Course (Part 554)

Course: GMAT Complete Math | Score Higher for Top Business Schools

Lesson

Topic: Math Principles — Fundamental Concepts

Date: 2025-11-15

Learning Outcomes

• Identify the five foundational domains: Numbers & Operations, Algebraic Expressions, Geometry, Functions, and Statistics & Probability.
• Apply place-value, fraction, decimal, percentage, exponent, and root concepts to solve problems.
• Use prime factorization, HCF/LCM, permutations & combinations, and basic probability in contextual problems.
• Interpret graphs (line, bar, histogram, box-plot) and convert units reliably.
• Employ problem-solving strategies and set up equations from word problems.

Conceptual Explanation

Overview

Mathematics is organized into interconnected domains. Mastery of core procedures (place value, operations, solving linear equations, factoring, manipulating exponents/roots) enables progress to advanced topics (quadratics, distributions, adaptive algorithms, GMAT-style data sufficiency).

Key Definitions & Formulae

Essential formulas and quick reminders

Concept	Formula / Rule
Mean	mean = (Σxi) / n
Median	middle value (or average of middle two when n even)
Standard deviation (population)	σ = √[Σ(xi − μ)² / N]
Quadratic formula	x = [−b ± √(b² − 4ac)] / (2a)
Distance / Rate / Time	Distance = Rate × Time
Area (rectangle / triangle / circle)	Rect: A = l × w; Tri: A = ½ × b × h; Circ: A = πr²
Permutation / Combination	P(n,r)=n!/(n−r)!   C(n,r)=n!/[r!(n−r)!]

Worked concept map (brief)

1. Numbers & Operations — place value, integers, fractions, decimals, percents, divisibility, prime factorization.
2. Algebra — variables, expressions, linear equations, inequalities, factoring, quadratics (completing the square, quadratic formula).
3. Geometry — lines, angles, triangles, quadrilaterals, circles, area & volume formulas.
4. Functions — definitions, types (linear, quadratic, exponential), graphing, domain & range.
5. Statistics & Probability — descriptive stats (mean, median, mode, sd), distributions (normal), probability rules, data interpretation (charts, histograms, box plots).

Common Problem-Solving Techniques

• Translate words → equations (define variables, write relations, solve, check).
• Use prime factor trees for HCF/LCM and fraction simplification.
• For rate/work problems, sum rates when workers work together: R_total = R1 + R2 + ...
• In absolute-value inequalities, convert to compound inequalities: |x| < a → −a < x < a; |x| > a → x < −a or x > a.
• For GMAT data-sufficiency style problems, test statements independently before combining.

Bilingual (EN → বাংলা)

Number সংখ্যা (উদাহরণ: স্বাভাবিক, পূর্ণ, সংখ্যা, রেশিও) Place value অঙ্কের স্থান-মান (উদাহরণ: 345 = 3×100 + 4×10 + 5×1) — স্থান-মান বোঝা মৌলিক। Fraction ভাগ (উপাংশ / সমগ্র): ¾ = তিন ভাগের মধ্যে তিনটি অংশ নয়; ½ = অর্ধেক। Mean / Median / Mode গড় / মধ্যমা / বহুলক (Mean = যোগফল ÷ সংখ্যা; Median = মধ্য মান; Mode = সর্বাধিক ঘনঘনতা) Quadratic Formula দ্বিঘাত সূত্র: x = [−b ± √(b² − 4ac)] / (2a) — সমাধান করতে ব্যবহৃত। Probability সম্ভাব্যতা: P(E) = অনুকূল ফলাফল / সম্ভাব্য মোট ফলাফল

সংক্ষিপ্ত নির্দেশ: উপরের ধারণাগুলি বাংলা ভাষায় অনুবাদ করা হয়েছে যাতে বাংলাভাষী শিক্ষার্থীরা দ্রুত ধারণা পায়।

Important Notes

• Practice transforms knowledge into skill — drill on variability: fractions, decimals, percent conversions, and exponents.
• Watch for units (rates, distances) and align units before calculating.
• When a single extreme value skews the mean, report the median as a robust central measure.
• Use graphs to check work: plotting a quadratic quickly reveals vertex and intercepts; histograms show distribution shape.

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Exercises

True / False

1. Every square is a rectangle, but not every rectangle is a square.
   Answer

   True. A square has all properties of a rectangle but with equal sides.
2. The mean is unaffected by extremely large outliers.
   Answer

   False. The mean is sensitive to outliers; the median is more robust.
3. If two lines have equal slopes and different intercepts, they intersect at one point.
   Answer

   False. Equal slopes with different intercepts means they are parallel and never intersect.

Fill in the Blanks

1. The sum of interior angles of a quadrilateral equals _____.
   Answer

   360 degrees.
2. For the quadratic ax²+bx+c, the discriminant is _____.
   Answer

   b² − 4ac.
3. The probability of an impossible event equals _____.
   Answer

   0.

Multiple Choice (MCQ)

1. What is the least common multiple (LCM) of 12 and 18?
   1. 36
   2. 54
   3. 72
   4. 18
   Answer

   A: 36. (Prime factors: 12=2²·3, 18=2·3² → LCM = 2²·3² = 36.)
2. Which expression represents the area of a circle?
   1. 2πr
   2. πr²
   3. πd
   4. r²/π
   Answer

   B: πr².
3. If |x − 4| < 3, which interval is the solution?
   1. x < 1 or x > 7
   2. 1 < x < 7
   3. −3 < x < 3
   4. 4 < x < 7
   Answer

   B: 1 < x < 7. (|x−4|<3 → −3 < x−4 < 3 → 1 < x < 7.)

Matching

Match the concept (L) with the formula or description (R).

1. Mean
2. Standard deviation (population)
3. Permutation P(n,r)

1. σ = √[Σ(xi − μ)² / N]
2. n! / (n−r)!
3. (Σxi) / n

Answer

1 → C, 2 → A, 3 → B.

Descriptive

Problem:

Given a rectangle with fixed perimeter 60 m, determine the dimensions that maximize the area. Explain and compute the maximum area.

Answer

Let length = L and width = W. Perimeter: 2L + 2W = 60 → W = 30 − L. Area A = L·W = L(30 − L) = 30L − L². To maximize A, take derivative: dA/dL = 30 − 2L = 0 → L = 15 m. Then W = 15 m. Maximum area = 15 × 15 = 225 m². (A square maximizes area for fixed perimeter.)

Creative

Design a short word problem that involves both percentages and compound interest. Provide the solution.

Answer

Problem: Lina invests $2,000 in an account paying 6% annual interest, compounded annually. After 3 years she withdraws 20% of the balance. What is the remaining balance immediately after the withdrawal?

Solution: Balance after 3 years: A = 2000(1.06)³ ≈ 2000 × 1.191016 = 2382.03. Withdrawal: 20% of 2382.03 ≈ 0.20 × 2382.03 = 476.406. Remaining balance = 2382.03 − 476.406 ≈ $1905.62.

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Activities

1. Group task: Create a one-page summary (English/Bangla) for the place-value system and present three classroom examples.
2. Lab exercise: Use a small dataset to compute mean, median, mode, variance, and standard deviation; produce a histogram and a box plot.

Vocabulary

• Factorial (n!) — n × (n−1) × ... × 1
• Discriminant — b² − 4ac
• Quartile / IQR — measures for spread
• Permutation vs Combination — order matters vs not

Homework

1. Prime-factorize: 360 and 1001. Provide HCF and LCM of the pair (360, 1001).
   Answer

   360 = 2³·3²·5; 1001 = 7·11·13. HCF = 1; LCM = 360 × 1001 = 360360 (since coprime).
2. Graph the quadratic y = −2x² + 4x + 1: find vertex, axis, intercepts and plot (hand-sketched).
   Answer

   Vertex x = −b/(2a) = −4/(−4) =1 → y = −2(1)+4+1 =3. Vertex (1,3). Axis x=1. y-intercept (0,1). x-intercepts via formula: x = [−4 ± √(16 − 4(−2)(1))]/(2(−2)) = [−4 ± √(24)]/(−4) → numeric roots ≈ (−4 ± 4.89898)/(−4) → x ≈ 0.224 and x ≈ −2.224; plot accordingly.

Practice Task

Choose any three concepts from this sheet and create two practice questions each. Exchange with a peer and grade each other.

Instructor notes: Use the GMAT-analysis approach for timed practice—record time used per question and update your analysis sheet after each practice set.

Prepared by: Math Principles Course Team