2.1 Basic Arithmetic, Order of Operations & Number Properties

Basic arithmetic forms the cornerstone of quantitative reasoning in the Nigeria Police Exam. A solid grasp of fundamental mathematical concepts is non-negotiable for success. In this section, we delve deeply into basic operations, the order of operations, and fundamental number properties, providing you with the tools needed to tackle arithmetic problems swiftly and accurately.

The Four Basic Operations

The four fundamental arithmetic operations are addition, subtraction, multiplication, and division. While these may seem elementary, the exam often tests them in the context of large numbers or complex word problems where mistakes are easily made under pressure. Let us review the principles of each operation.

Addition is the process of calculating the total of two or more numbers. When adding large numbers, it is crucial to align the digits according to their place value (units, tens, hundreds, thousands). Subtraction is the inverse of addition, representing the process of finding the difference between numbers. Borrowing or regrouping is often required when subtracting a larger digit from a smaller one in a specific place value column.

Multiplication can be thought of as repeated addition. For instance, multiplying a number by five is equivalent to adding that number to itself five times. Division is the process of splitting a number into equal parts or groups. It is the inverse of multiplication. Understanding concepts like dividends, divisors, quotients, and remainders is essential. The dividend is the number being divided, the divisor is the number you divide by, the quotient is the result, and the remainder is what is left over if the division is not exact.

Order of Operations (BODMAS)

One of the most frequently tested concepts in quantitative reasoning is the order of operations. When an expression contains multiple operations, there is a specific sequence in which they must be performed to arrive at the correct answer. The acronym BODMAS is universally used to remember this sequence:

• Brackets: Perform operations inside brackets first. If there are nested brackets, start with the innermost ones.
• Orders (or Of): This refers to powers, roots, or operations denoted by 'of' (which usually implies multiplication in fractions).
• Division and Multiplication: Perform these operations from left to right as they appear in the expression. They hold equal precedence.
• Addition and Subtraction: Perform these operations from left to right. They also hold equal precedence.

Consider the expression: $15 + 3 \times (8 - 2) \div 2$. Following BODMAS, we first solve the operation inside the bracket: $8 - 2 = 6$. The expression becomes $15 + 3 \times 6 \div 2$. Next, we perform multiplication and division from left to right. First, $3 \times 6 = 18$. The expression is now $15 + 18 \div 2$. Then, we perform the division: $18 \div 2 = 9$. Finally, we perform addition: $15 + 9 = 24$. Ignoring BODMAS would lead to entirely different and incorrect results, a trap commonly set by examiners.

Number Properties

Understanding number properties is like having a mathematical toolkit that allows you to simplify and rearrange expressions efficiently. Let us explore the most important properties.

The Commutative Property states that the order of numbers does not change the result in addition and multiplication. For addition: $a + b = b + a$. For multiplication: $a \times b = b \times a$. Note that this property does not apply to subtraction or division (e.g., $5 - 3 \neq 3 - 5$).

The Associative Property states that the grouping of numbers does not change the result in addition and multiplication. For addition: $(a + b) + c = a + (b + c)$. For multiplication: $(a \times b) \times c = a \times (b \times c)$. This is particularly useful when adding or multiplying a long sequence of numbers, allowing you to group them in a way that makes mental calculation easier.

The Distributive Property relates multiplication to addition (or subtraction). It states that multiplying a sum by a number is the same as multiplying each addend by the number and then adding the products. The formula is: $a \times (b + c) = (a \times b) + (a \times c)$. For example, $5 \times (10 + 2) = (5 \times 10) + (5 \times 2) = 50 + 10 = 60$. This property is immensely helpful for mental math and algebraic simplification.

Prime Numbers, Factors, and Multiples

A deeper understanding of arithmetic requires knowledge of prime numbers, factors, and multiples. A prime number is a whole number greater than 1 whose only divisors are 1 and itself. Examples include 2, 3, 5, 7, 11, 13, 17, 19, and so on. Note that 2 is the only even prime number. Numbers with more than two divisors are called composite numbers.

A factor is a number that divides another number exactly, leaving no remainder. For instance, the factors of 12 are 1, 2, 3, 4, 6, and 12. Identifying the Greatest Common Factor (GCF) or Highest Common Factor (HCF) is crucial for simplifying fractions.

A multiple is the result of multiplying a number by an integer. The multiples of 5 are 5, 10, 15, 20, 25, etc. Finding the Least Common Multiple (LCM) is essential for adding and subtracting fractions with different denominators. Understanding how to find prime factorizations of numbers helps significantly in determining both GCF and LCM efficiently.

In conclusion, mastering these basic arithmetic concepts and rules is foundational. Practice applying BODMAS strictly, use number properties to simplify calculations, and become comfortable identifying factors and multiples. This strong foundation will support your success throughout the quantitative reasoning section of the Nigeria Police Exam.