Rules for Signs in Algebra & BODMAS Calculation Principles

1. Fundamental Rules for Signs in Algebra

In commercial mathematics and financial modeling, incorrect sign handling leads to balance sheet discrepancies and erroneous profit/loss projections:

Matrix of Rules for Signs across Arithmetic Operations

Addition & Subtraction of Signed Numbers

Same Signs: Add absolute values and keep the common sign.
(+5) + (+7) = +12 | (-5) + (-7) = -12
Different Signs: Subtract the smaller absolute value from the larger, and keep the sign of the larger.
(+8) + (-3) = +5 | (-8) + (+3) = -5
Double Negatives: Subtracting a negative turns into addition.
a - (-b) = a + b (e.g., 10 - (-4) = 10 + 4 = 14)

Multiplication & Division of Signed Numbers

Like Signs Produce Positive:
(+) × (+) = (+) | (-) × (-) = (+)
(+12) ÷ (+3) = +4 | (-12) ÷ (-3) = +4
Unlike Signs Produce Negative:
(+) × (-) = (-) | (-) × (+) = (-)
(+12) ÷ (-3) = -4 | (-12) ÷ (+3) = -4
Even vs Odd Negatives: An even number of negative factors yields positive; an odd number yields negative.

2. The BODMAS Calculation Hierarchy

When a mathematical expression contains multiple operations, evaluating them arbitrarily from left to right yields incorrect results. The BODMAS rule establishes an authoritative sequence:

The BODMAS Execution Hierarchy

B — Step 1 Brackets (Parentheses):

Evaluate innermost brackets first in strict nested order: Bar / Line (Vinculum) → Round brackets ( ) → Curly braces { } → Square brackets [ ].

O — Step 2 Orders (Of, Powers, Roots):

Evaluate exponents, powers, square roots, and the mathematical operator "of" (multiplication with higher priority than division).

D & M — Step 3 Division & Multiplication:

Carry equal rank; evaluated from left to right as they appear in the expression.

A & S — Step 4 Addition & Subtraction:

Final priority tier; evaluated from left to right among terms of like algebraic power.

3. Step-by-Step Practice Problems

Problem 1: Complex Nested Bracket Simplification

Question: Evaluate the following expression:

45 - [ 38 - { 60 ÷ 3 - (6 - 9 ÷ 3) } ]

Step-by-Step Solution:

Innermost Round Brackets (6 - 9 ÷ 3):
- Division first: 9 ÷ 3 = 3
- Then subtract: 6 - 3 = 3
Curly Braces { 60 ÷ 3 - 3 }:
- Division first: 60 ÷ 3 = 20
- Then subtract: 20 - 3 = 17
Square Brackets [ 38 - 17 ]:
- 38 - 17 = 21
Final Outer Subtraction:
- 45 - 21 = 24
- Final Answer = 24

Problem 2: Mixed Commercial Accounting Formula

Question: Calculate net profit from the revenue expression:

P = 15,000 + 4 × 2,500 - 18,000 ÷ 3 + 2 × (1,200 - 400)