Averages look like the easiest chapter in CAT Quant. Many aspirants still lose marks here because they average two averages, miss a count, or use the wrong speed formula. The most important CAT averages questions are not hard. They are slippery, and a small slip costs a question you could have solved in a minute.
This guide covers the question types that matter, the formulas behind them, and solved examples. You will also find a free CAT averages questions PDF with 35 questions and solutions, and a formula sheet.
Why CAT Averages Questions Matter
CAT 2026 is scheduled for 29 November 2026, so check the official notification at iimcat.ac.in for any change. The expected pattern is 68 questions in 120 minutes, with 40 minutes per section. The Quantitative Aptitude section is expected to have 22 questions, and arithmetic is commonly reported to make up roughly half of it.
Averages work as a foundation topic. They appear directly, and they sit inside weighted averages, mixtures and alligation, data interpretation sets, and time-speed-distance. A student who is solid on averages is faster on all of these.
CAT Averages Questions at a Glance
| Question type | Core idea | Difficulty |
|---|---|---|
| Basic average and series | Sum = Average × Number of terms | Easy |
| Weighted / combined average | Weight each group by its size | Easy–Moderate |
| Adding or removing an item | Track totals, not averages | Moderate |
| Replacement | Change in total = n × change in average | Moderate |
| Average speed | Total distance ÷ total time | Moderate |
| Min/max with distinct integers | Push other values to extremes | Hard |
| Alligation-based averages | Ratio = distance of the mean from each end | Moderate–Hard |
The Formulas You Must Know for CAT Averages Questions
Remember one rule above all: work with totals. Convert every average into a sum, do the arithmetic, then convert back.
| Concept | Formula |
|---|---|
| Average | Sum ÷ Number of items |
| First n natural numbers | (n + 1) / 2 |
| First n odd numbers | n |
| First n even numbers | n + 1 |
| Equally spaced terms (AP) | (First + Last) / 2 |
| Combined average of two groups | (n₁A₁ + n₂A₂) / (n₁ + n₂) |
| New item added to n items, average changes by d | New item = Old average + (n + 1) × d |
| One item replaced in a group of n | New item = Old item + n × change in average |
| Average speed, equal distances | 2xy / (x + y) |
Most Important CAT Averages Questions with Solutions
1. Weighted average questions
Many “average of averages” questions test whether you weight by group size.
Question: 30 students average 60 marks and 20 students average 75. What is the average of all 50?
Solution: Total = 30 × 60 + 20 × 75 = 1800 + 1500 = 3300. Average = 3300 / 50 = 66.
If you had averaged 60 and 75 directly, you would get 67.5, which is wrong. The larger group is the 60-mark group, so the answer sits closer to 60.
2. Replacement questions
Question: The average weight of 8 people rises by 2.5 kg when a 65 kg person is replaced. What is the new person’s weight?
Solution: The total rises by 8 × 2.5 = 20 kg. New weight = 65 + 20 = 85 kg.
3. Batsman average questions
Question: A batsman scores 85 in his 17th innings and his average rises by 3. What is his average now?
Solution: New average = 85 − (16 × 3) = 37. Check: the old average is 34, so 34 × 16 = 544. Then 544 + 85 = 629, and 629 / 17 = 37.
4. Average speed questions
This is where the simple mean fails.
Question: A car travels uphill at 60 km/h and returns at 40 km/h over the same road. What is its average speed?
Solution: For equal distances, average speed = 2 × 60 × 40 / (60 + 40) = 48 km/h. It is not 50 km/h, because the car spends more time at the slower speed.
Use the simple mean only when the times are equal.
5. Maximum and minimum value questions
These questions test reasoning more than calculation.
Question: The average of 5 distinct positive integers is 20. What is the largest value the biggest integer can take?
Solution: The total is 100. To maximise one number, make the other four as small as possible: 1, 2, 3 and 4, which sum to 10. The largest is 100 − 10 = 90.
6. Alligation-based average questions
Question: A class averages 62 marks. Boys average 60 and girls average 65. What is the ratio of boys to girls?
Solution: Boys : Girls = (65 − 62) : (62 − 60) = 3 : 2. The class mean is closer to the boys’ mean, so there are more boys.
This links averages to mixtures and alligation for CAT.
CAT Averages Questions PDF: Free Practice Set
To practise under exam conditions, download these two files:
- 35 Most Important CAT Averages Questions with Solutions (PDF): 10 foundation, 12 core and 13 CAT-level questions, with step-by-step solutions and an answer key.
- CAT Averages Formula Sheet (PDF): formulas, shortcuts, common mistakes and worked examples.
Attempt each section against a timer, then compare your method with the written solution.
Common Mistakes in CAT Averages Questions
| Mistake | What to do instead |
|---|---|
| Averaging two averages directly | Use a weighted average with group sizes |
| Using (x + y)/2 for speed over equal distances | Use 2xy / (x + y) |
| Forgetting n + 1 after adding an item | Recount items after every change |
| Double-counting a shared term | Add group totals and subtract the overall total |
| Ignoring “distinct” or “integer” conditions | Test your answer with a real example |
How to Prepare: A Simple Plan
- Days 1–3: learn the formulas and solve the foundation questions.
- Days 4–7: practise weighted average and replacement questions daily.
- Week 2: add average speed and min/max reasoning.
- Ongoing: put 5 averages questions into every mock test, and keep an error log.
Pair this with CAT arithmetic practice
Frequently Asked Questions
What are the most important CAT averages questions?
Weighted average, replacement, adding or removing items, average speed, and min/max problems. They are tested directly and inside other arithmetic topics.
Is averages easy in CAT?
The concepts are basic. Most errors come from carelessness, such as wrong counts or averaging averages, not from difficult maths.
What is the formula for average in CAT?
Average = Sum of observations ÷ Number of observations. For combined groups, use (n₁A₁ + n₂A₂) / (n₁ + n₂).
How do I solve weighted average questions quickly?
Convert averages to totals and add the totals. For two groups, use alligation: the ratio of group sizes is the inverse of the distances from the mean.
Where can I get a CAT averages questions PDF?
Download the 35-question practice set with solutions and the formula sheet from this page.
What is the average speed formula for equal distances?
2xy / (x + y), where x and y are the two speeds.
Conclusion
The most important CAT averages questions reward one habit: think in totals. Learn the formulas, spot the question type in the first few seconds, and watch the common traps. Download the CAT averages questions PDF and formula sheet, practise under a timer, and averages can become a dependable source of marks in your Quant section.
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