Viral Maths — Ch05: Division

📌 What This Chapter Covers

Division is one of the most important fundamentals of arithmetic — it allows breaking large numbers into smaller groups of equal size.
The book advises daily practice: observe numbers around you and divide them verbally.
There are ten named approaches, mostly based on fraction shortcuts (5, 0.5, 0.25, 0.125, 25 and 125), along with special pattern-recognition types such as the 11 Family, 9 Family, Broken Heart and Half Pattern.

⚡ QUICK RECALL

Core idea: when dividing by a convenient fractional value (5, 25, 125, 0.5, 0.25 or 0.125), convert the operation to multiplication and shift the decimal. This makes division as fast as multiplication.

🗂️ Chapter Index

Tab	Types Covered	Core Trick
2	Type 1: ÷5	×2, shift decimal 1 place
3	Type 2,3,4: ÷0.5, ÷0.25, ÷0.125	×2, ×4, ×8 respectively
4	Type 5,6: ÷125, ÷25	×8/×4, shift decimal 3/2 places
5	Type 7: ÷11,22,33,44...	Add first+last digit, divide by single digit
6	Type 8: ÷9,99,999...	Repeating decimal pattern
7	Type 9: Broken Heart Approach	Split into clean-divide + known % fraction
8	Type 10: Half-Pattern	Special numerator:denominator ratio → answer is always 5
9	Master Table	All 10 types summarized

Type 1 — Division of Number with 5

Rule: 5 = 10/2, so ÷5 = ×2, then shift decimal 1 place left

Example: 36 ÷ 5 36 × 2 = 72 → shift decimal 1 place → 7.2 Example: 323 ÷ 5 323 × 2 = 646 → shift decimal 1 place → 64.6

⚡ QUICK RECALL

Double the number, then move the decimal point one place to the left. Simple and always works for ÷5.

Type 2 — Division of Number with 0.5

Rule: 0.5 = 1/2, so ÷0.5 = ×2

36 ÷ 0.5 = 36×2 = 72 362 ÷ 0.5 = 362×2 = 724

Type 3 — Division of Number with 0.25

Rule: 0.25 = 1/4, so ÷0.25 = ×4

3 ÷ 0.25 = 3×4 = 12 134 ÷ 0.25 = 134×4 = 536

Type 4 — Division of Number with 0.125

Rule: 0.125 = 1/8, so ÷0.125 = ×8

18 ÷ 0.125 = 18×8 = 144 132 ÷ 0.125 = 132×8 = 1056

⚠ EXAM TRAP

Dividing by a number LESS than 1 always INCREASES the result — 18÷0.125=144 is much bigger than 18. Students often confuse ÷0.125 with ÷125 (which decreases the result).

Type 5 — Division of Number with 125

Rule: 125 = 1000/8, so ÷125 = ×8, shift decimal 3 places left

18 ÷ 125 = 18×8 = 144 → shift 3 places → 0.144 132 ÷ 125 = 132×8 = 1056 → shift 3 places → 1.056

Type 6 — Division of Number with 25

Rule: 25 = 100/4, so ÷25 = ×4, shift decimal 2 places left

18 ÷ 25 = 18×4 = 72 → shift 2 places → 0.72 134 ÷ 25 = 134×4 = 536 → shift 2 places → 5.36

⚡ QUICK RECALL — Whole Family

÷0.5=×2 | ÷0.25=×4 | ÷0.125=×8 | ÷25=×4(shift 2) | ÷125=×8(shift 3)
The multiplier depends on which fraction (1/2, 1/4, 1/8) the divisor equals; the decimal shift depends on how many zeros/place-values are in the divisor.

Type 7 — Division with 11, 22, 33, 44...

Applies to numbers where first digit + last digit = middle digit (e.g. 495: 4+5=9)

Example: 132 ÷ 22

Step 1: Pick first and last digits of 132: "1" and "2" → 12 Step 2: Divide by 2 (since 22 = 11×2): 12/2 = 6

Example: 682 ÷ 55

Step 1: Pick first and last digits: "6" and "2" → 62 Step 2: Divide by 5 (since 55 = 11×5): 62/5 = 12.4

⚠ EXAM TRAP

For divisors 22/33/44/55..., the last digit gives the number by which the first-and-last-digit combination must be divided (22→÷2, 33→÷3, 44→÷4, 55→÷5). This works only for specific sandwich-digit numbers.

Type 8 — Division of Number with 9, 99, 999...

Rule: Match number of 9's in divisor to digit-length of numerator (pad with zeros if needed)

4 ÷ 9 (one 9, one digit) → 0.444... 23 ÷ 99 (two 9s, two digits) → 0.232323... 56 ÷ 999 (three 9s, but 56 is 2-digit) → pad to 3-digit: 056 → 0.056056056...

⚠ EXAM TRAP

If the numerator has fewer digits than the string of 9s in the divisor, LEFT-PAD it with zeros before repeating: 56÷999 = 0.056056, NOT 0.56056.

Type 9 — Broken Heart Approach

Break the numerator into two parts: one that divides evenly by the denominator, and a leftover expressed via a known percentage fraction

Example: 33 ÷ 16

Step 1: Break 33 = 32+1 Step 2: 32÷16 = 2 Step 3: 1÷16 → since 6.25% = 1/16 → 0.0625 Step 4: Add: 2+0.0625 = 2.0625

Example: 7862 ÷ 7

Step 1: Break 7862 = 7861+1 Step 2: 7861÷7 = 1123 Step 3: 1÷7 → since 14.28% = 1/7 → 0.1428 Step 4: Add: 1123+0.1428 = 1123.1428

⚡ QUICK RECALL

This approach is fast only when the fraction↔percentage table is memorized (1/7=14.28%, 1/16=6.25%, 1/8=12.5%, 1/3=33.33%, etc.). It is directly connected to the Percentage chapter.

Type 10 — Special Ratio Pattern (Remember This!)

When the numerator is constructed as exactly 5× the denominator (often with trailing zeros), the answer is always 5

Division	Answer
130 ÷ 26	5
120 ÷ 24	5
160 ÷ 32	5
270 ÷ 54	5
640 ÷ 128	5

⚡ QUICK RECALL

In an exam, when the numerator appears to be exactly five times the denominator, including trailing-zero patterns, write 5 as the answer without calculation to save time.

📋 Master Table — All Division Approaches

#	Approach	Core Rule
1	÷ 5	×2, shift decimal 1 place left
2	÷ 0.5	×2
3	÷ 0.25	×4
4	÷ 0.125	×8
5	÷ 125	×8, shift decimal 3 places left
6	÷ 25	×4, shift decimal 2 places left
7	÷ 11/22/33/44/55...	First+last digit combo ÷ (divisor's last digit)
8	÷ 9/99/999...	Repeating decimal, pad zeros to match 9-count
9	Broken Heart	Split numerator: clean-divide part + known-% part
10	Half-Pattern	Numerator = 5×denominator (with zeros) → answer is 5

🔑 Approach Selection Flowchart

Divisor is 5 → Type 1
Divisor is 0.5/0.25/0.125 → Type 2/3/4
Divisor is 25/125 → Type 6/5
Divisor is 11/22/33/44/55 AND number has the "sandwich digit" pattern → Type 7
Divisor is 9/99/999 → Type 8 (repeating decimal)
Awkward divisor, but numerator can be split into clean-part + 1 → Type 9 (Broken Heart)
Numerator looks like a clean multiple with zeros → check Type 10 first