Averages Formulas for CAT 2026: 12 Solved Questions
A standalone CAT 2026 averages cheatsheet: the 6 core formulas, the weighted-average cross-multiplication shortcut, the "remove one element" replacement trick, moving averages, the average-of-averages trap CAT plants every cycle, and 12 solved CAT-level questions with stepwise working. Built so averages stops being the under-respected topic that quietly costs 4 marks.
Averages Formulas for CAT 2026: 12 Solved Questions
Averages is the most under-respected topic in CAT Quant. Aspirants mark it "easy", skip ahead, then burn 3 minutes on a 60-second weighted-average trap. The averages formulas CAT aspirants need are six. Six. Yet every cycle, the 30-second question costs careless solvers 4 marks because the average-of-averages trap waits in the choices.
This is the standalone averages cheatsheet: 6 formulas, the weighted-average shortcut, the remove-one-element trick, moving averages, the average-of-averages trap, and 12 solved questions. Pair with the percentages formulas guide.
Six formulas cover every CAT averages question: simple, weighted, mean-shift, replacement, AP shortcut, deviation. Cross-multiplication shortcut drops weighted-average solve time from 90s to 30s. Replacement formula (new avg = old + (y − x)/n) is the highest-ROI trick. Average-of-averages trap: never average two averages unless group sizes match. 12 solved Qs below.
The 6 Core Averages Formulas Every CAT Aspirant Must Know
Six formulas cover roughly 95% of CAT averages questions across the last 10 cycles.
Simple arithmetic mean
Average = (Σx) / n
Foundational definition. Roughly 15% of CAT averages questions are pure simple-average; the rest build on it.
Weighted mean across unequal groups
Weighted Avg = (w₁v₁ + w₂v₂ + ... + wₙvₙ) / (w₁ + w₂ + ... + wₙ)
Used whenever items carry different weights (group sizes, frequencies, marks). The highest-frequency averages formula in CAT — in roughly 40% of averages questions, directly or via alligation.
New average after adding one element
New Avg = (n · A + x) / (n + 1)
If n items with average A get a new item x, gives the new average. Saves 25s vs recomputing the sum.
New average when one element is replaced
New Avg = Old Avg + (y − x) / n
When x is removed and y added in n items, the average shifts by (y − x) / n. The most useful CAT averages shortcut.
Average of consecutive integers or AP
Average = (first + last) / 2
For any AP, the average is the midpoint of first and last terms. Turns a 30s sum into a 5s mental step.
Assumed mean + average deviation
Average = Assumed Mean + (Σd) / n
Pick an anchor near the centre, average the deviations, add back. Useful when numbers (487, 491, 498) cluster near a midpoint.
The reference table maps each formula to use case, shortcut, frequency, and example number.
| Formula | Use Case | Shortcut | CAT Frequency | Difficulty | Example No. |
|---|---|---|---|---|---|
| Simple average | Mean of listed values | Σx / n | Med | Easy | Q1, Q2 |
| Weighted average | Unequal groups, marks, populations | w1:w2 = (A2−M):(M−A1) | High | Med | Q3, Q4, Q5 |
| Add one element | One value added | (nA + x) / (n+1) | Med | Easy | Q6 |
| Replacement | One out, one in | A + (y−x)/n | High | Med | Q7, Q8 |
| AP shortcut | Consecutive integers, AP | (first + last) / 2 | Med | Easy | Q9, Q10 |
| Deviation method | Messy numbers near centre | AM + Σd/n | Low | Med | Q11, Q12 |
Weighted Average: Formula, Shortcut, and Two CAT Traps
Weighted average is the most-tested averages concept in CAT. Standard form: Weighted Avg = Σ(wₙvₙ) / Σwₙ. Two-group shortcut: if A1 and A2 with weights w1 and w2 combine to give M, then w1 : w2 = (A2 − M) : (M − A1). 90s becomes 30s.
Cross-multiplication / alligation form
w₁ : w₂ = (A₂ − M) : (M − A₁)
Used when M is given and the ratio is asked, or vice versa. Denominator (M − A₁) is the distance from the lower average to M; numerator (A₂ − M) is the distance from M to the higher.
Trap 1 — simple-for-weighted. With unequal group sizes, the combined average is the weighted average, not the simple mean. CAT plants the wrong simple-mean in the choices every time.
Trap 2 — ratio direction. w1 : w2 corresponds to A1 and A2 in the same order. Swap them mid-solve and the answer flips. Label the lower average A1 and write M between A1 and A2 on scratch paper before applying the shortcut. See the ratio and proportion guide.
The "Remove One Element" Trick: When the Average Changes
The replacement formula is the highest-ROI averages shortcut in CAT. For n items with average A, if x is removed and y added, new average = A + (y − x) / n. No need to recompute the sum.
Three sub-cases share the logic. Removing only: (nA − x) / (n − 1). Adding only: (nA + x) / (n + 1). Pure replacement: A + (y − x) / n. CAT favours replacement because the setter can give the "before" average and ask for y, or reverse it.
Label original count n, original average A, removed element x, added element y.
Count stays at n: one out, one in; set size is unchanged.
New average = A + shift. To back-solve for y: y = x + n · (New Avg − A).
Classic CAT phrasing: "Average weight of 11 cricketers was 65 kg. A player weighing 58 kg is replaced and the team average rose by 0.4 kg. New player's weight?" Solve in 15s: 0.4 × 11 = 4.4, so y = 58 + 4.4 = 62.4 kg.
How CAT typically tests this. Replacement appears in roughly one of every two averages sets across the last 10 cycles, usually wrapped in a cricket-team, batting-average, class-marks, or employee-salary cover story. The setter gives n, A, the removed x, and the average shift, then asks for y — or hides one of those values and asks aspirants to back-solve. Mocks that recycle this shortcut keep n small (11, 20, 25) so the (y − x) / n arithmetic stays clean, while the careless solver recomputes the full sum and burns 90 seconds on what should be a 25-second question.
Moving Average and Running Sum: The Sequence-Style Average
Moving average is a sliding-window average across a sequence. Simple average computes once over the full set; moving average computes across every k consecutive terms as the window slides. CAT uses it in sequence problems where the setter wants recursive thinking.
Key insight: if the k-term moving average is constant, then aₙ = aₙ₋ₖ — the sequence repeats every k terms. If the moving average is itself an AP, the original sequence is a higher-order AP. Most CAT problems crack by writing 4-5 terms and spotting the pattern.
Running sum is the cousin formula: Sₙ = a₁ + ... + aₙ. The average over any window from position i to j equals (Sₘ − Sₕ) / (j − i) — the cleanest way to handle window-average questions without recomputing the underlying sum every time the window shifts.
Do not confuse moving average with the simple average of the whole sequence. The moving average of (10, 20, 30, 40, 50) with window 3 is the sequence (20, 30, 40) — three values, not one. CAT picks examples where the two coincidences break.
Where this trap usually appears in CAT mocks. Moving-average problems surface roughly once every three CAT cycles, typically inside DI sets that report a 3-month or 4-quarter rolling figure and ask aspirants to back out one missing entry. The setter rarely names the term — the question simply says "the average of any three consecutive months was 52" and expects the aspirant to spot that the underlying sequence repeats every three terms. Aspirants who skip moving averages because the topic looks niche routinely lose 4 marks on DI sets that hinge on this single recurrence. Recognising the rolling-window phrasing early is the entire battle; the arithmetic is trivial once the structure is correctly named.
The "Average of Averages" Trap (and Why CAT Loves It)
The average-of-averages trap is the classic CAT misdirection. A naive solver takes the arithmetic mean of two group averages instead of the weighted average. Example: class A has 30 students with average 60; class B has 20 students with average 80. The combined average is not (60 + 80) / 2 = 70. Correct weighted average: (60 × 30 + 80 × 20) / 50 = 3400 / 50 = 68.
The naive value 70 is always among the choices, planted for the careless solver. Rule: the simple mean of two averages equals the weighted average only when group sizes are identical. The moment sizes differ, the weighted average shifts toward the larger group, proportional to the size imbalance.
CAT plants this trap in roughly 1 of every 3 averages problems. Defence: whenever you see two averages, check if group sizes are equal. If not, write the weighted formula first. The same trap hides inside DI sets and profit-loss problems — see the profit and loss formulas guide.
Where this trap usually appears in CAT mocks. Setters favour three settings: two-class score combinations (boys-girls, section-A vs section-B), milk-water and alloy mixtures, and salary or age averages across departments or age bands. The naive simple-mean is always planted in the choices, and a fourth distractor sits equidistant on the opposite side to catch aspirants who flip the ratio mid-solve. The fastest defence is the alligation diagram: write A1, M, A2 on scratch paper in that order, then read the weight ratio directly off the differences before computing anything. The diagram doubles as a sanity check on the final answer and stops careless errors in their tracks.
Which Quant topics carry the most weight, and which deserve a quick brush-up?
Sharpen My Quant Topic Priority12 Solved CAT-Level Averages Questions (with shortcut working)
Twelve solved questions, each pinned to a formula above. Benchmark: 60-90s per question.
Average of 5 numbers is 27. Adding a sixth drops it to 25. Find the sixth.
Solution. 6 × 25 − 5 × 27 = 150 − 135 = 15.
Answer: 15
30 students average 72 marks. Exclude the top 5 (avg 95). Average of the remaining 25?
Solution. (2160 − 475) / 25 = 1685 / 25 = 67.4.
Answer: 67.4
Class A: 40 students, average 65. Class B: 60 students, average 75. Combined average?
Solution. (65 × 40 + 75 × 60) / 100 = 7100 / 100 = 71. Naive 70 is the planted trap.
Answer: 71
Two milk-water solutions, 20% and 50% milk, mixed to 35% milk. Mixing ratio?
Solution. w1 : w2 = (50 − 35) : (35 − 20) = 15 : 15 = 1 : 1.
Answer: 1 : 1
8 men average 35 years, 12 women average 28 years, 5 children average 12 years. Group average age?
Solution. (280 + 336 + 60) / 25 = 676 / 25 = 27.04.
Answer: 27.04 years
A batsman averages 42 over 20 innings. Runs needed in the 21st to lift the average to 45?
Solution. 21 × 45 − 20 × 42 = 945 − 840 = 105.
Answer: 105 runs
11 cricketers average 65 kg. A 58 kg player is replaced and the average rises by 0.4 kg. New player's weight?
Solution. y = 58 + 0.4 × 11 = 58 + 4.4 = 62.4 kg.
Answer: 62.4 kg
25 employees average Rs 35,000. One earning Rs 60,000 leaves; a new hire joins and the average drops Rs 500. New hire's salary?
Solution. y = 60,000 + 25 × (−500) = 60,000 − 12,500 = Rs 47,500.
Answer: Rs 47,500
Average of all even numbers from 12 to 88 inclusive?
Solution. (12 + 88) / 2 = 50.
Answer: 50
Average of the first 25 multiples of 7?
Solution. (7 + 175) / 2 = 91, or 7 × 26 / 2 = 91.
Answer: 91
Average of 6 numbers is 18.5. Four are 12, 19, 22, 25; the other two are equal. Find each.
Solution. (6 × 18.5 − 78) / 2 = 33 / 2 = 16.5.
Answer: 16.5 each
Boys average 70 marks, girls average 80, combined class average 74. Boys-to-girls ratio?
Solution. boys : girls = (80 − 74) : (74 − 70) = 6 : 4 = 3 : 2.
Answer: 3 : 2
- 10+ solved inside 90s each: averages is exam-ready.
- 6-9 solved but missed Q3 or Q12: drill the weighted-average shortcut.
- Fewer than 6: restart from the 6-formula list and rework the table.
- Sub-60s on Q1, Q2, Q9, Q10 is the speed-floor for 90-percentile QA.
- Pair with the CAT Quant score improvement guide.
How Averages Connects to the Rest of CAT Quant
Averages is rarely tested in isolation — it is embedded in mixtures, alligation, profit-loss, average-speed (a weighted harmonic mean: total distance / total time), and DI sets. The SI and CI guide, the geometry formulas guide (centroid = average of three vertex coordinates), and the linear equations word problems guide show similar embedded usage.
- Memorise the 6 formulas before drilling problems.
- Unequal group sizes: write the weighted formula first.
- Cross-multiplication on every alligation or mixture problem.
- "One in, one out": shift = (y − x) / n; never recompute the sum.
- For any AP, use (first + last) / 2.
- Two averages? If their simple mean is in the choices, it is a trap.
Six formulas. Three shortcuts. One trap. The entire averages chapter for CAT 2026.
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