Official Syllabus Item: 7, 45 Arithmetic & Number Theory

Simplification & VBODMAS Rule with Brackets

Master VBODMAS rule, removal of brackets (vinculum, round, curly, box), fraction simplification, and changing of units for RIMC Stage-I descriptive exam.

Overview & Fundamental Concept

VBODMAS strictly governs the hierarchy of operations in arithmetic and algebraic expressions: Vinculum, Brackets (Parentheses, Braces, Box brackets), Of, Division, Multiplication, Addition, and Subtraction. In RIMC descriptive papers, examiners test inside-out bracket elimination, rigorous sign distribution rules, fractional reciprocals, and unit harmonization (Syllabus Item 45).

Key Theoretical Principles

Brackets are removed inside-out: Vinculum first, then (), then {}, then [].
A minus sign preceding a bracket reverses every sign inside upon removal.
The 'Of' operator has higher priority than division and regular multiplication.
All mixed fractions must be converted to improper fractions before applying VBODMAS.
Dissimilar physical units (kg/g, rupees/paise, km/m) must be harmonized before calculating.

Illustrated Conceptual Diagrams & Visual Schematics

The VBODMAS Hierarchy of Operations Pyramid

Strict VBODMAS Order of Precedence: Always remove brackets from the inside out: Vinculum / Bar ($overline{a - b}$) $ o$ Round $( ) o$ Curly ${ } o$ Square $[ ]$. 'Of' indicates multiplication but executes before standalone division. Division & multiplication have equal tier and resolve from left to right.

Topper Strategy & Descriptive Step Breakdown

Mastering RIMC Mathematics Standard

Never perform mental leaps. Write each level of bracket removal on a separate line. When removing brackets preceded by a negative sign, highlight the inverted signs explicitly to secure step marks.

Step 1: Given Information & Variable Declaration (1-2 Marks)

Clearly write down the data provided in the question. State any assumption (e.g., 'Let the cost price be ₹ x').

Step 2: Formula & Conceptual Foundation (1-2 Marks)

State the exact mathematical theorem or formula in standard algebraic notation before entering numeric values.

Step 3: Systematic Calculation & Reduction (2-4 Marks)

Carry out each step on a fresh line. Avoid combining three operations in one step so examiners can award partial credit.

Step 4: Final Statement with Proper Units (1 Mark)

End with a concluding statement highlighting the answer in a neat box or underline with explicit units (e.g., '₹ 1,500', '45 km/h').

Key Formulas & Conversion Cheat Sheet

The VBODMAS Precedence

$$\text{Vinculum} \to \text{Brackets } [ \{ ( \overline{\quad} ) \} ] \to \text{Of} \to \div \to \times \to + \to -$$

Sign Distribution Rule

$$-(a - b + c - d) = -a + b - c + d$$

Fraction Division Reciprocal

$$\frac{a}{b} \div \frac{c}{d} = \frac{a}{b} \times \frac{d}{c}$$

Mixed Fraction Conversion

$$a\frac{b}{c} = \frac{ac + b}{c}$$

Solved Model Examples (Descriptive Format with Step Mark Distribution)

Solved Model 1 Model Solution

Simplify: 18 - [6 - {4 - (8 - 6 + 3̄)}].

Step-by-Step Model Solution:

Step 1: Vinculum 6 + 3 = 9 => 18 - [6 - {4 - (8 - 9)}].\nStep 2: Round brackets 8 - 9 = -1 => 4 - (-1) = 5 => 18 - [6 - 5].\nStep 3: Square brackets 6 - 5 = 1 => 18 - 1 = 17.\nFinal Answer: 17.

Solved Model 2 Model Solution

Simplify: 4 1/2 - [2 1/3 ÷ {1 1/4 - 1/2(2 1/2 - 1/4 - 1/6̄)}].

Step-by-Step Model Solution:

Step 1: Mixed to improper: 9/2 - [7/3 ÷ {5/4 - 1/2(5/2 - 1/12)}].\nStep 2: Vinculum 1/4 - 1/6 = 1/12. Inside (): 5/2 - 1/12 = 29/12.\nStep 3: Multiply by 1/2: 29/24. Inside {}: 5/4 - 29/24 = 1/24.\nStep 4: Division inside []: 7/3 ÷ 1/24 = 7/3 × 24 = 56.\nStep 5: 9/2 - 56 = -103/2 = -51 1/2.\nFinal Answer: -51 1/2.

Solved Model 3 Model Solution

Simplify and express in kilograms: 4.5 kg - [2500 g + {1 1/4 kg - (850 g - 350 g)}].

Step-by-Step Model Solution:

Step 1: Convert all to kg: 4.5 - [2.5 + {1.25 - (0.85 - 0.35)}].\nStep 2: Inside (): 0.85 - 0.35 = 0.50 kg.\nStep 3: Inside {}: 1.25 - 0.50 = 0.75 kg.\nStep 4: Inside []: 2.5 + 0.75 = 3.25 kg.\nStep 5: Final subtraction: 4.5 - 3.25 = 1.25 kg.\nFinal Answer: 1.25 kg (or 1 1/4 kg).

RIMC Practice & Drill Exercises

Attempt these questions on paper following proper step-wise conventions, then click Reveal Solution to verify your answers.

Question 1 5 Marks

Simplify using bracket removal rules: 25 - [20 - {10 - (7 - 5 - 3̄)}].