In a 40-minute CAT Quant section, shaving 30 seconds off each question reclaims about 11 minutes - enough to revisit four skipped problems. So calculation speed matters. But most "50 shortcuts for CAT" lists fail at the thing that actually decides whether a shortcut helps you: under exam pressure, a rule you have memorised without understanding is a rule you will misapply or abandon.
So this is eight techniques rather than fifty, and each one comes with the algebra that generates it. Eight understood techniques beat fifty half-remembered ones, because understanding is what lets you reconstruct a rule you have half-forgotten and recognise when it does not apply.
1. Squaring any number ending in 5
The rule: for a two-digit number ending in 5, take the tens digit n, compute n × (n+1), and append 25.
Worked: 75² → 7 × 8 = 56 → 5625. And 35² → 3 × 4 = 12 → 1225.
Why it works: any such number is 10n + 5, so
(10n + 5)² = 100n² + 100n + 25 = 100 · n(n+1) + 25
The "append 25" is the + 25, and the "n(n+1)" is multiplied by 100 - which is why it lands in the hundreds place. Once you have seen the identity, you also know it extends past two digits: 105² = 100 · (10 × 11) + 25 = 11025.
Where it appears: Geometry, constantly - anything with a radius or side ending in 5, and Pythagorean checks.
2. The successive-squares identity (how to get every square from one)
The rule: n² = (n−1)² + (2n−1). Each square is the previous one plus the next odd number.
Worked: you know 25² = 625. Then 26² = 625 + 51 = 676, and 24² = 625 − 49 = 576 (going down, subtract 2n−1 for the upper number).
Why it works: expand it. (n−1)² + 2n − 1 = n² − 2n + 1 + 2n − 1 = n². The gap between consecutive squares is always the odd number 2n−1, which is also why the squares 1, 4, 9, 16, 25 have gaps 3, 5, 7, 9.
Why this beats memorising a table: you only need the anchors - 15² = 225, 20² = 400, 25² = 625, 30² = 900 - and you can reach any square within a few steps of one. That is four facts instead of thirty, and it degrades gracefully: forget one anchor and you can still walk from the next.
3. Numbers near a base (the one identity that replaces a dozen tricks)
This is the highest-leverage item on the list, because it subsumes several rules that listicles present separately.
The rule: pick a convenient base B (50, 100, 1000). Write both numbers as B + a and B + b, where a and b may be negative. Then
(B + a)(B + b) = B² + B(a + b) + ab
Worked, base 50: 53 × 57 → 2500 + 50(3 + 7) + (3 × 7) = 2500 + 500 + 21 = 3021.
Worked, base 100 with a negative: 97 × 103 → 10000 + 100(−3 + 3) + (−3 × 3) = 10000 + 0 − 9 = 9991.
Worked, both below: 96 × 98 → 10000 + 100(−4 − 2) + (−4 × −2) = 10000 − 600 + 8 = 9408.
Why it works: it is nothing more than expanding the product. The reason it feels fast is that B² and B(a+b) are both trivial when B is a power of ten or half of one, so the only real multiplication left is ab - two small numbers.
What this replaces: the separate "multiply by 99", "multiply by 101", "both numbers near 100" and "both near 50" rules are all this one identity with different values of B, a and b. Learn the identity and you stop needing to remember which rule applies.
4. Factor rebalancing (halve one, double the other)
The rule: repeatedly halve one factor and double the other until one becomes a round number.
Worked: 36 × 25 → 18 × 50 → 9 × 100 = 900. Also 14 × 35 → 7 × 70 = 490.
Why it works: multiplication is associative, so 36 × 25 = (18 × 2) × 25 = 18 × (2 × 25) = 18 × 50. You are moving a factor of 2 across the multiplication sign, which cannot change the product.
The generalisation worth having: you can move any factor, not just 2. 15 × 24 = 15 × 8 × 3 = 120 × 3 = 360, or 45 × 12 = 45 × 4 × 3 = 180 × 3 = 540. Look for the factor that makes one side land on a multiple of 10.
5. Vertical and crosswise for two-digit multiplication
The rule: for AB × CD, the digits of the answer are [A×C] [A×D + B×C] [B×D], carrying where a column exceeds 9.
Worked: 34 × 52. Right column: 4 × 2 = 8. Middle: (3 × 2) + (4 × 5) = 6 + 20 = 26 → write 6, carry 2. Left: (3 × 5) = 15, + 2 = 17. Answer 1768.
Why it works: write the numbers as (30 + 4)(50 + 2) and expand: 1500 + 60 + 200 + 8. The three columns are just those four terms grouped by place value - hundreds (1500), tens (60 + 200), units (8). "Crosswise" is the two middle terms; there is no magic in it.
When to skip it: if either number is close to a base, technique 3 is faster and involves less carrying. Crosswise is the fallback for awkward pairs like 34 × 52 where no base helps.
6. Percentage-fraction equivalents, and reversing a percentage change
The rule: know the reciprocals cold. 1/6 = 16.67%, 1/7 ≈ 14.28%, 1/8 = 12.5%, 1/9 ≈ 11.11%, 1/11 ≈ 9.09%, 1/12 ≈ 8.33%.
The question this actually answers: "a price rises 16.67% - what decrease returns it to the original?" The tempting answer, 16.67%, is wrong.
Worked: a rise of 16.67% is a multiplication by 7/6. To undo it you multiply by 6/7, which is a decrease of 1/7 ≈ 14.28%. No arithmetic required once you see the fractions.
Why it works, generally: an increase of x% needs a decrease of 100x/(100 + x)% to reverse it, because you are dividing by (1 + x/100) rather than subtracting. The fraction form makes this visible: rise by 1/n means multiply by (n+1)/n, so reversing means multiply by n/(n+1), a fall of 1/(n+1).
Where it appears: profit and loss, discount chains, and successive-percentage questions - which is a large share of the arithmetic on any paper.
7. Complement multiplication for numbers just under a round figure
The rule: n × (B − k) = nB − nk.
Worked: 47 × 99 = 4700 − 47 = 4653. And 47 × 98 = 4700 − 94 = 4606. And 63 × 101 = 6300 + 63 = 6363.
Why it works: plain distribution. It is worth listing separately from technique 3 only because it applies when just one number is near a base, where technique 3 needs both.
The habit to build: before multiplying anything, glance at both numbers for proximity to 10, 100, 25, 50 or 1000. The recognition is the skill; the arithmetic afterwards is trivial.
8. Divisibility by 7, and why the rule is not arbitrary
The rule: strip the last digit, double it, subtract from the rest. If the result is divisible by 7, so was the original.
Worked: 343 → 34 − (2 × 3) = 28. Divisible by 7, so 343 is. And 1729 → 172 − 18 = 154 → 15 − 8 = 7. Divisible.
Why it works: write the number as 10a + b. Then
10(a − 2b) = 10a − 20b = (10a + b) − 21b
21b is a multiple of 7, so 10(a − 2b) and the original number leave the same remainder mod 7. And since 10 shares no factor with 7, dropping it changes nothing. That is the whole proof - which also tells you the rule is repeatable, as the 1729 example shows.
Related, and worth knowing for the same reason: the unit digits of cubes cycle 1→1, 2→8, 3→7, 4→4, 5→5, 6→6, 7→3, 8→2, 9→9, 0→0. Large-power questions in Number Systems frequently need only this.
When a shortcut is the wrong choice
This is the part most shortcut lists omit, and it costs marks.
A shortcut is only worth using if you recognise its applicability in under two seconds. If you spend eight seconds deciding whether a number is "near enough" to a base, you have lost more than the technique saves, and you have spent working memory you needed for the actual problem. Long multiplication you can execute without thinking beats a clever method you have to deliberate over.
The second failure is subtler: shortcuts that produce approximations. Knowing 1/7 ≈ 14.28% is fine for eliminating options, but if a question's answer choices are close together, an approximation can point you confidently at the wrong one. Notice when you have approximated.
Both of these are triage decisions rather than calculation ones, which is why they belong with the rest of your in-exam strategy - see our in-exam time management guide for how question selection interacts with this.
Turning eight techniques into reflexes
Knowing these in theory is not the same as firing one in three seconds under pressure. That gap closes with repetition and feedback, which is what Karma Yogi's Learn section is built for: reference grids for Tables, Squares and Cubes, plus a flip-card quiz with adjustable ranges, so you can drill squares 15-30 only, or tables 13-19 only.
Drill in both directions. Forward is "what is 17 × 14?"; reverse is "which two numbers between 13 and 19 multiply to 238?". Reverse is where automaticity is actually built, because CAT rarely asks you to multiply - it hands you a product and expects you to factor it. Our mental math guide goes further into structuring those drills.
Fifteen minutes a day beats an hour on Sunday: five minutes on the grids, ten on the quiz, at a fixed time so starting needs no willpower. Log each session so the heatmap tells you whether you are actually drilling or only intending to - the same principle as tracking mocks at section level, which in a published log of 20 real mocks turned out to be the difference between noticing a plateau and missing it for two months.
See how Karma Yogi tracks CAT preparation, or start free - it takes 60 seconds.
End of essay
- Anish Guruvelli