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Time, Speed & Distance
Every CAT Time, Speed & Distance formula on one page — relative speed, trains crossing poles and platforms, boats and streams, and delay problems — each with a worked example.
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Averages
CAT'26 QUANT CHEATSHEET
Every average formula and CAT shortcut you need for CAT 2026 — on one page.
Averages look simple and then quietly eat time on CAT, because the exam rarely asks for a plain mean. It buries the idea inside combined groups, weighted mixes, replacement and age problems where a careless student recomputes everything from scratch. The fix is one habit: turn every average into a total, solve on the totals, then convert back. This sheet lays out every formula you need for the CAT quant section — the core definition, adding and removing members, combined and weighted averages, alligation, consecutive-number and arithmetic-progression shortcuts, and age problems — each with a worked example in real numbers. Keep it open while you drill, and after a mock check your standing on the CAT score predictor to see where averages are leaking marks.
Dev note: the canonical route
/cheatsheets/optima-learn-averages-cheatsheet is not live yet
(no /cheatsheets hub in the current sitemap). Built production-ready as a drop-in once it ships.Averages: every formula you need
1Core Idea
The balance point of all the values.
Average = (Total Sum) ÷ (Number of Terms)
Example: marks 40, 50, 60, 70, 80 → (40+50+60+70+80)/5 = 60.
CAT Hack: Whenever an average is given, convert it to a total immediately — totals are easier to work with.
2Fundamental Formulae
Three rearrangements of the same relation.
Sum = Avg × Num | Avg = Sum ÷ Num | Num = Sum ÷ Avg
Example: Avg = 72 over 50 students → Total = 72 × 50 = 3600.
CAT Hack: Most average questions get easy once you rewrite the average as a total.
3Change in Average
Shift the total and the average shifts in proportion.
Change in Avg = (Change in Total) ÷ n
Example: 40 students, one score rises by 20 → change in avg = 20/40 = 0.5.
CAT Favourite: Avg salary rises ₹500 for 100 employees → total salary rises 500 × 100 = ₹50,000.
4Adding a New Member
One extra member joins the group.
New Avg = (nA + x) ÷ (n + 1)
Example: 10 students avg 50, new score 70 → (10·50 + 70)/11 = 51.82.
CAT Insight: x above the average pulls it up; x below the average pulls it down.
5Removing a Member
One member leaves the group.
New Avg = (nA − x) ÷ (n − 1)
Example: 20 staff avg ₹40,000; one earning ₹60,000 leaves → (20·40000 − 60000)/19 = 38,947.
CAT Insight: Removing a value above the average lowers it; removing one below raises it.
6Replacement Problems
One value is swapped for another.
Change in Avg = (New − Old) ÷ n
Example: 50 students, one score 40 → 70 → change in avg = (70 − 40)/50 = +0.6.
Common Mistake: Do not recompute the whole average — only the net shift from the swapped value matters.
7Combined Average
Merging two groups into one average.
Combined Avg = (n1A1 + n2A2) ÷ (n1 + n2)
Example: 30 avg 60 and 20 avg 75 → (30·60 + 20·75)/50 = 3300/50 = 66.
CAT Insight: The combined average always lies between the two individual averages.
8Alligation Shortcut
Split a mix into a ratio around the mean.
Ratio = (A2 − M) : (M − A1) (M = overall avg)
Example: avg 40 and avg 70 give overall 55 → (70 − 55):(55 − 40) = 15:15 = 1:1.
CAT Hack: Reach for alligation whenever the question asks for a ratio of quantities.
9Consecutive Numbers
Evenly spaced numbers average to their middle.
Average = middle term | first n naturals = (n + 1) ÷ 2
Example: 21, 22, 23, 24, 25 → average = 23; and 1 to 100 → (100+1)/2 = 50.5.
CAT Hack: For consecutive numbers, take the middle term instead of adding everything up.
10AP Average
Any arithmetic progression averages its ends.
Average = (First + Last) ÷ 2
Example: 5, 10, 15, 20, 25 → (5 + 25)/2 = 15.
CAT Favourite: Any evenly spaced set is solved instantly with (first + last)/2 — no summing needed.
11Age Problems
Every age rises together over time.
After x years → New Avg = Old Avg + x
Example: average age 25; after 4 years → 25 + 4 = 29.
CAT Insight: For entry/exit age problems, convert the average to a total age first, then adjust.
12Weighted Average
Averages weighted by how many are in each group.
Weighted Avg = Σ(wx) ÷ Σw
Example: 40 avg 70 and 60 avg 90 → (40·70 + 60·90)/100 = 8200/100 = 82.
CAT Hack: The larger group pulls the combined average toward its own value.
CAT exam shortcuts, traps & revision
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CAT Exam Shortcuts
- Average × Number = Total; Sum ÷ Number = Average
- Combined average lies between the two group averages
- Consecutive numbers → middle term
- AP average = (First + Last) ÷ 2
- Average age after x years → add x
- Use alligation whenever a ratio is asked
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Most Common CAT Traps
- Averaging averages directly: 10 avg 40 and 100 avg 80 is not (40+80)/2 — weight by group size.
- Ignoring group sizes when combining two averages.
- Recomputing the whole average on a change instead of tracking only the net shift.
- Solving in averages instead of converting average → total → solve → convert back.
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30-Second Revision Box
- Average = Sum ÷ Number; Sum = Average × Number
- Change in Average = Change in Total ÷ Number
- Combined Avg = (n1A1 + n2A2) ÷ (n1 + n2)
- Alligation ratio = (A2 − M) : (M − A1)
- Consecutive = middle term; AP avg = (first + last)/2
- Age problems → convert to total age first
Averages reward setup over arithmetic — spot whether a question is really about weighting, a net shift, or a total, and the numbers fall out in one line. Drill this sheet until converting average to total is automatic, then test it on full sets and track progress with the CAT score predictor. For more topic guides, browse the Optima Learn blog or explore every study guide, and work through the full CAT exam hub for section-wise strategy. When you want structured, mentor-led prep, the team at Optima Learn can map out your plan — book a free CAT 2026 call and line up your next eight weeks.
