Viral Maths — Ch04: Multiplication

📌 What This Chapter Covers

Multiplication is the method of finding the product of two or more numbers — this is the LARGEST chapter in Viral Maths with 26 named approaches.
Book ki advice: daily practice karo, surroundings mein numbers dhundo aur verbally multiply karo.
Approaches ko groups mein organize kiya hai: Repeat-family (11/99/101/1001), general break methods, Miya/consecutive-number tricks, ×5 family, fraction-shortcut family (25/125/0.5 etc.), Ice Cream (repeat-digit), Double & Partition, aur special-pattern methods (zero-middle, ends-in-1).

⚡ QUICK RECALL

Sabse pehle dekho: kya multiplier 11, 99, 101, 1001 hai? Ya ek number 5/25/125/0.5/1.5/2.5/12.5 hai? Ya dono numbers consecutive/close hain? Iske hisaab se sahi tab choose karo.

🗂️ Chapter Index (26 Types Grouped into 9 Families)

Tab	Types Covered	Family
2	Type 1, 2, 14, 15	×11, ×99/999/9999, ×101 (Chhoti Machine), ×1001 (Badi Machine)
3	Type 3, 21	Any-number×any-number break method, Table Approach
4	Type 4, 5, 6, 18	Bade Miya-Chote Miya (even×ends5, ending-75), Padosan (consecutive), Trishul (consecutive even/odd)
5	Type 7, 8	Even×5, Odd×5
6	Type 9,10,11,12,13,16,17,19	×25, ×0.5, ×125, ×1.25, ×12.5, ×1.5, ×15, ×2.5
7	Type 20	Ice Cream Approach (repeat-digit × single digit)
8	Type 22, 23, 24	Number×its double, Product Partition (a) and (b)
9	Type 25, 26	Triple-digit with 0 in middle, Two-digit numbers ending in 1
10	—	Master Table — all 26 types summarized

Type 1 — Multiplication with 11 (Space Method)

2-digit example: 11 × 63

Step 1: Leave a space between digits: 6 _ 3 Step 2: Add them: 6+3 = 9 Step 3: Place in middle: 693

3-digit example: 11 × 876

Step 1: 8 _ _ 6 (outer digits fixed) Step 2: Add adjacent pairs from right: 7+6=13, 8+7=15 Step 3: Place with carries: 8(15)(13)6 → carries roll left Step 4: Final answer: 9636

⚠ EXAM TRAP

Jab bhi adjacent-digit sum ≥10 ho, carry ko turant left mein add karo before finalizing — warna answer galat aayega.

Type 2 — Multiplication with 99 / 999 / 9999

Example: 999 × 648

Step 1: Subtract 1 from 648: 647 Step 2: 999 − 647 = 352 Step 3: Combine: 647352

⚡ QUICK RECALL

n × 99...9 (k nines) = (n−1) | (10^k − n) — left part is n−1, right part is the complement.

Type 14 — Chhoti Photocopy Machine Approach (× 101)

2-digit number × 101 → just repeat the number

21 × 101 = 2121 35 × 101 = 3535

Type 15 — Badi Photocopy Machine Approach (× 1001)

3-digit number × 1001 → repeat the number

352 × 1001 = 352352 546 × 1001 = 546546

2-digit number × 1001 → repeat with a 0 in the middle

45 × 1001 = 45045 31 × 1001 = 31031

⚡ QUICK RECALL — "Photocopy" Family

×101 = Chhoti (small) machine, repeats a 2-digit number.
×1001 = Badi (big) machine, repeats a 3-digit number (or 2-digit with 0 padding).

Type 3 — Any Number × Any Number (Place-Value Break)

Example: 39 × 17

Step 1: Break 39 as (30+9) Step 2: (30×17) + (9×17) = 510 + 153 Step 3: Answer = 663

Example: 251 × 12

Break 251 = 200+50+1 (200×12)+(50×12)+(1×12) = 2400+600+12 = 3012

Type 21 — Table Approach (Number 1–99 × Single Digit)

Example: 56 × 7

Step 1: Break 56 as (50+6) Step 2: 50×7 = 350 Step 3: 6×7 = 42 Step 4: Add: 350+42 = 392

⚡ QUICK RECALL

Type 3 aur Type 21 same core idea hain — bas Type 21 specifically single-digit multiplier ke liye highlight kiya gaya hai kyunki yeh sabse common exam pattern hai.

Type 4 — Bade Miya-Chote Miya Approach (Even × Ends-in-5)

Example: 35 × 26

Step 1: Double the number ending in 5: 35×2 = 70 Step 2: Half the even number: 26/2 = 13 Step 3: Multiply results: 70×13 = 910

Type 18 — Bade Miya-Chote Miya Extended (× Number Ending in 75)

Base pattern: 75×4=300, 175×4=700, 275×4=1100, 375×4=1500...

Example: 28 × 175

Step 1: Divide 28 by 4: 7 Step 2: Multiply 175 by 4: 700 Step 3: Multiply results: 7×700 = 4900

⚠ EXAM TRAP

Yeh method sirf tab kaam karta hai jab first number 4 se cleanly divide ho jaaye — warna fraction aa jayega aur approach slow ho jayega.

Type 5 — Padosan Approach (Consecutive Numbers)

Requires knowing squares 1–100 by heart

Example: 24 × 25

Method A: Square of smaller + smaller: 24²+24 = 576+24 = 600 Method B: Square of bigger − bigger: 25²−25 = 625−25 = 600

Type 6 — Trishul Approach (Consecutive Even/Odd Numbers)

Example: 12 × 14 (middle = 13)

12 is 1 less than 13, 14 is 1 more Answer = 13² − 1² = 169−1 = 168

Example: 47 × 53 (middle = 50)

47 is 3 less, 53 is 3 more than 50 Answer = 50² − 3² = 2500−9 = 2491

⚡ QUICK RECALL

Trishul = find the middle number, square it, subtract the square of the gap. Works for BOTH consecutive-even and consecutive-odd pairs, and even wider symmetric gaps.

Type 7 — Multiplication of Even Number with 5

Example: 56 × 5

Step 1: Half of 56 = 28 Step 2: Add a 0 at the end: 280

Example: 648 × 5

Half of 648 = 324 → 3240

Type 8 — Multiplication of Odd Number with 5

Example: 57 × 5

Step 1: Subtract 1: 57−1 = 56 (now even) Step 2: Half of 56: 28 Step 3: Add 5 at the end: 285

Example: 457 × 5

457−1=456 → half=228 → append 5 → 2285

⚠ EXAM TRAP

Even number×5 → append 0. Odd number×5 → append 5. Don't mix these up.

Type 9 — × 25 (= 100/4)

Example: 648 × 25

Divide by 4: 648/4 = 162 Add 00: 16200

Type 10 — × 0.5 (= 1/2)

Example: 246 × 0.5

Just halve it: 246/2 = 123

Type 11 — × 125 (= 1000/8)

Example: 128 × 125

Divide by 8: 128/8 = 16 Add 000: 16000

Type 12 — × 1.25 (= 10/8)

Example: 48 × 1.25

Divide by 8: 48/8 = 6 Add 0: 60

Type 13 — × 12.5 (= 100/8)

Example: 128 × 12.5

Divide by 8: 128/8 = 16 Add 00: 1600

⚠ EXAM TRAP

×1.25, ×12.5, ×125 — sab ÷8 hain. Sirf zeros ki ginती alag hai: 1.25→one 0, 12.5→two 0s, 125→three 0s.

Type 16 — × 1.5 (Add Half of Itself)

Example: 42 × 1.5

Half of 42 = 21 42+21 = 63

Type 17 — × 15 (Add Half, Then ×10)

Example: 48 × 15

Step 1: Add half of itself: 48+24 = 72 Step 2: Append a 0: 720

Type 19 — × 2.5 (= 10/4)

Example: 36 × 2.5

Divide by 4: 36/4 = 9 Add 0: 90

Example: 248 × 2.5

248/4 = 62 → 620

⚡ QUICK RECALL — Whole Fraction-Shortcut Family

×0.5=÷2 | ×2.5=÷4+0 | ×25=÷4+00 | ×1.25=÷8+0 | ×12.5=÷8+00 | ×125=÷8+000 | ×1.5=+half | ×15=+half then ×10

Type 20 — Ice Cream Approach (Repeated-digit Number × Single Digit)

Use for: 22, 44, 55, 555, 666... × single digit

Example: 44 × 3

Step 1: Units product: 4×3 = 12 Step 2: One "4" left over → leave 1 space Step 3: Place sum of end digits (1+2=3) in between Answer: 1(3)2 = 132

Example: 666 × 8

Step 1: Units product: 6×8 = 48 Step 2: Two "6"s left → leave 2 spaces Step 3: Fill with sum (4+8=12), carry 1 Answer: 5328

⚠ EXAM TRAP

Jitni baar digit repeat hoti hai, utni hi spaces beech mein banti hain — carries ko dhyan se left mein propagate karo.

Type 22 — Number × Its Own Double

Example: 14 × 28

Square of smaller: 14² = 196 Double it: 196×2 = 392

⚡ QUICK RECALL

n × 2n = 2n² — requires knowing squares 1-100 fluently (connects to Ch01 Important Products).

Type 23 — Product Partition Approach (a) — Clean 2-digit Split

Example: 1224 × 4

Step 1: Break into 12 (Part 1) and 24 (Part 2) Step 2: 12×4 = 48 Step 3: 24×4 = 96 Step 4: Since part 2 has exactly 2 digits, combine directly: 4896

Type 24 — Product Partition Approach (b) — With Carry

Example: 2568 × 2

Step 1: Break into 256 (Part 1) and 8 (Part 2) Step 2: 256×2 = 512 Step 3: 8×2 = 16 Step 4: Part 2 should be 1 digit only — use 6, carry the 1 to Part 1 Step 5: 512+1 = 513, combine with 6: 5136

⚠ EXAM TRAP

Right-partition ka result hamesha utne hi digits ka hona chahiye jitna original right-part tha — extra digit ho to carry LEFT mein daalo, jaise column-addition mein hota hai.

Type 25 — Triple-Digit × Triple-Digit with Zero in Middle

Pattern: a0b × c0d

Example: 503 × 408

Step 1: 5×4 = 20 (left block) Step 2: 5×8 + 4×3 = 40+12 = 52 (middle block) Step 3: 3×8 = 24 (right block) Step 4: Combine (with carries): 20|52|24 → 205224

⚡ QUICK RECALL

Left = a×c, Middle = a×d+b×c, Right = b×d — teen blocks banao, phir right-to-left carry adjust karo.

Type 26 — Two-digit Numbers Ending in 1

Example: 51 × 41

Step 1: Units product: 1×1 = 1 Step 2: Sum of tens digits: 5+4 = 9 Step 3: Product of tens digits: 5×4 = 20 Combine: 20 | 9 | 1 → 2091

⚠ EXAM TRAP

Yeh sirf tab kaam karta hai jab DONO numbers ka units digit 1 ho — 51×42 pe apply nahi hoga.

📋 Master Table — All 26 Multiplication Types

#	Approach	Rule
1	× 11	Space method: add adjacent digits, place in middle
2	× 99/999/9999	(n−1) \| complement to power of 10
3	Any × Any	Break by place value, multiply each part, add
4	Bade Miya-Chote Miya	Even × ends-5: double the 5-ender, halve the even
5	Padosan (consecutive)	smaller²+smaller OR bigger²−bigger
6	Trishul (consecutive even/odd)	middle² − gap²
7	Even × 5	Half the number, append 0
8	Odd × 5	Subtract 1, half, append 5
9	× 25	÷4, append 00
10	× 0.5	÷2
11	× 125	÷8, append 000
12	× 1.25	÷8, append 0
13	× 12.5	÷8, append 00
14	Chhoti Machine (×101)	Repeat 2-digit number
15	Badi Machine (×1001)	Repeat 3-digit (or 2-digit with 0 pad)
16	× 1.5	Add half of itself
17	× 15	Add half of itself, append 0
18	Bade Miya (ends-75)	÷4 first number, ×4 second number, multiply
19	× 2.5	÷4, append 0
20	Ice Cream (repeat-digit×1digit)	units product, spaced digit-sum insert
21	Table Approach	Break tens+units, multiply single digit, add
22	Number × its double	2 × (smaller number)²
23	Product Partition (a)	Split clean, multiply each, combine
24	Product Partition (b)	Split with carry when right part overflows
25	Zero-middle triple×triple	a×c \| a×d+b×c \| b×d
26	Ends-in-1 (2-digit)	tens×tens \| tens sum \| 1

🔑 Approach Selection Flowchart

Multiplier is 11 → Type 1
Multiplier is 99/999/9999 → Type 2
Multiplier is 101 → Type 14 | 1001 → Type 15
One number even, other ends in 5 → Type 4 (or Type 18 if ends in 75)
Two consecutive numbers → Type 5
Two consecutive even OR odd numbers → Type 6 (Trishul)
Multiplier is 5 → Type 7/8 (even/odd)
Multiplier is 25/125/0.5/1.25/12.5/1.5/15/2.5 → Fraction-Shortcut family (Tab 6)
Number is a repeated digit (22,44,555...) × single digit → Type 20 (Ice Cream)
Number × its own double → Type 22
Big number × single digit, no special pattern → Type 23/24 (Product Partition)
Triple-digit × triple-digit, zero in middle → Type 25
Both numbers end in 1 → Type 26