Viral Maths — Ch12: Approximation

📌 What This Chapter Covers

Approximation means converting a decimal number to a whole number that is close to, but not exactly equal to, the original value by rounding it off.
There are four rounding cases. Decide the required action according to the value of the digit.
There is also a separate technique for root approximation: locate a nearby perfect square or cube and estimate the root.

⚡ QUICK RECALL

This chapter connects to Chapter 7 (Square Roots) and Chapter 8 (Cube Roots). Root approximation uses the same Guess Approach, now applied to non-perfect numbers.

🗂️ 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

In Case 3, the number is rounded neither UP nor DOWN; it remains exactly as it was (54.5 remains 54.5, not 54 or 55). A common misconception is that an exact 5 is always rounded up, but that is not the rule used in this book.

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

This technique works only when the number is VERY close to a perfect square or cube. Exams use such questions so that the approximation is clean.

⚠ EXAM TRAP

The squares from 1 to 100 and cubes from 1 to 25, from Chapters 1 and 8, should be memorized fluently; otherwise finding the nearby perfect square or cube will be slow.

📋 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