Arithmetic Operations
🟢 Lite — Quick Review (1h–1d)
Rapid summary for last-minute revision before your exam.
Arithmetic operations form the foundation of quantitative reasoning. Master the four operations (+, −, ×, ÷), understand the order of operations (BODMAS/PEMDAS), and know the properties of operations — commutative, associative, and distributive. In the UI entrance test, arithmetic questions test speed and accuracy.
Order of Operations (BODMAS):
- Brackets first: ( ), [ ], { }
- Of / Order: indices, roots, powers
- Division: ÷
- Multiplication: × (same level as division, left to right)
- Addition: +
- Subtraction: − (same level as addition, left to right)
Example: 8 ÷ 2(2+2) = 8 ÷ 2 × 4 = 4 × 4 = 16. NOT 8 ÷ 8 = 1. The ambiguity in “2(2+2)” is resolved by treating it as 2 × (2+2) = 8, then 8 ÷ 8 = 1 IF the bracket means the 2 multiplies the bracket result before division. The safest interpretation: 8 ÷ 2 × (2+2) = 8 ÷ 2 × 4 = 16 (left to right).
Key Properties:
- Commutative: a + b = b + a; a × b = b × a
- Associative: (a + b) + c = a + (b + c); (a × b) × c = a × (b × c)
- Distributive: a(b + c) = ab + ac
- Identity: a + 0 = a; a × 1 = a
- Inverse: a + (−a) = 0; a × (1/a) = 1 (for a ≠ 0)
⚡ Exam Tip: In multiple-choice UI entrance questions, estimate first to eliminate obviously wrong answers before calculating precisely. For example, 47 × 53 ≈ 50 × 50 = 2,500, so if one answer is 2,491 and another is 3,200, you can quickly eliminate the latter. Watch for questions testing BODMAS — they often place operations in an order designed to trap you into calculating left-to-right without considering brackets.
🟡 Standard — Regular Study (2d–2mo)
Standard content for students with a few days to months.
The Four Operations — Detailed
Addition: Combine quantities. When adding integers, account for signs. Example: (−7) + (+3) = −4. Subtraction: Find the difference. Subtracting a negative: (−5) − (−3) = −5 + 3 = −2. Multiplication: Repeated addition. Rules for signs: (+)(+) = +; (+)(−) = −; (−)(+) = −; (−)(−) = +. Division: The inverse of multiplication. Rules for signs: same as multiplication.
Working with Negative Numbers
Number line: negative numbers extend left from zero. −3 is 3 units left of zero.
Addition: −4 + (−7) = −11 (same sign, add magnitudes, keep negative). Addition: −4 + 7 = 3 (different signs, subtract magnitudes: 7 − 4 = 3, result takes sign of larger magnitude: positive).
Multiplication: (−2) × (−3) = +6. (−2) × 3 = −6. Division: (−12) ÷ (−3) = +4. (−12) ÷ 3 = −4.
Fractions — All Operations
Addition/Subtraction: Need common denominator. Example: 3/5 + 1/3 = 9/15 + 5/15 = 14/15. Multiplication: Multiply numerators and denominators. Example: (2/3) × (4/5) = 8/15. Division: Multiply by the reciprocal of the divisor. Example: (3/4) ÷ (2/3) = (3/4) × (3/2) = 9/8 = 1 1/8.
Mixed numbers: convert to improper fractions first. 2 3/5 = (2×5+3)/5 = 13/5.
Decimals
Adding/subtracting: align decimal points. Multiplying decimals: multiply as integers, then place decimal. 0.3 × 0.4 = 12 → 0.12 (two decimal places total). Dividing decimals: if divisor is decimal, multiply both by power of 10 to make divisor whole. 1.5 ÷ 0.3 → 15 ÷ 3 = 5.
Powers and Roots
Powers: 2³ = 2 × 2 × 2 = 8. 5² = 25. Square roots: √25 = 5 (principal positive root). (−5)² = 25, but √25 = 5 (not −5 by convention). Cube roots: ³√8 = 2 because 2³ = 8.
Laws of indices: a^m × a^n = a^(m+n); a^m ÷ a^n = a^(m−n); (a^m)^n = a^(mn).
Order of Operations — Worked Examples
Example: 3 + 2 × (4² − 6) ÷ 5 Step 1 (brackets): (4² − 6) = (16 − 6) = 10. So 3 + 2 × 10 ÷ 5. Step 2 (order): 2 × 10 ÷ 5 = 20 ÷ 5 = 16. Step 3 (addition): 3 + 16 = 19.
Example: Evaluate 2 + 3 × 4² − 8 ÷ 2 = 2 + 3 × 16 − 8 ÷ 2 = 2 + 48 − 4 = 46.
Factor and Multiple Concepts
Factors of 12: 1, 2, 3, 4, 6, 12 (divide evenly into 12). Multiples of 4: 4, 8, 12, 16, 20, … (multiply 4 by integers). HCF (Highest Common Factor): the largest number dividing two or more numbers. HCF of 12 and 18 is 6. LCM (Lowest Common Multiple): the smallest number divisible by two or more numbers. LCM of 12 and 18 is 36.
Prime Numbers and Factorisation
Prime: only divisible by 1 and itself. First primes: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29. Prime factorisation: express a number as a product of primes. 60 = 2² × 3 × 5. Using prime factorisation: √72 = √(2³ × 3²) = √(2² × 2 × 3²) = 2 × 3 × √2 = 6√2.
Problem-Solving Strategies:
- For BODMAS questions, copy the expression and insert brackets around the operation you’re doing at each step
- For large products, use approximation to check whether your answer is in the right ballpark
- When finding factors of a number, divide systematically: try 1, 2, 3, … up to √n
- For HCF/LCM problems involving real-world scenarios (e.g., two bells ringing together), LCM is the answer
Common Mistakes:
- Doing subtraction before addition when not following BODMAS: 10 − 3 + 2 = 9 (not 5)
- Misplacing the decimal point in multiplication/division of decimals
- Forgetting that subtracting a negative is addition: 5 − (−3) = 8, not 2
- In division, dividing the divisor into the dividend with the wrong sign
🔴 Extended — Deep Study (3mo+)
Comprehensive coverage for students on a longer study timeline.
Mathematical Induction Proofs — Sum of Series
Prove that 1 + 2 + 3 + … + n = n(n+1)/2 for all positive integers n.
Step 1 (base case): n = 1. LHS = 1. RHS = 1(2)/2 = 1. ✓ Step 2 (inductive step): Assume true for n = k: 1 + 2 + … + k = k(k+1)/2. For n = k+1: 1 + 2 + … + k + (k+1) = k(k+1)/2 + (k+1) = (k(k+1) + 2(k+1))/2 = (k+1)(k+2)/2. This equals (k+1)((k+1)+1)/2 — the formula with n replaced by (k+1). ✓ Therefore true for all n by induction.
HCF and LCM via Prime Factorisation
For two numbers A and B: HCF = product of common prime factors, each raised to the lowest power appearing in either factorisation. LCM = product of all prime factors, each raised to the highest power appearing in either factorisation.
Example: Find HCF and LCM of 72 and 120. 72 = 2³ × 3²; 120 = 2³ × 3 × 5. HCF: common primes = 2³ (min power of 2), 3¹ (min power of 3) = 2³ × 3 = 8 × 3 = 24. LCM: all primes with highest power = 2³ × 3² × 5 = 8 × 9 × 5 = 360. Check: HCF × LCM = 24 × 360 = 8,640 = 72 × 120. ✓
Modular Arithmetic — Remainders
When a number is divided by a divisor, the remainder is the number modulo the divisor. Example: 47 ÷ 5 gives remainder 2, so 47 ≡ 2 (mod 5). Properties: (a + b) mod n = ((a mod n) + (b mod n)) mod n. Example: What is the remainder when 7¹⁰⁰ is divided by 8? 7 ≡ −1 (mod 8). So 7¹⁰⁰ ≡ (−1)¹⁰⁰ ≡ 1 (mod 8). Remainder is 1.
Number Base Systems
We work in base 10 (decimal). In base b, digits range from 0 to b−1. Converting from base b to decimal: (digits) × b^(position from right, starting 0). Example: 3 × 10² + 7 × 10¹ + 2 × 10⁰ = 300 + 70 + 2 = 372 in base 10. Converting from decimal to base: divide by the new base repeatedly, recording remainders. Example: 37 in base 2: 37 ÷ 2 = 18 r 1; 18 ÷ 2 = 9 r 0; 9 ÷ 2 = 4 r 1; 4 ÷ 2 = 2 r 0; 2 ÷ 2 = 1 r 0; 1 ÷ 2 = 0 r 1. Reading remainders bottom to top: 100101₂. Check: 32+4+1=37. ✓
Rational and Irrational Numbers
Rational: can be expressed as p/q where p, q are integers (q ≠ 0). Examples: 3/4, 0.75, 0.333… (repeating), 2 = 2/1. Irrational: cannot be expressed as p/q. Examples: √2 ≈ 1.4142…, π ≈ 3.14159…, e ≈ 2.71828… Sum of rational and irrational is irrational. Product of non-zero rational and irrational is irrational.
UI Entrance Exam Patterns
Arithmetic questions test:
- BODMAS evaluation (often in a multiple-choice format with tempting wrong answers)
- Negative number operations
- Fraction arithmetic
- HCF/LCM problems (e.g., “Three bells ring at intervals of 12s, 18s, and 30s. If they ring together at 0s, when will they next ring together?”)
- Percentage applications
- Prime factorisation and roots
⚡ Exam Strategy: Use the answer choices to your advantage. If a BODMAS question has answer choices that differ significantly, do only enough calculation to eliminate wrong answers. If the question asks for a rough value, round numbers first.
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