Viral Maths — Ch12: Approximation

📌 What This Chapter Covers

Approximation = converting a decimal number to a complete (whole) number, but not exactly equal — round off karna.
4 cases hain rounding ke — digit ki value ke hisaab se decide karo kya karna hai.
Ek separate technique bhi hai: root approximation — nearby perfect square/cube dhundh ke root nikalna.

⚡ QUICK RECALL

Yeh chapter Ch07 (Square Roots) aur Ch08 (Cube Roots) se connected hai — root approximation wahi guess-approach hai, bas ab non-perfect numbers ke liye use ho raha hai.

🗂️ Chapter Index

Tab	Content
2	4 Cases of Decimal Rounding
3	Root Approximation (using nearby perfect square/cube)
4	Master Table

Case 1 — Digit After Decimal is MORE than 5

Example: 54.8

Digit right of decimal (8) > 5 Add 1 to the digit left of decimal: 54 → 55 Answer: 55

Case 2 — Digit After Decimal is LESS than 5

Example: 54.3

Digit right of decimal (3) < 5 No change to the left-side digit Answer: 54

Case 3 — Digit After Decimal is EXACTLY 5

Example: 54.5

Digit right of decimal is exactly 5 No change to the original number Answer: 54.5 (stays as-is)

⚠ EXAM TRAP

Case 3 mein number ko round UP ya DOWN nahi karte — number bilkul same rehta hai jaisa tha (54.5 hi rehta hai, na 54 na 55). Yeh common misconception hai ki exactly-5 hamesha round up hota hai — is book ke rule mein aisa nahi hai.

Case 4 — Digit is 5, AND Next Digit is MORE than 5

Example: 34.58

Step 1: Digit right of decimal is 5, next digit (8) > 5 Step 2: Since next digit > 5, treat the 5 as incremented to 6 Step 3: Now apply Case 1 logic (6 > 5): add 1 to left digit 34 → 35 Answer: 35

⚡ QUICK RECALL — All 4 Cases At a Glance

Digit >5 → round UP | Digit <5 → round DOWN | Digit =5 exactly (nothing after) → NO CHANGE | Digit=5 but next digit>5 → treat as 6, then round UP

Root Approximation — Nearest Perfect Square/Cube

Find the nearby value which IS a perfect square or perfect cube, then compute its root

Example: √37

Nearby perfect square: 36 (very close to 37) √37 ≈ √36 = 6

Example: √126

Nearby perfect square: 125 is a perfect CUBE (5³), not square Actually nearby perfect square: 121 (11²) √126 ≈ √121 = 11

⚡ QUICK RECALL

Yeh technique tabhi kaam karti hai jab number kisi perfect square/cube ke BAHUT paas ho — exam mein aise hi questions banaye jaate hain taaki approximation clean nikle.

⚠ EXAM TRAP

Squares 1–100 aur cubes 1–25 (Ch01 aur Ch08 se) fluent yaad hone chahiye — warna "nearby perfect square/cube" dhundhna slow ho jaayega.

📋 Master Table — Approximation Rules

Case	Condition	Action
Case 1	Digit right of decimal > 5	Round UP (add 1 to left digit)
Case 2	Digit right of decimal < 5	Round DOWN (no change)
Case 3	Digit right of decimal = 5 exactly	NO CHANGE — number stays as-is
Case 4	Digit = 5, next digit > 5	Treat 5 as 6, then round UP
Root Approximation	Any non-perfect square/cube	Use nearest perfect square/cube's root