3. Shortcuts & Speed Tricks

Trick 1 — Deviation (Assumed Mean) Method: For any set of close-valued numbers, don’t add them directly. Pick a round number near the expected average, find deviations, average the deviations, and add back. Example: Average of 97, 101, 103, 99 → assume A=100 → deviations: −3,+1,+3,−1 → sum=0 → average = 100.

Trick 2 — Replacement Difference Shortcut: Whenever a member is replaced and the average changes by d, directly compute New = Old ± (n×d). Never re-derive from total sums. Example: Average of 15 workers rises by 4 kg when a 60 kg worker is replaced. New worker’s weight = 60 + 15×4 = 120 kg.

Trick 3 — Join/Leave Shortcut Using “Excess/Deficit over old average”: When someone joins and the average changes by d, the joining value = old average + (n+1)×d. This avoids computing two separate totals. Example: Average age of 20 people is 30. A new person joins, average becomes 31. New person’s age = 30 + 21×1 = 51.

Trick 4 — Combined Average by Alligation (skip the weighted formula for two groups): Combined average divides the gap between two averages in the inverse ratio of group sizes. This is often faster than plugging into the weighted average fraction, especially with awkward numbers. Example: 25 students average 80, 15 students average 60. Ratio 25:15 = 5:3, so combined average splits gap (80−60=20) in ratio 3:5 (inverse), from 60 side: 60 + (5/8)×20 = 60+12.5 = 72.5.

Trick 5 — Average Speed Recognition Rule: Before computing, ask “equal distance or equal time?” If distance is equal → use 2xy/(x+y). If time is equal → use simple average (x+y)/2. This single question prevents 90% of average-speed errors.

Trick 6 — Sum of Consecutive Numbers Using Average × Count: To quickly sum a long consecutive series (e.g., 45 to 89), find the average = (45+89)/2 = 67, count = 89−45+1 = 45, sum = 67×45 = 3015 — much faster than adding.

Trick 7 — Age-Shift Shortcut: If “x years later/ago” is mentioned and the group composition is unchanged, just add/subtract x directly to/from the average — never recompute individual ages unless a member has joined, left, or been born in the interval.

Trick 8 — Symmetric Cancellation in Deviation Method: When numbers are symmetric around a central value (like 41,43,47,49,53 around 47 with equal spread on both sides except one skew), the positive and negative deviations partially cancel — spot this visually before calculating to estimate the answer instantly and check for silly mistakes.

Trick 9 — Using “Total = Average × Number” to Skip a Step: In multi-part average questions (e.g., “average of first 6 is X, average of last 6 is Y, common/overlap element found”), directly write total sums using Average × Count for each part rather than working backward from individual values. Example: 11 numbers average 10.9; first six average 10.5, last six average 11.4. The 6th number (common to both groups) = (Sum of first 6 + Sum of last 6) − Total sum = (63+68.4)−119.9 = 131.4−119.9 = 11.5.

Trick 10 — Percentage-Change Shortcut for Average Speed with Round-trip: For a to-and-fro journey where return speed is k times the onward speed, average speed = 2k/(1+k) × (onward speed). This avoids recomputing 2xy/(x+y) from scratch when speeds are given as a ratio. Example: Onward speed 30, return speed is 2× onward = 60. Average speed = 2×2/(1+2) × 30 = (4/3)×30 = 40 km/h. (Verify: 2×30×60/90 = 3600/90=40 ✓)

Trick 11 — Constant Add/Multiply Shortcut: If every value in a data set is increased by a fixed amount k, the average simply increases by k (no recalculation of totals needed). If every value is scaled by a factor k, the average scales by the same factor k. This is useful in “if each employee’s salary is hiked by 10%, find the new average salary” type questions — just apply 10% directly to the old average. Example: Average salary ₹25,000, hiked by 10% for everyone. New average = 25,000 × 1.10 = ₹27,500 — no need to know individual salaries or headcount.

Trick 12 — Spotting “Middle Term = Average” for Odd-Count Consecutive Series: Whenever a series has an odd number of consecutive terms (numbers, evens, odds, or any AP), the average always equals the middle term directly — skip the (first+last)/2 computation and just identify the middle term by counting. Example: Average of 7 consecutive multiples of 5 is required, given the smallest is 40. The 7 terms are 40,45,50,55,60,65,70; middle (4th) term = 55 = average directly, confirmed by (40+70)/2=55 ✓. For odd-count series this saves you from computing the last term at all if the middle is directly identifiable from context.