Course (Part 553)

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

Lesson

Math Principles: Fundamentals & Practice

Topic

Overview of numbers and operations, algebraic expressions, geometry, functions, statistics & probability, place value, and applied practice (including GMAT‑style questions).

Learning Outcomes

• Classify numbers (natural, whole, integer, rational, irrational) and apply divisibility rules.
• Manipulate algebraic expressions and solve linear/quadratic equations (including completing the square and quadratic formula).
• Recognize key geometric figures, compute area/perimeter/volume, and graph quadratics and other functions.
• Understand place value and decimal operations; convert between fractions, decimals and percentages.
• Apply descriptive statistics (mean, median, mode, standard deviation) and basic probability; interpret common charts.
• Use problem‑solving strategies and GMAT‑style analysis for adaptive test prep.

Conceptual Explanation

This lesson consolidates foundational math concepts. Start with the nature of numbers (classification and operations), then move to symbolic manipulation (algebra), spatial reasoning (geometry and place value), and relational concepts (functions and their graphs). Statistics and probability provide tools to describe and reason about data. Throughout, emphasize exact computation methods (factorization, prime factorization, HCF/LCM), rules (order of operations, divisibility), and problem‑solving heuristics (working backward, creating equations, checking units). For exam contexts like the GMAT, couple content knowledge with strategic practice and analysis of error patterns.

Bilingual (EN → বাংলা)

English	বাংলা	Short Explanation / উদাহরণ
Natural number	প্রাকৃতিক সংখ্যা	1, 2, 3,... (ওপরে গোনার সংখ্যা)
Integer	পূর্ণ সংখ্যা	..., -2, -1, 0, 1, 2, ...
Rational	রাশিমিক সংখ্যা	ভগ্নাংশ/ভাগফল যা p/q আকারে লাগে
Prime factorization	প্রাথমিক গুণনীয়ক বিভাজন	60 = 2² × 3 × 5
Mean / Median / Mode	গড় / মধ্যক / সর্বাধিক বারপাওয়া	Central measures of data / ডেটার কেন্দ্র নির্দেশ করে
Percent / Percentage	শতাংশ	Part / whole × 100 (উদাহরণ: 15% of 200 = 30)

Important Notes

• Always follow the order of operations (PEMDAS/BODMAS) when evaluating expressions.
• When manipulating inequalities: multiplying/dividing by a negative number reverses the inequality sign.
• For quadratic equations, pick the fastest method depending on structure: factor if simple, complete the square for vertex form, use the quadratic formula for general case.
• In probability, clearly define the sample space before counting favorable outcomes.
• When preparing for adaptive tests (GMAT), track error types and timing; use an analysis sheet to guide study focus.

Exercises

True / False

1. Every even number is divisible by 2.
Answer
True — definitionally, an even number ends in 0,2,4,6,8 and is divisible by 2.
2. All prime numbers are odd.
Answer
False — 2 is prime and even. All other primes are odd.
3. The mode of a dataset is always unique.
Answer
False — a dataset can be bimodal, multimodal, or have no mode at all.

Fill in the Blanks

1. The sum of interior angles of a quadrilateral equals ___ degrees.
Answer
360 degrees.
2. For the quadratic ax² + bx + c, the vertex x‑coordinate is ___.
Answer
x = −b/(2a).
3. The prime factorization of 84 is ___.
Answer
84 = 2² × 3 × 7.

Multiple Choice Questions (MCQ)

1. What is the least common multiple (LCM) of 8 and 12?
   1. 24
   2. 48
   3. 96
   4. 12
   Answer

   A: 24. LCM uses highest powers: 8 = 2³, 12 = 2²×3 ⇒ LCM = 2³×3 = 24.
2. If a number's digits sum to 18, it is divisible by:
   1. 2
   2. 3
   3. 9
   4. Both B and C
   Answer

   D: Both 3 and 9. Sum 18 is divisible by 3 and 9, so the number is divisible by both.
3. Which function corresponds to a parabola opening downward?
   1. f(x) = 2x² + 3x + 1
   2. f(x) = −x² + 4x − 5
   3. f(x) = 3x + 2
   4. f(x) = x³ − x
   Answer

   B: coefficient of x² is negative ⇒ parabola opens downward.

Matching

Match the left column (concept) with the right column (formula/description).

A	Concept	B	Match
1	Standard deviation (population)	a	√(Σ(x−μ)²/N)
2	Area of a circle	b	πr²
3	Permutation nPr	c	n!/(n−r)!
4	Combination nCr	d	n!/(r!(n−r)!)

Answer

1→a, 2→b, 3→c, 4→d.

Descriptive

Explain why a circle gives maximum area for a given perimeter among simple closed planar curves (isoperimetric principle — intuitive answer).

Answer

A circle minimizes the boundary length required to enclose a given area, or equivalently maximizes area for a fixed boundary. Intuitively, the circle distributes perimeter uniformly around the center, avoiding indentations or elongations that waste perimeter. Formally this is the isoperimetric inequality: for a closed curve with perimeter P and area A, 4πA ≤ P², with equality only for a circle.

Creative

Design a short GMAT-style data‑sufficiency prompt related to prime factorization and provide the solution key.

Answer

Prompt: What is the number of distinct prime factors of N = 2² × 3³ × 5 × 7² ?

Statement (1): N is divisible by 3.

Statement (2): N is divisible by 11.

Solution Key: The prime factorization is given explicitly in the prompt: primes are 2,3,5,7 ⇒ 4 distinct prime factors. Statement (1) is true but redundant (confirming divisibility by 3). Statement (2) is false (11 is not a factor). Each statement alone is not necessary since the prompt already gives full factorization. If this were a data sufficiency format without the factorization provided, using only (1) would be insufficient and (2) insufficient. But given the full N expression the answer is 4 (both statements irrelevant).

Activities, Vocabulary, Homework, Practice Task

Activities

• Group: create an analysis sheet template to log practice test results (section scores, question type, time per question, error reason).
• Hands‑on: prime factorize ten random integers using both factor tree and division methods; compare speeds.
• Graphing lab: plot y = ax² + bx + c for three different a values to observe vertex and opening direction.

Vocabulary

• Factorial, permutation, combination
• Mean, median, mode, standard deviation, variance
• Discriminant, vertex, axis of symmetry
• Divisibility, HCF (GCD), LCM
• Absolute value, inequality, percentile

Homework

1. Prime factorize: 360, 231, 1001; compute HCF and LCM for pairs (360, 231) and (360, 1001). Provide steps.
Answer

360 = 2³×3²×5; 231 = 3×7×11; 1001 = 7×11×13. HCF(360,231) = 3. LCM(360,231) = 2³×3²×5×7×11 = 27720. HCF(360,1001)=1. LCM(360,1001)=360×1001=360360 (since coprime).

2. Solve: Complete the square and solve x² − 4x − 5 = 0. Graph the parabola and label vertex and intercepts.
Answer

Complete square: x² − 4x = 5 ⇒ x² − 4x + 4 = 9 ⇒ (x − 2)² = 9 ⇒ x − 2 = ±3 ⇒ x = 5 or x = −1. Vertex (2, −9 + 4? check): f(2)=4−8−5=−9. So vertex (2,−9). X‑intercepts (−1,0) and (5,0).

3. Statistics: Given data {4,8,6,5,3}, compute mean, median, mode, variance, and standard deviation (population).
Answer

Mean=5.2. Median=5. Mode=none (no repeat). Variance (population)=2.96. Standard deviation ≈1.72.

Practice Task

Prepare a one‑page GMAT review: choose 5 Quant questions (one each: arithmetic, algebra, geometry, data sufficiency, probability) and provide timed solutions plus notes on common traps.

Suggested approach / Answer

Use analysis sheet to record: question, correct answer, time taken, trap encountered (misread, arithmetic slip, wrong assumption), and corrective action (practice, formula review).