Study Guide · Chapter 15
4. Common Mistakes Aspirants Make
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- Averaging speeds by simple mean when distances (not times) are equal. Correction: use Harmonic Mean 2xy/(x+y) whenever equal distances are covered at different speeds; simple average (x+y)/2 applies only when times are equal.
- Confusing the three consecutive-number formulas. Correction: natural numbers → (n+1)/2; even numbers → n+1; odd numbers → n. Write them down before attempting the question if under exam stress.
- Forgetting that “replaced by” and “joined by” are different mechanisms. Correction: replacement keeps group size constant (use n×d rule directly); joining/leaving changes the group size (use (n+1) or (n−1) in the formula).
- Applying the age-shift rule (+x years) even when the group size has changed (e.g., a baby is born, or a member has left/joined during the time gap). Correction: re-derive using total sums whenever group composition changes across the time period.
- Treating weighted average as a simple average of the given averages, ignoring the differing group sizes/weights. Correction: always multiply each average by its own weight (count/quantity) before adding, then divide by the sum of weights.
- Sign errors in the deviation method — adding all deviations as positive. Correction: retain the sign (+/−) of each deviation relative to the assumed mean; the final sum of deviations can be negative, and this reduces the assumed average.
- Using n instead of (n+1) or (n−1) in join/leave problems. Correction: when someone joins, the new count becomes (n+1); when someone leaves, the new count becomes (n−1). Always identify the correct denominator matching the new average given in the question.
- Ignoring that average of squares/cubes formulas apply only to “first n natural numbers”, not to an arbitrary set of consecutive numbers. Correction: for average of squares of numbers from, say, 5 to 15, compute directly (sum of squares from 1 to 15 minus sum of squares from 1 to 4, divided by 11) rather than misapplying the “first n” formula.
- Rushing calculation for combined average of 3+ groups by pairing only two averages at a time incorrectly. Correction: always use the full weighted-average formula with all groups’ weights and averages included simultaneously — or apply alligation pairwise only after first combining two groups into a single effective group (with the combined weight and combined average), then alligate that combined group with the third.
- Assuming “average speed” equals “average of the two given speeds” whenever a question merely mentions two speeds, without checking the underlying condition (distance/time) at all. Correction: always read the question stem carefully — words like “same distance,” “returns via the same route,” or “for half the journey” typically signal equal distance, while “for equal durations” or “for 2 hours each” signal equal time.
- Forgetting to double-check whether a “found excluded/included” element is being double-counted, as in problems where the average of the “first k” and “last k” elements of a list overlap in one common term. Correction: explicitly check whether the groups described in the question share a common member before adding their totals; if they do, subtract the grand total to isolate the shared value (see Trick 9).
- Blindly applying “new average” formulas without first confirming whether the count of members has genuinely changed. For instance, in “the average marks of the class change because one student’s marks were recorded wrong and later corrected,” the number of students stays the same — this is a correction-of-error case (use Trick: recompute total with correct value swapped in), not a join/leave or replacement-of-person case, even though the surface wording can look similar. Correction: identify whether the scenario is (i) a data-entry correction (same person, corrected value), (ii) a replacement (different person swapped in), or (iii) a join/leave (change in headcount) before choosing a formula.
- Rounding intermediate values too early, especially in weighted average and combined average calculations involving fractions like 1/3 or 2/7. Correction: carry fractions through the entire calculation and round only the final answer, matching it to the closest MCQ option — SSC/RRB options are often designed to penalise premature rounding.
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