The one idea
Average = total ÷ number of items, so total = average × number. Almost every average problem is solved by working with totals: add or remove an item, and the total changes by that amount.
Worked examples
Example 1. The average of five numbers is 24. When one number is removed, the average of the other four is 22. The removed number: total 120 − 88 = 32.
Example 2 (weighted average). A class of 30 has an average score of 60 and another class of 20 has an average of 75. Combined average: (1,800 + 1,500) ÷ 50 = 66.
Example 3. The average age of a group of 4 is 30. A new member joins and the average becomes 25. The new member's age: new total = 5 × 25 = 125; old total = 4 × 30 = 120; so the new member is 5 years old.
Example 4. A batter has an average of 40 after 10 innings. How many must he score in the 11th to raise the average to 42? Needed total = 11 × 42 = 462; current = 400; so 62 runs.
Example 5 (deviation method). The average of 48, 52, 47, 55 and 53. Take 50 as the base: deviations −2, +2, −3, +5, +3 sum to +5. The average is 50 + 5 ÷ 5 = 51.
Example 6 (speed). A car goes 60 km at 30 km/h and returns at 60 km/h. The average speed is total distance ÷ total time = 120 ÷ (2 + 1) = 40 km/h, not 45.
Common traps
- Averaging averages without weights.
- Averaging speeds when distances are equal: use total distance over total time (the harmonic mean).
- Forgetting that the count changes when an item is added or removed.
- Age problems: remember that everyone ages together; after n years the average rises by n if the group is unchanged.
Speed tips
- Use the deviation method for numbers close together.
- For an item that changes the average, multiply the change in average by the new count.
- For consecutive numbers, the average is the middle value.
A practice routine
- Day 1: basic totals and missing numbers, ten questions.
- Day 2: weighted averages and mixed groups.
- Day 3: age and cricket-style problems.
- Day 4: average speed and a timed set of 12 questions at 45 seconds each.
Where it appears
Averages appear in SSC CGL, banking, railway mathematics, CLAT quantitative techniques and inside data interpretation sets. Ratio problems pair well: see ratio and proportion.
Practice tests below are published by institutes and educators on TestMint.
