8.1 Mental Arithmetic, Order of Operations & Number Properties

Mental Arithmetic, Order of Operations & Number Properties

Core Concept: In the Edutest Mathematics subtest, speed and computational accuracy are inextricably linked. Edutest's Mathematics test lasts 30 minutes, and Edutest states that actual tests contain between 40 and 60 questions, and calculators are not permitted, so candidates average about 30 to 45 seconds per question. Navigating this pacing ceiling requires replacing brute-force pencil calculations with systematic order-of-operations mechanics, instant divisibility diagnostics, prime factorisation shortcuts, and algebraic mental calculation tricks.

While the Numerical Reasoning test evaluates pattern recognition and reasoning, Edutest describes the Mathematics test as measuring year-level mathematical knowledge across numbers, measurement, algebra, space and data. Calculators are not permitted in any Edutest test. A candidate who relies on standard long division and multi-line column multiplication will quickly encounter time exhaustion. Strong results require rapid mental triage and proven calculation shortcuts.

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Exam Architecture & The Strict No-Calculator Constraint

Edutest's Mathematics test lasts 30 minutes, and Edutest states that actual tests contain between 40 and 60 questions. This yields an average of only 30 to 45 seconds per question.

[30-Minute Total Time Window]
  ├── Total Questions: 40 to 60 items (Edutest's stated range)
  ├── Average Time per Item: 30–45 seconds
  ├── Target Time on Direct Arithmetic: 15–20 seconds (to bank time for multi-step word problems)
  ├── Calculator Policy: NOT PERMITTED (no separate working out paper either)
  └── Triage Rule: If stuck on a computation past 30 seconds, estimate, eliminate extremes, and guess.

Because there is no negative marking, unanswered questions represent forfeited marks. The primary pacing objective on arithmetic questions is to execute calculations in 15 to 20 seconds, banking surplus time for extended geometry, probability, and multi-step algebraic word problems.

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The Order of Operations Hierarchy (BODMAS / PEMDAS)

Under timed pressure, the most common source of arithmetic error is misapplying operational precedence. The universal order of operations follows the BODMAS (or PEMDAS) convention:

1. B / P (Brackets / Parentheses): Evaluate expressions inside grouping symbols first, working from the innermost grouping outward: parentheses $( )$, brackets $[ ]$, and braces $\{ \}$.
2. O / E (Orders / Exponents): Powers, roots, and indices (e.g., $3^3$, $\sqrt{49}$).
3. DM (Division & Multiplication): These two operations share strictly equal precedence. They must be resolved in order from left to right as they appear in the expression.
4. AS (Addition & Subtraction): These two operations also share strictly equal precedence and must be resolved from left to right.

  Precedence Level 1:   ( )  [ ]  { }        Brackets (Innermost first)
  Precedence Level 2:   x²   √x   xⁿ         Orders / Indices
  Precedence Level 3:   ÷    ×               Division & Multiplication (Strictly Left-to-Right)
  Precedence Level 4:   +    -               Addition & Subtraction (Strictly Left-to-Right)

The Vinculum as an Implicit Grouping Symbol

A frequent trap in scholarship examinations is the vinculum—the horizontal fraction bar. The vinculum acts as an implicit set of brackets enclosing both the entire numerator and the entire denominator:

$\frac{a + b}{c - d} \equiv (a + b) \div (c - d)$

Never divide an isolated term in the numerator by the denominator before fully evaluating the sum or difference within each tier.

Worked Example: Multi-Tier Expression with Nested Brackets and Vinculum

$\text{Evaluate: } \frac{48 - 4 \times (3^3 - 19)}{2^4 - 3 \times 2^2}$

• Step 1 (Evaluate innermost brackets in numerator): $3^3 - 19 = 27 - 19 = 8$
• Step 2 (Evaluate numerator multiplication and subtraction): $48 - 4 \times 8 = 48 - 32 = 16$
• Step 3 (Evaluate denominator powers and operations): $2^4 = 16, \quad 2^2 = 4$ $2^4 - 3 \times 2^2 = 16 - 3 \times 4 = 16 - 12 = 4$
• Step 4 (Resolve the vinculum division): $\frac{16}{4} = 4$

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Comprehensive Divisibility Rules (2 to 11)

Divisibility rules allow you to determine whether a large integer is divisible by a given divisor without performing long division. They are invaluable for simplifying fractions, finding common factors, and testing prime candidates.

The Rule for 11 in Missing-Digit Problems

Selective entry questions frequently feature an unknown digit within a number: "The number $52x,394$ is divisible by 11. Find the value of digit $x$."

Using the alternating sum rule:

• Sum of odd positions (from right: 1st, 3rd, 5th): $4 + 3 + 2 = 9$
• Sum of even positions (from right: 2nd, 4th, 6th): $9 + x + 5 = 14 + x$
• Difference: $(14 + x) - 9 = 5 + x$
• For $5 + x$ to be a multiple of $11$, where $x \in \{0, 1, 2, \dots, 9\}$: $5 + x = 11 \implies x = 6$

Composite Divisibility Testing

To test divisibility by composite numbers not listed above, decompose the composite number into coprime factors (factors that share no common divisor other than 1):

• Divisible by 12 $\iff$ Divisible by 3 AND 4 (Do NOT test $2$ and $6$, as they are not coprime).
• Divisible by 15 $\iff$ Divisible by 3 AND 5.
• Divisible by 18 $\iff$ Divisible by 2 AND 9.
• Divisible by 24 $\iff$ Divisible by 3 AND 8.

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Prime Numbers, Factorisation & Index Notation

A prime number is an integer strictly greater than 1 that has exactly two distinct positive divisors: 1 and itself.

• The integer 1 is neither prime nor composite (it has only one divisor).
• 2 is the only even prime number, and the smallest prime.

The 25 Primes Under 100

Candidates should memorize the 25 prime numbers below 100 to avoid wasting time testing them:

$\{2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97\}$

The Prime Testing Theorem

To verify whether an integer $N$ is prime, you only need to test for divisibility by prime numbers up to $\lfloor\sqrt{N}\rfloor$. If no prime $p \le \sqrt{N}$ divides $N$, then $N$ is unconditionally prime.

Example: Is $173$ prime?

• $\sqrt{173} \approx 13.15$ (since $13^2 = 169$ and $14^2 = 196$).
• Test only primes $\le 13$: $\{2, 3, 5, 7, 11, 13\}$.
• $173$ is odd (not div by 2); digit sum $1+7+3 = 11$ (not div by 3); doesn't end in 0 or 5 (not div by 5); $173 \div 7 = 24.7$ (not div by 7); alternating sum $(3+1)-7 = -3$ (not div by 11); $173 = 13 \times 13 + 4$ (not div by 13).
• Therefore, $173$ is prime.

Canonical Prime Factorisation & Factor Counting

Every integer $N > 1$ can be uniquely decomposed into prime factors in index notation:

$N = p_1^{a_1} \cdot p_2^{a_2} \cdots p_k^{a_k}$

Shortcut for Total Number of Factors: The total number of positive factors of $N$ is given by adding 1 to each exponent and multiplying the results:

$\text{Total Factors} = (a_1 + 1)(a_2 + 1) \cdots (a_k + 1)$

Example: How many positive divisors does $360$ have?

• Prime factorise: $360 = 36 \times 10 = (2^2 \times 3^2) \times (2 \times 5) = 2^3 \times 3^2 \times 5^1$
• Total divisors: $(3 + 1)(2 + 1)(1 + 1) = 4 \times 3 \times 2 = 24$ divisors.

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HCF and LCM: High-Speed Algorithms

• Highest Common Factor (HCF / GCD): The greatest integer that divides each of two or more given integers without a remainder.
• Lowest Common Multiple (LCM): The smallest positive integer that is a multiple of each of two or more given integers.

The Prime Factorisation Method

1. Express all numbers in prime factor index notation.
2. HCF: Take the minimum exponent for each common prime factor.
3. LCM: Take the maximum exponent for every prime factor present in any number.

Worked Example: HCF and LCM of 72 and 120

• $72 = 2^3 \times 3^2$
• $120 = 2^3 \times 3^1 \times 5^1$

$\text{HCF} = 2^{\min(3,3)} \times 3^{\min(2,1)} = 2^3 \times 3^1 = 8 \times 3 = 24$ $\text{LCM} = 2^{\max(3,3)} \times 3^{\max(2,1)} \times 5^{\max(0,1)} = 2^3 \times 3^2 \times 5^1 = 8 \times 9 \times 5 = 360$

The Fundamental Product Identity

For any two positive integers $a$ and $b$:

$\text{HCF}(a, b) \times \text{LCM}(a, b) = a \times b$

If a question states that two numbers have an HCF of 12 and an LCM of 180, and one number is 36, the other number is immediately:

$b = \frac{\text{HCF} \times \text{LCM}}{a} = \frac{12 \times 180}{36} = \frac{180}{3} = 60$

The Euclidean Algorithm Shortcut for Large Numbers

When finding the HCF of two large numbers where prime factorisation is too cumbersome, apply Euclidean division: $\gcd(a, b) = \gcd(b, a \bmod b)$.

Find the HCF of 357 and 221:

1. $357 \div 221 = 1$ with remainder $136$ ($357 - 221 = 136$)
2. $221 \div 136 = 1$ with remainder $85$ ($221 - 136 = 85$)
3. $136 \div 85 = 1$ with remainder $51$ ($136 - 85 = 51$)
4. $85 \div 51 = 1$ with remainder $34$ ($85 - 51 = 34$)
5. $51 \div 34 = 1$ with remainder $17$ ($51 - 34 = 17$)
6. $34 \div 17 = 2$ with remainder $0$

The last non-zero remainder is 17. Hence, $\text{HCF}(357, 221) = 17$.

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Mental Calculation Shortcuts for Timed Tests

1. Doubling and Halving

When multiplying two numbers where one is even and the other ends in 5 (or is awkward), halve the even number and double the other number. The product remains unchanged:

$a \times b = \left(\frac{a}{2}\right) \times (2b)$

• $16 \times 35 = 8 \times 70 = 560$
• $24 \times 25 = 12 \times 50 = 6 \times 100 = 600$
• $18 \times 4.5 = 9 \times 9 = 81$

2. Difference of Two Squares ($a^2 - b^2$)

The algebraic identity $a^2 - b^2 = (a-b)(a+b)$ provides two high-speed computation tricks:

• Rapid Squaring Difference: $68^2 - 32^2 = (68 - 32)(68 + 32) = 36 \times 100 = 3,600$
• Multiplying Symmetric Numbers: To multiply two numbers that are equidistant from a round number $(M - d)(M + d) = M^2 - d^2$: $47 \times 53 = (50 - 3)(50 + 3) = 50^2 - 3^2 = 2500 - 9 = 2,491$ $76 \times 84 = (80 - 4)(80 + 4) = 80^2 - 4^2 = 6400 - 16 = 6,384$

3. Squaring Numbers Ending in 5

Any two-digit number ending in 5, expressed as $10n + 5$, has a square equal to $n(n+1)$ followed by $25$:

$(10n + 5)^2 = 100n(n+1) + 25$

• $35^2: 3 \times 4 = 12 \implies 1,225$
• $65^2: 6 \times 7 = 42 \implies 4,225$
• $85^2: 8 \times 9 = 72 \implies 7,225$
• $115^2: 11 \times 12 = 132 \implies 13,225$

4. Fast Multiplication Shortcuts (9, 11, 25)

• Multiplying by 9: Multiply by 10 and subtract the original number: $9x = 10x - x$. $43 \times 9 = 430 - 43 = 387$
• Multiplying by 11: Write the outer digits and insert the sum of the digits between them (carrying 1 if sum $\ge 10$): $53 \times 11 = 5\,[5+3]\,3 = 583$ $78 \times 11 = 7\,[7+8]\,8 = 7\,[15]\,8 = (7+1)\,5\,8 = 858$
• Multiplying by 25: Multiply by 100 and divide by 4: $25x = \frac{100x}{4}$. $64 \times 25 = \frac{6400}{4} = 1,600$ $78 \times 25 = \frac{7800}{4} = 1,950$

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High-Yield Traps in Arithmetic & Number Properties

1. The Left-to-Right Precedence Trap

A student reads $12 - 4 \times 2 + 5$ and computes from left to right: $12 - 4 = 8$, $8 \times 2 = 16$, $16 + 5 = 21$. This is completely incorrect. Multiplication must be evaluated first: $4 \times 2 = 8$, then evaluate additions and subtractions left to right: $12 - 8 + 5 = 4 + 5 = 9$.

2. Prime Classification Mistakes

Candidates often assume all odd numbers are prime or mistake common composites for primes. High-frequency non-primes that look prime on exam papers include:

• $51 = 3 \times 17$
• $57 = 3 \times 19$
• $87 = 3 \times 29$
• $91 = 7 \times 13$
• $119 = 7 \times 17$
• $133 = 7 \times 19$ Always check divisibility by 3 (digit sum) and 7 before assuming a two-digit or small three-digit number is prime.

3. Confusing HCF and LCM in Real-World Word Problems

In word problems:

• If you are asked to split items into equal groups, divide tiles into maximum-sized squares, or cut rods into equal pieces without waste, find the HCF.
• If you are asked when two traffic lights will blink together again, when two runners on a circular track will meet at the start line, or to find the minimum number of items to pack into identical boxes, find the LCM.