Mathematics and Logical Reasoning

🟢 Lite — Quick Review (1h–1d)

Rapid summary for last-minute revision before your exam.

The quantitative and logical reasoning section tests your ability to handle numbers, interpret data, and think systematically. Focus on arithmetic shortcuts, basic algebra, percentage calculations, averages, ratios, and common logical reasoning patterns. These skills are used in everyday professional life as well as in exam contexts.

High-Yield Facts for Qimiyah:

• Percentage: To find X% of Y: (X/100) × Y. To express A as % of B: (A/B) × 100
• Average: Sum of values / number of values — even if the distribution is unequal, the average lies between min and max
• Ratio: If A:B = 3:4 and B = 20, then A = 15 (3/4 × 20)
• Simple interest: I = PRT/100 (P = principal, R = rate, T = time in years)
• Probability: P(event) = Number of favorable outcomes / Total number of possible outcomes
• ⚡ Exam tip: When solving percentage problems, use the fraction method — 25% = 1/4, 20% = 1/5, 10% = 1/10 — this is faster than decimal multiplication

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🟡 Standard — Regular Study (2d–2mo)

Standard content for students with a few days to months.

Mathematics and Logical Reasoning — Qimiyah Exam (Saudi) Study Guide

Arithmetic — Core Skills

Percentages

Finding percentages:

• 10% of 80 = 8 (simply move the decimal point: 10% × 80 = 0.10 × 80 = 8)
• 15% of 200 = 30 (10% = 20, 5% = 10, so 15% = 30)
• 7.5% of 400 = 30 (7.5% = 75/1000; 75/1000 × 400 = 30)

Percentage increase/decrease:

• Original price = 80; new price = 100 → Increase = (100-80)/80 × 100 = 25%
• Original = 200; new = 160 → Decrease = (200-160)/200 × 100 = 20%

Percentage shortcuts:

• X% of Y = Y% of X ( commutative property of multiplication): 8% of 25 = 25% of 8 = 2
• To increase by X%: multiply by (100+X)%; to decrease by X%: multiply by (100–X)%
• A is X% more than B → A = B × (1 + X/100)

Compound percentage:

• A value increases by 10% each year: multiply by 1.10 each year
• Starting value = 1000; after 3 years at 10% compound: 1000 × 1.1³ = 1331

Averages

Formula: Average = (Sum of all values) / (Number of values)

Example: Test scores: 70, 80, 85, 90, 75

• Sum = 400; Number = 5 → Average = 80

Weighted average:

• If group A has average 60 with 10 people; group B has average 80 with 5 people
• Combined average = (60×10 + 80×5) / 15 = (600 + 400) / 15 = 1000/15 ≈ 66.7

Common trap: The average of averages is NOT the overall average unless all groups have the same size.

Ratios and Proportions

Ratio A:B expresses the relative size of A to B.

• If A:B = 3:4 and A + B = 140 → A = 3/7 × 140 = 60; B = 4/7 × 140 = 80

Direct proportion: When one value increases, the other increases at the same rate (A ∝ B) Inverse proportion: When one value increases, the other decreases (A ∝ 1/B)

Sharing in a ratio: If money is divided in ratio A:B:C = 2:3:5 and total = 1000:

• Total parts = 2+3+5 = 10; A = 2/10 × 1000 = 200; B = 300; C = 500

Profit and Loss

Profit % = (Profit / Cost Price) × 100 Loss % = (Loss / Cost Price) × 100 Selling Price (SP) = Cost Price (CP) + Profit = CP – Loss

Example: CP = 80, SP = 100 → Profit = 20 → Profit % = 20/80 × 100 = 25% Example: SP = 90, profit = 10 → CP = SP – profit = 80 → Profit % = 10/80 × 100 = 12.5%

Discount: Marked Price (MP) → Discount% → Selling Price

• MP = 200; discount 15% → SP = 200 × 0.85 = 170

Simple and Compound Interest

Simple Interest: I = P × R × T / 100

• P = 1000, R = 5%, T = 3 years → I = 1000 × 5 × 3 / 100 = 150
• Total amount = P + I = 1150

Compound Interest: Amount = P × (1 + R/100)^T

• P = 1000, R = 5%, T = 2 years → Amount = 1000 × 1.05 × 1.05 = 1102.5
• CI = 1102.5 – 1000 = 102.5 (vs simple interest of 100)

Algebra — Basic Operations

Linear Equations

• Solve for x: 3x + 7 = 22 → 3x = 15 → x = 5
• Word problem: “A number plus 5 equals 12” → x + 5 = 12 → x = 7

Quadratic Equations

Standard form: ax² + bx + c = 0 Quadratic formula: x = [–b ± √(b² – 4ac)] / 2a

Example: x² – 5x + 6 = 0

• Factor: (x – 2)(x – 3) = 0 → x = 2 or x = 3
• Or: a=1, b=–5, c=6 → discriminant = 25 – 24 = 1 → x = (5 ± 1)/2 → x = 3 or 2

Simultaneous Equations

Solve: 2x + y = 10 and x + y = 6

• Subtract second from first: x = 4 → substitute: 4 + y = 6 → y = 2

Exponents and Powers

• aᵐ × aⁿ = aᵐ⁺ⁿ
• aᵐ / aⁿ = aᵐ⁻ⁿ
• (aᵐ)ⁿ = aᵐⁿ
• a⁰ = 1 (for a ≠ 0)
• a⁻¹ = 1/a

Geometry — Essential Facts

Angles

• Sum of angles in a triangle = 180°
• Sum of angles in a quadrilateral = 360°
• Straight angle = 180°
• Angles around a point = 360°
• Complementary angles sum to 90°; supplementary angles sum to 180°

Triangle Types

• Equilateral: All sides equal (60° each)
• Isosceles: Two sides equal; base angles equal
• Right triangle: One angle = 90°; Pythagorean theorem applies

Pythagorean Theorem

• For a right triangle with legs a and b, hypotenuse c: a² + b² = c²
• Classic 3-4-5 triangle: 3² + 4² = 5² → 9 + 16 = 25
• Other common Pythagorean triples: 5-12-13, 8-15-17, 7-24-25

Circles

• Circumference: C = 2πr = πd
• Area: A = πr²
• Arc length: (θ/360) × 2πr where θ is the angle in degrees
• Sector area: (θ/360) × πr²

Areas and Volumes

Shape	Area Formula	Volume Formula
Rectangle	l × w	—
Triangle	½ × base × height	—
Circle	πr²	—
Cube	6a²	a³
Cylinder	2πr(r+h)	πr²h
Sphere	4πr²	⁴⁄₃πr³

Probability and Statistics

Probability

• P(event) = Favorable outcomes / Total outcomes
• P(always true) = 1; P(impossible) = 0
• P(not A) = 1 – P(A) (complement rule)
• P(A or B) = P(A) + P(B) – P(A and B) (addition rule — only if events are not mutually exclusive)

Independent events: P(A and B) = P(A) × P(B)

• Example: Probability of getting two heads in two coin tosses = ½ × ½ = ¼

Mutually exclusive events: Cannot happen at the same time → P(A and B) = 0

Statistics

Mean (Average): Sum / Count Median: The middle value when data is arranged in order (if two middle values, average them) Mode: Most frequently occurring value (can be no mode, one mode, or multiple modes) Range: Maximum value – minimum value

Standard deviation: Measures spread from the mean — higher SD means more spread

Logical Reasoning

Number Series

Find the pattern:

• 2, 4, 8, 16, 32 → Pattern: multiply by 2 (powers of 2)
• 1, 1, 2, 3, 5, 8 → Pattern: Fibonacci (sum of previous two)
• 3, 6, 11, 18, 27 → Pattern: differences are 3, 5, 7, 9 (odd numbers)

Coding and Decoding

• If “APPLE” is coded as “BQQMF” → each letter shifted by +1 in the alphabet
• “BANANA” in this code would be “CBOBOF”
• Reverse alphabet coding: A=26, B=25… Z=1 → “APPLE” would be… (A=26, P=11…)

Blood Relations

• “A is B’s brother” → A and B share at least one parent; A is male
• “A is B’s sister” → A and B share at least one parent; A is female
• “A is B’s mother” → A is female; one generation above B
• “A is B’s daughter” → A is female; one generation below B
• Logical deduction: If all X are Y, and Z is X → Z is Y. If some X are Y, and Z is X → Z might be Y.

Syllogisms

Structure: Premise 1 + Premise 2 → Conclusion

Example:

• Premise 1: All mammals are warm-blooded
• Premise 2: All whales are mammals
• Conclusion: All whales are warm-blooded ✓

Invalid example:

• Premise 1: All A are B
• Premise 2: All C are B
• Conclusion: All A are C ✗ (just because A and C are both B doesn’t mean A = C)

Data Interpretation

• Read tables, charts, and graphs carefully
• Identify the trend (increasing, decreasing, constant)
• Calculate percentages or ratios as needed
• Check units — ensure consistency

Common Exam Traps

1. Misreading the decimal point in percentage calculations — always double-check
2. Forgetting that average of averages is not the overall average (need weighted average)
3. Compound interest vs simple interest — compound grows faster over time; make sure you use the right formula
4. Assuming a pattern in number series continues the same way — always verify by checking all given terms
5. Confusing median and mode — median is the middle value; mode is the most frequent value
6. Forgetting to square the difference in the quadratic formula — the discriminant is b² – 4ac (not b – 4ac)

⚡ Exam tip: In the heat of the exam, many students forget that “percentage points” and “percentage change” are different. A change from 20% to 25% is a 5 percentage point increase AND a 25% increase relative to the original (5/20 × 100 = 25%). Watch for this trick.

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