Week 19: Vedic Maths & Speed Calculation Techniques

1. Why This Belongs in an Aptitude Course

Every technique in Weeks 1–18 tells you which calculation a question needs. None of them make the arithmetic itself faster once you know that — and on a timed test, a student who instantly recognizes "this is a percentage problem" but then spends 40 seconds long-multiplying two 2-digit numbers loses the same time as a student who didn't recognize the problem at all.

Vedic Maths is a set of 16 sutras (aphorisms), attributed to Bharati Krishna Tirthaji, that give shortcut algorithms for arithmetic that would otherwise require a full written method. This week covers four of the most immediately useful ones — not as a curiosity, but as the speed layer underneath everything the rest of this course already taught you to recognize.

These are shortcuts for specific number patterns, not a universal replacement

Each sutra below applies cleanly to a recognizable situation — a number ending in 5, two numbers near the same base, any two 2-digit numbers. Part of getting fast is recognizing which situation you're in, not trying to force every sutra onto every problem.

2. Ekadhikena Purvena — Squaring Numbers Ending in 5

"By one more than the previous" — this sutra squares any number ending in 5 in exactly two steps, with no multiplication of two-digit numbers at all.

the rule: for a number "X5", compute X × (X+1), then append 25

65^2:
  X = 6, X+1 = 7
  6 x 7 = 42
  Answer: 4225  (42, then append 25)

95^2:
  X = 9, X+1 = 10
  9 x 10 = 90
  Answer: 9025

105^2:
  X = 10, X+1 = 11
  10 x 11 = 110
  Answer: 11025

This works because (10a+5)² = 100·a(a+1) + 25 for any leading digit(s) a — the algebra behind the shortcut is simple, but the shortcut itself needs none of that to use: just multiply the leading digit(s) by one more than themselves, and append 25.

3. Nikhilam — Multiplying Near a Base

"All from 9 and the last from 10" — this sutra multiplies two numbers that are both close to the same power of 10 (10, 100, 1000) using their small deviations from that base instead of the numbers themselves.

both numbers below the base — deviations are negative

98 x 97   (base 100)
  98 is -2 from 100
  97 is -3 from 100

  Left part:  98 + (-3) = 95   (or 97 + (-2), same result)
  Right part: (-2) x (-3) = 6, written as 06 (2 digits, since base=100)

  Answer: 95|06 = 9506

one number above the base — deviations can mix signs

103 x 98   (base 100)
  103 is +3 from 100
  98  is -2 from 100

  Left part:  103 + (-2) = 101
  Right part: (+3) x (-2) = -6

  Answer: 101|(-06) = 10100 - 6 = 10094

The two parts are the same idea seen in Week 1's remainder work and Week 14's index laws — deviation from a convenient reference point is easier to multiply than the original numbers, and this sutra formalizes exactly that.

4. Urdhva-Tiryagbhyam — General Multiplication

"Vertically and crosswise" is the general-purpose sutra — it multiplies any two numbers of any size, and is what most placement-test candidates use Nikhilam's base trick to avoid needing at all. For two 2-digit numbers AB × CD:

23 x 14, step by step

    2 3
  x 1 4
  -------
Step 1 (units x units):     3 x 4 = 12        -> write 2, carry 1
Step 2 (cross-multiply,
        add products):  (2x4)+(3x1) = 8+3 = 11
                          11 + carry 1 = 12    -> write 2, carry 1
Step 3 (tens x tens):        2 x 1 = 2
                          2 + carry 1 = 3      -> write 3

Answer, right to left: 3 2 2  =  322

Every step only ever multiplies single digits — the "carrying" replaces writing out a full multi-line long multiplication, and the same crosswise pattern extends to 3-digit-by-3-digit multiplication by adding one more crosswise term per step.

5. Digit-Sum Verification

Also called "casting out nines" — a fast way to catch an arithmetic slip without redoing the whole calculation. Reduce each number to a single digit by repeatedly summing its digits, then check that the same relationship holds for the digit sums.

checking 47 x 38 = 1786

Digit sum of 47: 4+7=11 -> 1+1=2
Digit sum of 38: 3+8=11 -> 1+1=2
Product of digit sums: 2 x 2 = 4

Digit sum of the answer, 1786: 1+7+8+6=22 -> 2+2=4

Both sides give 4 -- the answer is CONSISTENT (though this
does not prove it's exactly correct -- see the callout below)

A passing digit-sum check rules out most mistakes — it doesn't prove correctness

Digit-sum verification catches transposed digits, dropped digits and most common slips, but it can't detect an error that happens to change the answer by a multiple of 9 (e.g. writing 1795 instead of 1786 would also pass). Treat a pass as "very likely correct, move on," and a failure as "definitely wrong, redo it" — not as absolute proof either way.

6. Practice Problems

Practice

30 problems, no calculator — Easy, Medium & Tough

Easy problems should take under 15 seconds each once the pattern clicks; Medium under 30 seconds; Tough under 60 seconds.

Easy (1–10) — Ekadhikena Purvena (squaring numbers ending in 5)

25²
35²
45²
55²
65²
75²
85²
95²
105²
115²

Easy — Answers

1) 625. 2) 1225. 3) 2025. 4) 3025. 5) 4225. 6) 5625. 7) 7225. 8) 9025. 9) 11025. 10) 13225.

Medium (11–20) — Nikhilam (multiplication near a base)

98 × 96 (base 100)
103 × 104 (base 100)
99 × 87 (base 100)
102 × 97 (base 100)
994 × 998 (base 1000)
1008 × 1003 (base 1000)
9 × 7 (base 10)
13 × 12 (base 10)
96 × 96 (base 100)
101 × 99 (base 100)

Medium — Answers

11) 9408. 12) 10712. 13) 8613. 14) 9894. 15) 992012. 16) 1011024. 17) 63. 18) 156. 19) 9216. 20) 9999.

Tough (21–30) — Urdhva-Tiryagbhyam (general 2-digit multiplication) & verification

23 × 14 (crosswise method)
67 × 54 (crosswise method)
38 × 47 (crosswise method)
72 × 89 (crosswise method)
56 × 63 (crosswise method)
Use digit-sum verification to check whether 84 × 37 = 3108 is consistent.
Use digit-sum verification to check whether 56 × 49 = 2744 is consistent.
145² using Ekadhikena Purvena.
993 × 989 using Nikhilam (base 1000).
A test has 30 rows of 47 marks each. Find the total marks using any sutra above.

Tough — Answers

21) 322. 22) 3618. 23) 1786. 24) 6408. 25) 3528. 26) digit sums 8+4=12→3 and 3+7=10→1, product 3×1=3; 3108→3+1+0+8=12→3 — consistent. 27) digit sums 5+6=11→2 and 4+9=13→4, product 2×4=8; 2744→2+7+4+4=17→8 — consistent. 28) 21025. 29) 982077. 30) 30×47=1410 marks.

7. Knowledge Check

Four quick questions. Expand each to check your answer.

Why does Ekadhikena Purvena only apply to numbers ending in 5, and not any two-digit number?

The shortcut relies on the algebraic identity (10a+5)² = 100·a(a+1)+25, which only produces the clean "append 25" pattern because the last digit is exactly 5. A number ending in any other digit doesn't reduce to this same form, so the two-step shortcut simply doesn't apply — a different sutra (like Urdhva-Tiryagbhyam) is needed instead.

Why is Nikhilam faster than Urdhva-Tiryagbhyam for a problem like 98 × 97, but not for a problem like 23 × 14?

Nikhilam's speed comes from both numbers having small deviations from a convenient base (100 in this case), which turns the multiplication into a tiny product of single-digit deviations. 23 and 14 aren't close to any shared power-of-10 base, so their deviations would be large and unhelpful — Urdhva-Tiryagbhyam's crosswise method, which works for any two numbers regardless of proximity to a base, is the better fit there.

If a digit-sum check passes, does that guarantee the multiplication was done correctly? Why or why not?

No — a passing check strongly suggests the answer is correct, but it cannot detect an error that shifts the result by an exact multiple of 9 (since digit-sum reduction is arithmetic modulo 9). It's a fast filter for catching most mistakes cheaply, not an airtight proof, so a pass means "very likely right," not "certainly right."

Why does this course place Vedic Maths in Week 19, after every other topic, rather than in Week 1?

These sutras speed up arithmetic you're already doing correctly — they don't teach you which calculation a word problem needs in the first place. Learning them before the topics that generate the calculations (percentages, interest, DI) would mean practicing speed on problems you can't yet recognize, whereas learning them here lets you immediately apply the speed gain to every calculation the previous 18 weeks already taught you to set up.