Averages
The average, or arithmetic mean, is the most natural way to compress many numbers into one representative value, and CAT loves it precisely because it links so many other chapters together. An average is just the total divided by the count, so the single most useful re-reading of any average question is "total = average × count" — once you think in totals rather than means, age problems, marks problems, run-rate problems and salary problems all collapse into one-line equations. CAT rarely asks a plain "find the average of these five numbers"; instead it tests what happens when a value is added, removed or replaced, when two groups merge, or when speed changes over different stretches of a journey. This chapter builds that fluency in four moves: the arithmetic mean and its shift/deviation shortcuts, the weighted average and the alligation rule that makes mixture and group questions almost instant, average speed (where the harmonic mean quietly hides), and replacement problems where a person or value swaps out and the average jumps. Each topic comes with worked CAT-style examples, the fastest exam method, and the traps — like averaging two speeds the wrong way — that silently cost careless students marks.
Topics
⚡ CAT shortcuts & speed methods
The fastest ways to crack this chapter under time pressure — the techniques that separate a 95+ percentiler from the rest.
- Re-read every average question as total = average × count; solve for whichever of the three is missing.
- For any evenly spaced set the average is the middle term (or (first + last)/2) — never sum the list.
- Use the deviation method: pick an assumed mean, add only the small deviations, divide, add back.
- Mixing two groups? Use alligation: w₁ : w₂ = (A₂ − Avg) : (Avg − A₁) — the blend sits closer to the heavier group.
- Equal distances at two speeds ⇒ harmonic mean 2xy/(x+y). Equal times ⇒ simple mean (x+y)/2.
- Pure replacement (count fixed): new value = old value + (change in average) × count.
⚠️ Common mistakes & traps
CAT is designed so that careless errors here cost you marks. Internalise each trap before the exam.
- Averaging two speeds as (x+y)/2 when the distances are equal — the correct value is the harmonic mean.
- Multiplying the change in average by the wrong count in replacement problems (use the unchanged group size for a swap).
- Taking a simple mean of group averages when the groups differ in size — you need a weighted average.
- Forgetting that adding/removing a member changes the count, so total = average × count must be redone for both states.
- Reversing the alligation ratio — the weight is proportional to the distance from the OTHER value, not the nearer one.
📈 CAT exam insight & PYQ analysis
🎴 Flashcards — instant recall
Tap a card to reveal the answer. Drill these until they are automatic.
📌 Quick revision
Chapter test
🏆 Vidaara CAT success checklist
You have truly mastered Averages when you can tick every box below.
- Recall every formula in this chapter without looking them up
- Solve each topic’s practice set with at least 80% accuracy
- Use the chapter shortcuts to cut your solving time in half
- Spot and avoid every common trap listed above
- Score 80%+ on the timed chapter test
📋 Chapter mastery scorecard
Track where you stand. Aim for the target before moving to the next chapter.
| Skill checkpoint | Target |
|---|---|
| Concept theory & formulas understood | 100% |
| Topic practice sets attempted (4 topics) | 4/4 |
| Best topic-test score | — → 80%+ |
| Chapter test score | — → 80%+ |
| Flashcards drilled to instant recall | 12 cards |
Formula Reference Sheet
Core average relations
| Arithmetic mean | Average = (Sum of observations) / (Number of observations) |
|---|---|
| Sum from average | Sum = Average × Count |
| Average of first n natural numbers | (n + 1) / 2 |
| Average of an arithmetic progression | (First term + Last term) / 2 |
| Deviation (shift) method | Average = Assumed mean + (Sum of deviations) / Count |
Weighted, speed & replacement tools
| Weighted average | (w₁a₁ + w₂a₂ + …) / (w₁ + w₂ + …) |
|---|---|
| Alligation (ratio of weights) | w₁ : w₂ = (A₂ − Avg) : (Avg − A₁) |
| Average speed (whole journey) | Total distance / Total time |
| Equal-distance two speeds | 2xy / (x + y) (harmonic mean) |
| Change in average on replacement | New value = Old value ± (Change in average × Count) |