Quick-Revision Sheet – Average Problems Tricks
RRB Railway & Other CBTs – 2-min last look
Key Points (One-Liners)
- Average = Sum ÷ Number of items
- Sum = Average × Number of items
- Replace “all items” with “average item” in mind—only total changes
- If every number ↑/↓ by k, average ↑/↓ by k (no formula needed)
- If every number ×/÷ k, average ×/÷ k
- Adding a new value: new avg = (old sum + new value) ÷ (n + 1)
- Removing a value: new avg = (old sum – value) ÷ (n – 1)
- Combined average = (n₁A₁ + n₂A₂) ÷ (n₁ + n₂)
- Weights can be persons, days, groups—treat n as weight
- Age problems: difference remains constant; average age ↑ 1 yr every year for fixed group
- Cricket/batting: average = total runs ÷ innings; “0” innings still count
- Replacement: if avg ↑ by k after swap, new member = old member + k × n
- Assumed avg method: take 0 as base, add deviations, divide, then add back base
- Median ≠ average—don’t confuse in hurry
- Shortcut: balance the “excess” and “shortage” visually—cross-cancellation saves 30 s
- Never find LCM for average—work with totals only
- Two groups merged: difference in averages is inversely proportional to sizes
- Always write n, A, S (number, average, sum) triangle—fill two, get third
- When series is AP, average = (first + last) ÷ 2
- Round only final answer; keep decimals till last step to avoid ±1 trap
Important Formulas/Rules
| Formula/Rule | Application in one line |
|---|---|
| 1. A = S / n | Core definition |
| 2. S = A × n | Reverse calculation |
| 3. A_new = (S ± x) / (n ± 1) | Add/remove single item |
| 4. A_mix = (n₁A₁ + n₂A₂) / (n₁ + n₂) | Two-section merge |
| 5. Member_replaced = outgoing + nΔ | Replacement causes Δ change |
| 6. Deviation = value – assumed avg | Assumed-base method |
| 7. A_AP = (a₁ + aₙ)/2 | AP series shortcut |
| 8. If all ×k, A×k | Scaling rule |
| 9. Weighted A = Σ(wi Ai)/Σwi | General weighted case |
| 10. A_gear = total distance / total time | Speed-type average (harmonic if equal distance) |
Memory Tricks
- “SAN” triangle: Start with any two of Sum, Average, Number—third drops out.
- “KISS” – Keep It on Sum & n; ignore individual values.
- Replacement: “plus n delta” sounds like “police” – easy recall.
- AP average: “First + Last, cut in half—like slicing bread.”
- Weighted: “ heavier side pulls the see-saw” – bigger n dominates.
Common Mistakes
| Mistake | Correct Approach |
|---|---|
| 1. Averaging averages without weights | Multiply each avg by its n, then divide by total n |
| 2. Forgetting to count ‘0’ innings/scores | Include every listed item; 0 is a real score |
| 3. Using simple avg for speeds when distances equal | Use Harmonic Mean: 2ab/(a+b) |
| 4. Rounding intermediate values | Keep one extra decimal; round only the final choice |
| 5. Adding instead of replacing while using formula | Check keyword: “replaced” ⇒ n stays same |
Last-Minute Exam Tips
- Write n-A-S on rough sheet first; fill given data instantly.
- See “change in average” → immediately think ±nΔ shortcut.
- Options spaced >2 units? Estimate first, calculate only if tie.
- Age/weight problems: fix reference year → convert all data to same year.
- 30 s gone with no road? Skip; average questions hide in later sets too.
Quick Practice (5 MCQs)
BETA
