Arithmetic. Yet many students get confused because an average is not just a simple calculation—it represents the equal distribution of total value, and this makes it behave differently when observations change, when values combine, or when speeds vary across segments.
If you understand these three ideas:
✔ Total sum of observations
✔ Number of observations
✔ How values combine or change
Then you can solve ANY question effortlessly.
This Averages Guide for Competitive Exams covers all formulas, concepts, shortcuts, cases, examples, FAQs, and exam-oriented tips, making it the perfect one-stop resource for SSC, Banking, Railways, UPSC CSAT, and campus aptitude tests.
Quick Overview: Average Formulas
| Concept / Situation | Distance / Data Considered | Speed / Value Used | Formula & Definitions |
|---|---|---|---|
| Basic Average Formula | Total of all observations | Number of observations | Average = (Sum of observations) / (Number of observations) |
| Updating Average | Old average + added values | – | New Avg = (Old Sum ± Change) / New Count |
| Weighted Average | When groups combine | Values × Weights | Weighted Avg = (a₁w₁ + a₂w₂ + …) / (w₁ + w₂ + …) |
| Average Speed (equal distance) | Same distance covered | Harmonic mean | Average Speed = (2xy) / (x + y) where x & y are speeds |
Average Formulas (Foundation of All Problems)
Understanding average means understanding distribution. The average shows what each observation would be if all values were equal.
1. Basic Average Formula
The most fundamental formula:
Average = Sum of observations/Number of observations
Where:
- Sum = total of all values
- Number = count of values
This formula becomes the base of every type of averages question—be it marks, ages, runs, weights, or expenses.
Why this formula works
Because average assumes all values are evenly distributed.
If the average is known, you can always reconstruct the total:
Sum = Average×Number
This is extremely helpful for reverse-calculation problems.
Average Speed Formula (When Equal Distances Are Covered)
Average speed is not the simple average of speeds.
Most students make this mistake.
If a person covers equal distance at speeds:
- x km/hr
- y km/hr
Then:
Average Speed = 2xy/(x+y) km/hr
Why this formula works
The traveller spends more time at the lower speed and less time at the higher speed.
So total time is not balanced equally, and hence simple averages do not work.
The correct approach is:
Average Speed = Total Distance/Total Time
For equal distances, this simplifies to the formula:
2xy/x+y
This formula is widely used in SSC, RRB, Banking, and CSAT.
Concepts of Average
Average problems are easier when you clearly understand how the sum behaves.
1. When a new value is added
If a number is added to the group:
- The average increases only if the new value is greater than the existing average.
- The average decreases only if the new value is smaller than the existing average.
2. When a value is removed
Removing a value smaller than the average raises the average.
Removing a value larger than the average reduces it.
3. When two groups combine
Use the weighted average formula:
Combined Average = a1n1+a2n2/n1+n2
Where:
a₁ = average of group 1
n₁ = members in group 1
a₂, n₂ = same for group 2
4. When numbers change by constant values
If all observations increase or decrease by “k”:
New Average = Old Average+k
Smart Tips and Practical Tricks for Solving Average Problems
1. Always find the total first
Average questions simplify instantly when you convert the average into the total:
Sum = Average × Number
You can plug this into any situation ages, marks, expenses, mixture, etc.
2. Compare with average to decide increase or decrease
If a new observation is added:
- Greater than average → average increases
- Less than average → average decreases
This helps in reasoning-based questions.
3. Use the deviation method
Instead of calculating large totals:
- Check how far each value is from the average.
- Adjust deviations, sum them, and set total deviation to zero.
This trick reduces long calculations.
4. Don’t mix units
For average speed:
- Use km/hr for all speeds
- Use equal distance (NOT equal time)
- Only then apply 2xy/x+y
5. Average Speed is NOT the simple average
If speeds are x and y:
Wrong average = (x + y)/2
Correct average = 2xy/(x + y)
Always use harmonic mean for equal-distance travel.
6. Practice common patterns
Repeated questions include:
- Average age changes after years
- One value added/removed from group
- Students shifting between classes
- Runs scored in matches
- Average expenditure per day
- Average marks across subjects
- Travel scenarios using multiple speeds
FAQs About Average
Q1. Why is average speed not the simple average of two speeds?
Because time spent at each speed is not equal. With equal distance covered, the slower speed takes more time, making the harmonic mean the correct approach.
Q2. When do we use the formula 2xy/(x + y)?
Only when the distances covered at speeds x and y are exactly equal.
Q3. How does adding a number affect the average?
If the added number is greater than the current average, the average increases; otherwise, it decreases.
Q4. How to find the sum when average and number of items are given?
Use: Sum = Average × Number.
Q5. What is weighted average?
It is the average when different groups or values contribute with different weights or frequencies.
Q6. Why does removing a value change the average?
Because the removed value changes the total sum while also changing the number of observations.
Q7. How do averages help in reasoning questions?
They help in problems involving distribution, comparison, and equalization.
Q8. What is the deviation method in averages?
It involves comparing each value to the assumed average, adjusting deviations until the net deviation becomes zero.
Q9. Why is average used in calculations of age problems?
Because ages increase uniformly each year, making total sum and average predictable.
Q10. What is the simplest strategy to master average questions?
Convert every average into a total, identify how the total changes, and apply direct formulas using patterns.