3.1 Mental Arithmetic, Order of Operations, Integers & Number Properties

3.1 Mental Arithmetic, Order of Operations & Integers

Numerical proficiency forms the bedrock of quantitative aptitude testing for military recruit selection. In demanding operational settings, service personnel must rapidly calculate supply distributions, verify convoy fuel needs, tally equipment manifests, and adjust coordinates without relying on electronic devices. The Ghana Armed Forces recruit aptitude assessment evaluates these core proficiencies under strictly supervised conditions, and because GAF publishes no calculator rules, you should prepare to work entirely by hand. Success demands absolute mastery of arithmetic operational hierarchy, flawless precision with positive and negative integers, and rapid mental calculation strategies that bypass cumbersome manual scratch work.

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The Hierarchy of Operations: BODMAS and PEMDAS

When a mathematical expression contains multiple operations, resolving them in arbitrary order leads to erroneous answers. To ensure universal consistency, mathematics establishes a standard order of precedence, commonly taught using the acronyms BODMAS or PEMDAS.

The Left-to-Right Precedence Rule

A frequent mistake occurs when candidates treat Multiplication as taking absolute priority over Division, or Addition over Subtraction. In formal mathematics, Division and Multiplication form an equal priority tier; Addition and Subtraction form an equal priority tier. Whichever operation in the pair appears first when reading from left to right must be computed first.

Consider the expression:

$$24 \div 6 \times 2$$

Evaluating multiplication first yields $24 \div 12 = 2$, which is incorrect. Evaluating from left to right gives:

1. $24 \div 6 = 4$
2. $4 \times 2 = 8$

Similarly, in subtraction and addition:

$$15 - 8 + 4$$

Evaluating addition first gives $15 - 12 = 3$, an error. The correct left-to-right evaluation is $15 - 8 = 7$, followed by $7 + 4 = 11$.

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Directed Numbers: Principles of Integer Operations

Directed numbers carry positive ($+$) or negative ($-$) signs indicating direction or state relative to zero, representing inventory deficits, elevation changes, or opposing movements.

Addition and Subtraction of Integers

• Same Signs: Add absolute values and preserve the shared sign: $(+7) + (+5) = +12$; $(-8) + (-6) = -14$.
• Different Signs: Find the difference between absolute values and take the sign of the larger magnitude: $(-14) + (+9) = -5$; $(+18) + (-7) = +11$.
• Subtracting a Negative Number: Subtracting a negative integer is identical to adding a positive ($a - (-b) = a + b$): $12 - (-7) = 12 + 7 = 19$ $-15 - (-20) = -15 + 20 = 5$

Multiplication and Division of Integers

Multiplying or dividing integers with identical signs produces a positive result; opposing signs produce a negative result:

• Like Signs: $(+6) \times (+7) = +42$ and $(-6) \times (-7) = +42$; $(-36) \div (-4) = +9$.
• Unlike Signs: $(+6) \times (-7) = -42$ and $(-6) \times (+7) = -42$; $(-36) \div 4 = -9$.

Distributing Negative Multipliers

Distributing a negative factor across brackets reverses every enclosed sign:

$$-4(3a - 5b + 2) = -12a + 20b - 8$$

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Mental Calculation Shortcuts for Calculator-Free Testing

Working without a calculator is the safe assumption, so rapid mental decomposition techniques save crucial minutes during the test.

1. Distributive Splitting (Chunking)

Decompose complex factors into round components:

• Near-Decade Factors: $34 \times 19 = 34 \times (20 - 1) = 680 - 34 = 646$.
• Two-Digit Products: $16 \times 23 = (16 \times 20) + (16 \times 3) = 320 + 48 = 368$.

2. Doubling and Halving

When multiplying an even factor and a factor ending in $5$, double one and halve the other:

$$35 \times 18 = 70 \times 9 = 630$$

3. Compatible Number Grouping

Reorder terms in multi-number additions to form convenient multiples of $10$ or $100$:

$$37 + 84 + 63 + 16 = (37 + 63) + (84 + 16) = 100 + 100 = 200$$

4. Compensation in Subtraction

Subtract a convenient round number and adjust for the difference:

$$624 - 295 = 624 - 300 + 5 = 324 + 5 = 329$$

5. Estimation and Unit-Digit Elimination

• Unit-Digit Analysis: In $43 \times 27$, the unit digit is $3 \times 7 = 21$ (ends in $1$). If only one option ends in $1$, select it immediately.
• Order of Magnitude: In $4,982 \div 49 \approx 5,000 \div 50 = 100$, identifying options near $102$ eliminates distant distractors instantly.

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Number Properties: Factors, Multiples, Primes, HCF and LCM

Aptitude papers often test the building blocks of whole numbers, and these facts also speed up fraction work and ratio problems.

• A factor of a number divides it exactly: the factors of 18 are 1, 2, 3, 6, 9 and 18.
• A multiple is the number multiplied by a whole number: the multiples of 6 are 6, 12, 18, 24, and so on.
• A prime number has exactly two factors, 1 and itself: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, … The number 1 is not prime, and 2 is the only even prime.
• A composite number has more than two factors (4, 6, 8, 9, 10, …).

Quick Divisibility Tests

HCF and LCM by Prime Factors

Write each number as a product of primes:

$$72 = 2^3 \times 3^2, \qquad 60 = 2^2 \times 3 \times 5$$

• HCF (highest common factor): take the lowest power of each prime the numbers share: $2^2 \times 3 = 12$.
• LCM (lowest common multiple): take the highest power of every prime that appears: $2^3 \times 3^2 \times 5 = 360$.
• Check: $\text{HCF} \times \text{LCM} = 12 \times 360 = 4320 = 72 \times 60$.

Typical uses. The HCF answers "largest equal groups" questions: 72 rifles and 60 helmets can be shared into at most 12 identical kits with nothing left over. The LCM answers "when do events coincide" questions: if one patrol leaves every 12 minutes and another every 18 minutes, and both leave at 0600, they next leave together after $\text{LCM}(12, 18) = 36$ minutes, at 0636.

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Worked Examples with Step-by-Step Solutions

Example 1: Multi-Tier Order of Operations

Problem: Simplify: $30 - 2 \times [4 + 3 \times (6 - 8)^2] \div 8$

Step-by-step Solution:

1. Innermost brackets: $6 - 8 = -2$.
2. Exponent within brackets: $(-2)^2 = 4$.
3. Multiplication within brackets: $3 \times 4 = 12$.
4. Addition within brackets: $4 + 12 = 16$. The expression becomes: $30 - 2 \times 16 \div 8$.
5. Multiplication and division left to right: $2 \times 16 = 32$, then $32 \div 8 = 4$.
6. Subtraction: $30 - 4 = 26$. Final Answer: $26$.

Example 2: Operational Quartermaster Inventory Balance

Problem: An inventory log shows a starting deficit of $-18$ tool kits. The depot receives $7$ crates of $12$ kits each, issues $65$ kits to field units, and takes back $4$ damaged kits returned to the supplier. What is the net inventory balance?

Step-by-step Solution:

1. Assign signed values: Start: $-18$; Received: $+ (7 \times 12) = +84$; Issued: $-65$; Removed/Returned: $-4$.
2. Combine terms: $\text{Balance} = -18 + 84 - 65 - 4$.
3. Evaluate left to right: $-18 + 84 = +66$; $66 - 65 = +1$; $1 - 4 = -3$. Final Answer: A net balance of $-3$ kits (deficit of $3$).