Mathematical Operations: Logical Reasoning Notes for NSO (Science Olympiad)

Logical Reasoning – Mathematical Operations

NSO (Science Olympiad) – Practice & Revision Guide

Goal: Master the core concepts of mathematical operations and the rules that govern them so you can solve any operation‑based question quickly and accurately.

1. Simple Explanation

Mathematical operations are the “building blocks” of arithmetic.
They tell us how to combine numbers to get a result.
The four basic operations are:

Symbol	Operation	Example
`+`	Addition	`3 + 5 = 8`
`–`	Subtraction	`9 – 4 = 5`
`×` or `⋅`	Multiplication	`6 × 7 = 42`
`÷` or `/`	Division	`20 ÷ 4 = 5`

Beyond these, we encounter:

Exponentiation (a^b) – repeated multiplication
2^4 = 2×2×2×2 = 16
Roots (√a or a^(1/2)) – the inverse of exponentiation
√16 = 4
Factorials (n!) – product of all positive integers ≤ n
5! = 5×4×3×2×1 = 120
Combinations (C(n, k) = n!/(k!(n−k)!)) – ways to choose k items from n
Permutations (P(n, k) = n!/(n−k)!) – ways to arrange k items from n

These operations obey a set of rules that let us simplify expressions, compare values, and solve equations.

2. Key Points & Rules

Category	Rule	Why It Matters
Order of Operations	PEMDAS (Parentheses, Exponents, Multiplication/Division, Addition/Subtraction) or BODMAS (Brackets, Orders, Division/Multiplication, Addition/Subtraction)	Ensures you evaluate expressions the same way every time.
Associative Property	`(a + b) + c = a + (b + c)` and `(a × b) × c = a × (b × c)`	Allows regrouping when adding or multiplying. Useful for mental math.
Commutative Property	`a + b = b + a` and `a × b = b × a`	Lets you reorder terms to simplify or group like terms.
Distributive Property	`a(b + c) = ab + ac`	Breaks down complex products; key for factoring and expanding.
Inverse Operations	`a + b = c` ⇔ `c – b = a`; `a × b = c` ⇔ `c ÷ b = a`	Helps solve for unknowns or check work.
Division by Zero	Undefined	Avoids mistakes when simplifying fractions.
Fraction Simplification	Divide numerator & denominator by their greatest common divisor (GCD)	Keeps numbers small and easier to compare.
Decimal & Percent Conversion	`1% = 0.01`	Essential for estimates and scaling.
Factorial Growth	`n!` grows faster than exponential functions	Useful for bounding combinatorial quantities.
Binomial Theorem	`(a + b)^n = Σ C(n, k) a^(n−k) b^k`	Quickly expands powers of binomials.
Logarithm Rules	`log_b(xy) = log_b x + log_b y`; `log_b(x/y) = log_b x – log_b y`; `log_b(x^k) = k log_b x`	Needed for solving exponential equations.

3. Worked Example

Problem:
Evaluate the expression
[ \frac{(3^4 + 2^3) \times 5 - 12}{(6! / 4!)} + \sqrt{81} - 7 \times (2^3 - 1) ]

Step‑by‑Step Solution

Compute inner parentheses
- 3^4 = 81
- 2^3 = 8
- 3^4 + 2^3 = 81 + 8 = 89
Multiply by 5
- 89 × 5 = 445
Subtract 12
- 445 – 12 = 433
Evaluate factorial ratio
- 6! = 720
- 4! = 24
- 6! / 4! = 720 / 24 = 30
Divide the two results
- 433 ÷ 30 ≈ 14.4333…
- Keep as fraction 433/30 if exactness needed.
Compute the remaining terms
- √81 = 9
- 2^3 - 1 = 8 - 1 = 7
- 7 × 7 = 49
Combine all pieces
[ \frac{433}{30} + 9 - 49 = \frac{433}{30} + \frac{9 \times 30}{30} - \frac{49 \times 30}{30} ] [ = \frac{433 + 270 - 1470}{30} = \frac{433 + 270 - 1470}{30} ] [ = \frac{703 - 1470}{30} = \frac{-767}{30} ] [ = -25.566\overline{6} ]

Answer: -25.566… (exactly −767⁄30)

Quick Checks

Did you follow PEMDAS? Yes (Parentheses → Exponents → Multiplication/Division → Addition/Sub