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Chapter Review

Arithmetic Foundations

Basic Arithmetic Operations · Number Properties and Divisibility · Ratio and Proportion · Averages, Mean, and Medians

BODMAS / Order of Operations

BODMAS (Brackets, Orders, Division/Multiplication, Addition/Subtraction) defines the priority sequence for evaluating any arithmetic expression. Operations at the same priority level are resolved left to right.

Key Points

  • •
    Brackets first, then powers/roots, then × and ÷ (left to right), then + and − (left to right)
  • •
    Multiplication and division have equal priority — evaluate whichever comes first from left to right
  • •
    Nested brackets: evaluate innermost brackets first and work outward
  • •
    A minus sign before a bracket applies to only the bracket's result, not the entire remaining expression
  • •
    Always write each simplification step on a separate line to avoid errors

Fraction Operations

Fractions represent parts of a whole as numerator/denominator. The four operations each have distinct rules: addition/subtraction require a common denominator, multiplication goes straight across, and division uses the reciprocal.

Key Points

  • •
    Addition: find LCM of denominators, convert both fractions, then add numerators
  • •
    Multiplication: multiply numerators together and denominators together — cross-cancel first if possible
  • •
    Division: keep the first fraction, change ÷ to ×, flip the second fraction (reciprocal)
  • •
    Always simplify the final result by dividing numerator and denominator by their HCF
  • •
    Convert mixed numbers to improper fractions before operating
Formula

Decimals and Percentages

Decimals extend the base-10 place value system beyond the decimal point. Percentages express a ratio out of 100 — converting between fractions, decimals, and percentages is foundational for business calculations.

Key Points

  • •
    Decimal to fraction: write as numerator/10ⁿ and simplify
  • •
    Multiply decimals by treating them as whole numbers, then count total decimal places for the result
  • •
    Percentage = (Part / Whole) × 100
  • •
    x% of y = (x/100) × y — the word 'of' means multiply
  • •
    Percent change = ((New − Original) / Original) × 100
  • •
    A 10% increase followed by a 10% decrease does NOT return to the original value
Formula

Prime Numbers and Factorization

A prime number has exactly two factors: 1 and itself. Every integer greater than 1 has a unique prime factorization (Fundamental Theorem of Arithmetic), which is the basis for HCF, LCM, and divisibility analysis.

Key Points

  • •
    First primes: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47
  • •
    2 is the only even prime; 1 is neither prime nor composite
  • •
    Factorize by repeatedly dividing by the smallest prime that divides the number
  • •
    Total number of factors of N = p₁ᵃ × p₂ᵇ × ⋯ is (a+1)(b+1)⋯
  • •
    Check divisibility by all primes up to √N when testing primality
Formula

Divisibility Rules

Divisibility rules provide quick checks for whether one number divides another without performing long division. Each rule targets a specific digit pattern or digit-sum property.

Key Points

  • •
    By 2: last digit is even; By 5: last digit is 0 or 5; By 10: last digit is 0
  • •
    By 3: sum of digits divisible by 3; By 9: sum of digits divisible by 9
  • •
    By 4: last two digits form a number divisible by 4; By 8: last three digits divisible by 8
  • •
    By 6: must satisfy both rules for 2 and 3; By 12: must satisfy rules for 3 and 4
  • •
    By 11: alternate digit sum (odd-position minus even-position) is 0 or divisible by 11

HCF and LCM

HCF (Highest Common Factor) is the largest number dividing all given numbers exactly — take the minimum exponent of each common prime. LCM (Least Common Multiple) is the smallest number all given numbers divide into — take the maximum exponent of each prime.

Key Points

  • •
    HCF via prime factorization: take the minimum exponent of each common prime
  • •
    LCM via prime factorization: take the maximum exponent of each prime appearing in any number
  • •
    For two numbers: HCF(a,b) × LCM(a,b) = a × b (does NOT extend to three or more numbers)
  • •
    HCF of fractions = HCF of numerators / LCM of denominators (cross-swap)
  • •
    LCM of fractions = LCM of numerators / HCF of denominators (cross-swap)
  • •
    Fractions must be in lowest terms before applying the fraction HCF/LCM formulas
Formula

Ratios and Proportion

A ratio compares quantities by division (a : b). A proportion states that two ratios are equal, which enables finding unknown quantities through cross-multiplication.

Key Points

  • •
    Ratios are unitless and order matters — a : b ≠ b : a
  • •
    Simplify ratios by dividing both terms by their GCD
  • •
    Cross-multiplication: if a/b = c/d, then ad = bc
  • •
    To divide a quantity Q in ratio a : b, shares are Q × a/(a+b) and Q × b/(a+b)
  • •
    Mean proportional between a and c: b = √(a × c), giving continued proportion a : b = b : c
  • •
    Compound ratio of (a:b) and (c:d) is ac : bd — combines successive proportional changes
Formula

Direct and Inverse Proportion

Direct proportion: both quantities change by the same factor (ratio is constant). Inverse proportion: one increases as the other decreases (product is constant). Identifying which type applies is the critical first step.

Key Points

  • •
    Direct: x₁/y₁ = x₂/y₂ = k — 'more means more' at the same rate
  • •
    Inverse: x₁ × y₁ = x₂ × y₂ = k — 'more means less', total is fixed
  • •
    Unitary method: find value for 1 unit first, then scale up
  • •
    Partnership profits: share ∝ Investment × Time
  • •
    Mixture problems: component amount = Total × (its parts / total parts)
  • •
    Test: does doubling one quantity double the other (direct) or halve it (inverse)?
Formula

Mean, Median, and Weighted Average

The arithmetic mean is the sum divided by count. The median is the middle value in sorted data. The weighted average accounts for varying importance of each value. Choosing the right measure depends on the data and whether outliers are present.

Key Points

  • •
    Mean = Σxᵢ / n; the sum–mean link (Σxᵢ = n × x̄) solves most average problems
  • •
    Combined mean of two groups: x̄c = (n₁x̄₁ + n₂x̄₂) / (n₁ + n₂)
  • •
    Weighted average = Σ(wᵢ × xᵢ) / Σwᵢ — each value is scaled by its weight
  • •
    Median: for odd n, the middle value; for even n, the average of the two middle values
  • •
    Median is resistant to outliers — use it when data is skewed
  • •
    Missing value = n × x̄ − Σ(known values)
Formula

Average Speed and Remainder Patterns

Average speed over equal distances uses the harmonic mean, not the arithmetic mean. For number problems, powers produce cyclical patterns in their unit digits that can be exploited using modular arithmetic.

Key Points

  • •
    Same distance at two speeds: average speed = 2v₁v₂ / (v₁ + v₂) — the harmonic mean
  • •
    Same time at two speeds: average speed = (v₁ + v₂) / 2 — the arithmetic mean
  • •
    The harmonic mean is always less than the arithmetic mean when v₁ ≠ v₂
  • •
    Units digit of powers follows fixed cycles (e.g., 7ⁿ cycles: 7, 9, 3, 1 every 4 powers)
  • •
    Remainder arithmetic: (a × b) mod d = (a mod d) × (b mod d) mod d
Formula

Formulas

Fraction Addition

Cross-multiply to get a common denominator, then add numerators

Percentage of a Number

Convert percentage to decimal and multiply — 'of' means ×

Percent Change

Measures increase or decrease relative to the original value

HCF × LCM Identity

Product of two numbers equals product of their HCF and LCM (two numbers only)

Total Number of Factors

Product of (exponent + 1) for each prime in the factorization

Cross-Multiplication

Product of extremes equals product of means in a proportion

Mean Proportional

Middle term in continued proportion a : b = b : c

Harmonic Mean for Average Speed

Average speed when equal distances are covered at different speeds