1-Hour Quant Intensive | 3-Hour GMAT Crash Course

Hour 1

Number Properties

Integers, Factors, and Multiples

Factor: a divides evenly into b → a is a factor of b
Multiple: b is a multiple of a if b = a × k for some integer k
Prime: divisible only by 1 and itself (2, 3, 5, 7, 11, 13, 17, 19, 23…)
1 is NOT prime. 2 is the only even prime.
Divisibility rules: div by 3 → digit sum div by 3; div by 9 → digit sum div by 9; div by 4 → last 2 digits div by 4

// GCF and LCM formulas

GCF(a, b) × LCM(a, b) = a × b

// Example: GCF(12,18) = 6 → LCM(12,18) = 12×18/6 = 36

Prime factorization method:

12 = 2² × 3¹

18 = 2¹ × 3²

GCF = 2¹ × 3¹ = 6 (lowest powers)

LCM = 2² × 3² = 36 (highest powers)

Odd / Even Rules & Remainders

Odd × Odd = Odd; Even × anything = Even
Odd + Odd = Even; Odd + Even = Odd
Remainder when n divided by d: n = dq + r, where 0 ≤ r < d
Consecutive integers: sum of n consecutive integers = n × (middle term) when n is odd
Units digit cycles: 2→2,4,8,6; 3→3,9,7,1; 7→7,9,3,1; cycle length 4

Q1. What is the LCM of 8 and 12? +

Answer: 24
8 = 2³; 12 = 2² × 3. LCM = 2³ × 3 = 24. Alternatively: GCF(8,12) = 4, so LCM = 8×12/4 = 24.

Q2. Is 0 even or odd? +

Answer: Even
0 = 2 × 0, so 0 is divisible by 2 and is therefore even. This is a common GMAT trap — do not assume "even" means positive.

Q3. What is the units digit of 7^53? +

Answer: 7
The units digit of powers of 7 cycles with period 4: 7¹→7, 7²→9, 7³→3, 7⁴→1, 7⁵→7… 53 mod 4 = 1, so units digit = 7.

Q4. How many prime numbers are between 20 and 40? +

Answer: 4
Primes between 20 and 40: 23, 29, 31, 37. Check each is not divisible by 2, 3, 5, or 7 (since √40 < 7).

Q5. When integer n is divided by 7, the remainder is 4. What is the remainder when 3n is divided by 7? +

Answer: 5
n = 7q + 4, so 3n = 21q + 12 = 7(3q+1) + 5. Remainder = 5.
Shortcut: Multiply the remainder: 3×4 = 12 → 12 mod 7 = 5.

Hour 2

Algebra

Core Algebra Rules

FOIL: (a+b)(a−b) = a² − b²; (a+b)² = a²+2ab+b²
Factoring: ax²+bx+c — look for two numbers that multiply to ac and add to b
Systems: n equations needed for n unknowns (but watch for GMAT tricks)
Inequalities: flip the sign when multiplying/dividing by a negative number
Absolute value: |x| = a means x = a OR x = −a

Linear equation graph: y = 2x + 3 (slope-intercept form)

// Quadratic formula — memorize this!

x = (−b ± √(b²−4ac)) / 2a

// Discriminant tells you number of real roots:

b²−4ac > 0 → two distinct real roots

b²−4ac = 0 → one repeated real root

b²−4ac < 0 → no real roots (complex only)

// Slope formula

m = (y₂−y₁) / (x₂−x₁)

Exponent & Radical Rules

aᵐ × aⁿ = aᵐ⁺ⁿ; aᵐ / aⁿ = aᵐ⁻ⁿ; (aᵐ)ⁿ = aᵐⁿ
a⁰ = 1 (for a ≠ 0); a⁻ⁿ = 1/aⁿ
√(ab) = √a × √b; √(a/b) = √a/√b
Rationalize: multiply numerator and denominator by the conjugate
Fractional exponent: aᵐ/ⁿ = ⁿ√(aᵐ)

Q1. Solve for x: 2x − 5 = 3(x + 1) +

Answer: x = −8
2x − 5 = 3x + 3 → −5 − 3 = 3x − 2x → x = −8.

Q2. What are the roots of x² − 5x + 6 = 0? +

Answer: x = 2 and x = 3
Factor: (x−2)(x−3) = 0. Two numbers that multiply to 6 and add to −5 are −2 and −3.

Q3. If 3^(x+2) = 81, what is x? +

Answer: x = 2
81 = 3⁴, so x + 2 = 4 → x = 2. Always convert both sides to the same base.

Q4. Simplify: (2x²y³)³ +

Answer: 8x⁶y⁹
Cube each factor: 2³=8; (x²)³=x⁶; (y³)³=y⁹. Result: 8x⁶y⁹.

Q5. A line passes through (2, 5) and (6, 13). What is its equation? +

Answer: y = 2x + 1
Slope m = (13−5)/(6−2) = 8/4 = 2. Using point (2,5): 5 = 2(2) + b → b = 1. Equation: y = 2x + 1.

Hour 3

Word Problems

Rate, Time, Distance

Formula: Distance = Rate × Time (D = R × T)
Average speed for a round trip = 2r₁r₂ / (r₁+r₂) — NOT the arithmetic average
Combined rate: if A does job in a hours and B does it in b hours together: 1/a + 1/b = 1/t
Relative speed: same direction → subtract rates; opposite → add rates

Cover the variable you want to find: D=R×T, R=D/T, T=D/R

Mixture Problems

Component (amount × concentration) must balance: C₁V₁ + C₂V₂ = C_final × V_final
Allegation shortcut: ratio of quantities = |C_final − C₂| : |C_final − C₁|
Always set up a table: Solution | Volume | Concentration | Amount

Mixture: combine two solutions to reach a target concentration

Work & Age Problems

Work: rate × time = 1 job (or fraction of job)
If A takes a days and B takes b days: combined time t = ab/(a+b)
Age problems: set up two equations with present and future/past ages
Let x = current age; then x+n = future age, x−n = past age

Q1. A car travels 120 miles at 60 mph, then returns at 40 mph. What is the average speed for the entire trip? +

Answer: 48 mph
Use harmonic mean: 2(60)(40)/(60+40) = 4800/100 = 48 mph. Do NOT average 60 and 40 — that gives 50, which is wrong.

Q2. Pipe A fills a tank in 6 hours; Pipe B fills it in 12 hours. How long to fill together? +

Answer: 4 hours
Combined rate = 1/6 + 1/12 = 2/12 + 1/12 = 3/12 = 1/4. Time = 4 hours.

Q3. How many liters of 20% acid solution must be added to 10 liters of 50% acid solution to get a 30% solution? +

Answer: 20 liters
Let x = liters of 20% solution. 0.20x + 0.50(10) = 0.30(x+10)
0.20x + 5 = 0.30x + 3 → 2 = 0.10x → x = 20.

Q4. Two trains leave cities 360 miles apart heading toward each other at 60 mph and 80 mph. When do they meet? +

Answer: After 2.57 hours (≈ 2 hours 34 minutes)
Combined speed = 60 + 80 = 140 mph (toward each other). Time = 360/140 = 18/7 ≈ 2.57 hours.

Q5. Maria is 3 times as old as her daughter now. In 10 years, she will be twice as old. How old is Maria now? +

Answer: 30 years old
Let daughter = d; Maria = 3d. In 10 years: 3d+10 = 2(d+10) → 3d+10 = 2d+20 → d=10. Maria = 30.

Hour 4

Percentages & Statistics

Percentage Formulas

Percent change = (New − Old) / Old × 100%
Successive percent changes: multiply (not add) the multipliers
Example: 20% increase then 20% decrease = 1.2 × 0.8 = 0.96 → 4% net decrease
Markup vs margin: markup on cost; margin on selling price — know the base
Simple interest: I = P × r × t; Compound interest: A = P(1+r/n)^(nt)

Statistics: Mean, Median, Mode, Range, SD

Mean: sum / count; affected by outliers
Median: middle value when sorted; resistant to outliers
Mode: most frequent value
Range: max − min
Standard deviation: measures spread around the mean
Adding a constant to all values: mean shifts, SD unchanged
Multiplying all values by k: mean multiplies by k, SD multiplies by |k|

Bell curve: 68-95-99.7 rule for standard deviations

// Weighted average formula

Weighted Avg = (w₁x₁ + w₂x₂ + … + wₙxₙ) / (w₁ + w₂ + … + wₙ)

// Variance and Standard Deviation

Variance σ² = Σ(xᵢ − μ)² / n

Standard deviation σ = √(Variance)

// On the GMAT: you need to compare SDs, not calculate them

More spread out → higher SD; more clustered → lower SD

Q1. A jacket costs $80. After a 25% discount, what is the final price? +

Answer: $60
$80 × (1 − 0.25) = $80 × 0.75 = $60. Alternatively: 25% of $80 = $20; $80 − $20 = $60.

Q2. A stock rises 40% then falls 40%. What is the net percent change? +

Answer: −16% (a 16% net loss)
Multiply multipliers: 1.4 × 0.6 = 0.84 → net = −16%. Successive percentages are not additive.

Q3. Set A: {2, 4, 6, 8, 10}. Set B: {12, 14, 16, 18, 20}. Which has a larger SD? +

Answer: Neither — they are equal
Both sets have the same spacing (increments of 2) and the same number of elements. Adding 10 to every element shifts the mean but does NOT change the SD. SD(A) = SD(B).

Q4. The average of 5 numbers is 20. A 6th number is added and the average becomes 22. What is the 6th number? +

Answer: 32
Sum of 5 numbers = 5×20 = 100. New sum = 6×22 = 132. 6th number = 132 − 100 = 32.

Q5. $5,000 is invested at 6% annual simple interest for 3 years. What is the total amount at the end? +

Answer: $5,900
Interest = P × r × t = 5000 × 0.06 × 3 = $900. Total = $5,000 + $900 = $5,900.

Hour 5

Geometry: Lines & Triangles

Lines and Angles

Straight line = 180°; full rotation = 360°
Vertical angles are equal; supplementary angles sum to 180°
Parallel lines cut by transversal: alternate interior angles equal; corresponding angles equal; co-interior (same-side interior) sum to 180°
Triangle interior angles sum to 180°
Exterior angle of a triangle = sum of the two non-adjacent interior angles

Pythagorean triples to memorize: 3-4-5, 5-12-13, 8-15-17, 7-24-25

Triangle Type	Angles	Sides	GMAT Tip
45-45-90	45°, 45°, 90°	x, x, x√2	Half a square; diagonal = side × √2
30-60-90	30°, 60°, 90°	x, x√3, 2x	Short leg opposite 30°; hyp = 2 × short
Equilateral	60°, 60°, 60°	a, a, a	Height = a√3/2; Area = a²√3/4
Isosceles	Two equal	Two equal	Base angles are equal

Triangle Area, Perimeter, and Inequalities

Area = (1/2) × base × height (height must be perpendicular to base)
Heron's formula: A = √(s(s−a)(s−b)(s−c)) where s = (a+b+c)/2
Triangle inequality: each side must be less than the sum of the other two
Larger angle is opposite longer side

Q1. A right triangle has legs 5 and 12. What is the hypotenuse? +

Answer: 13
Pythagorean triple: 5-12-13. Verify: 5² + 12² = 25 + 144 = 169 = 13².

Q2. In a 30-60-90 triangle, the hypotenuse is 10. What is the length of the shorter leg? +

Answer: 5
In a 30-60-90 triangle, sides are x, x√3, 2x. Hypotenuse = 2x = 10 → x = 5. Short leg = 5, long leg = 5√3.

Q3. Two parallel lines are cut by a transversal. One angle is 65°. What are all 8 angles? +

Answer: Four 65° angles and four 115° angles
Angles alternate between 65° and 180°−65°=115°. Corresponding, alternate interior, and alternate exterior angles are all equal (65°); co-interior angles sum to 180° (115°).

Q4. An equilateral triangle has side length 6. What is its area? +

Answer: 9√3
Area = (√3/4) × a² = (√3/4) × 36 = 9√3 ≈ 15.59 square units.

Q5. Can a triangle have sides 3, 4, and 8? +

Answer: No
Triangle inequality: 3 + 4 = 7 < 8. The sum of any two sides must be greater than the third side. Since 3 + 4 is not greater than 8, no such triangle exists.

Hour 6

Circles & Polygons

Circle Formulas

Circumference = 2πr = πd
Area = πr²
Arc length = (θ/360°) × 2πr
Sector area = (θ/360°) × πr²
Inscribed angle = (1/2) × central angle subtending the same arc
Tangent is perpendicular to the radius at the point of tangency

Circle anatomy: radius, diameter, chord, arc, sector, tangent

Polygons

Sum of interior angles of an n-gon = (n−2) × 180°
Each interior angle of a regular n-gon = (n−2) × 180° / n
Quadrilateral: 360°; Pentagon: 540°; Hexagon: 720°
Rectangle: Area = l × w; Diagonal = √(l² + w²)
Square: Area = s²; Diagonal = s√2
Trapezoid: Area = (1/2)(b₁ + b₂) × h
Parallelogram: Area = base × height

Q1. A circle has circumference 20π. What is its area? +

Answer: 100π
Circumference = 2πr = 20π → r = 10. Area = πr² = π(10)² = 100π.

Q2. A sector has a central angle of 90° and radius 8. What is the area of the sector? +

Answer: 16π
Sector area = (90/360) × πr² = (1/4) × π × 64 = 16π.

Q3. What is the sum of the interior angles of a hexagon? +

Answer: 720°
(n−2) × 180° = (6−2) × 180° = 4 × 180° = 720°. Each interior angle of a regular hexagon = 720°/6 = 120°.

Q4. A rectangle has a diagonal of 13 and one side of 5. What is the area? +

Answer: 60
Using Pythagorean theorem: other side = √(13²−5²) = √(169−25) = √144 = 12. Area = 5 × 12 = 60.

Q5. A trapezoid has parallel sides of 6 and 10 and height 4. What is its area? +

Answer: 32
Area = (1/2)(b₁ + b₂) × h = (1/2)(6 + 10) × 4 = (1/2)(16)(4) = 32.

Hour 7

Data Sufficiency

The 5 Data Sufficiency Answer Choices (Memorize These!)

A

Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.

B

Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient.

C

BOTH statements TOGETHER are sufficient, but NEITHER alone is sufficient.

D

EACH statement ALONE is sufficient.

E

Statements (1) and (2) TOGETHER are not sufficient.

DS Strategy: The YES/NO Test

For value questions: sufficient = gives ONE unique value
For yes/no questions: sufficient = always YES or always NO (not "sometimes")
Work each statement independently — block out the other
If Statement 1 is sufficient → answer is A or D; if not → answer is B, C, or E
If Statement 2 is sufficient → answer is B or D; if not → combined check needed

DS Decision Flowchart: always evaluate each statement independently first

DS Trap	Description	Example
Statement contamination	Using info from (1) while evaluating (2)	Block out (1) completely when testing (2)
"Could be" vs "must be"	Sufficient needs to always be true	If x²=4, then x could be 2 or −2: NOT sufficient for value
Sufficiency ≠ correct	A wrong unique answer is still sufficient	The question asks if answer is determined, not if you can compute it
Rephrasing the question	Simplify before evaluating statements	"Is x>0?" may simplify from a complex expression

DS Q1. What is the value of x? (1) x² = 16 (2) x > 0 +

Answer: C
(1) alone: x = 4 or x = −4 → NOT sufficient. (2) alone: x > 0 → NOT sufficient. Together: x² = 16 AND x > 0 → x = 4. Sufficient. Answer: C.

DS Q2. Is n divisible by 6? (1) n is divisible by 2 (2) n is divisible by 3 +

Answer: C
(1) alone: n could be 4 (not div by 6) or 6 (div by 6) → NOT sufficient. (2) alone: n could be 9 or 6 → NOT sufficient. Together: divisible by both 2 and 3 → divisible by 6. Answer: C.

DS Q3. Is x > y? (1) x + y > 0 (2) x − y > 0 +

Answer: B
(1) alone: x=3, y=−1: x+y=2>0, and x>y (YES); but x=−1, y=3: x+y=2>0, x<y (NO) → NOT sufficient. (2) alone: x−y>0 directly means x>y → ALWAYS YES. Sufficient. Answer: B.

DS Q4. What is the area of circle O? (1) The diameter is 10 (2) The circumference is 10π +

Answer: D
(1) alone: diameter=10 → r=5 → Area = 25π. Sufficient. (2) alone: 2πr = 10π → r=5 → Area = 25π. Sufficient. Each statement alone is sufficient. Answer: D.

DS Q5. Is the integer k odd? (1) k/2 is not an integer (2) k−1 is even +

Answer: D
(1) alone: k/2 is not an integer means k is not divisible by 2, so k is odd → YES, always. Sufficient. (2) alone: k−1 is even → k is odd (odd−1=even) → YES, always. Sufficient. Answer: D.

Hour 8

Review & Timed Drills

GMAT Quant Pacing Guide

GMAT Focus Quant: 45 questions in 45 minutes → ~1 minute per question average
Spend up to 2 min on hard questions; bail if taking more than 2.5 min
Process of elimination (POE) is your friend — use it actively
Estimation often works: eliminate obviously wrong answers first
Check units, check whether the question asks for an approximation or exact value
Flag questions for review if GMAT Focus Edition allows; don't leave blanks

10-Question Speed Drill

Target: complete in 12 minutes. Click any question to reveal the answer below.

D1. What is 15% of 240?

D2. If 2x + 3 = 11, what is x?

D3. Area of a circle with radius 7?

D4. What is the GCF of 36 and 48?

D5. A triangle has angles 40° and 75°. What is the third angle?

D6. (x+3)(x−3) = ?

D7. What is the median of: 3, 7, 1, 9, 5?

D8. A square has area 64. What is its perimeter?

D9. What is 2/3 + 3/4?

D10. If n is even, is n² odd or even?

D1. What is 15% of 240? +

Answer: 36
10% of 240 = 24; 5% = 12; 15% = 24 + 12 = 36. Or: 0.15 × 240 = 36.

D2. If 2x + 3 = 11, what is x? +

Answer: x = 4
2x = 11 − 3 = 8; x = 4.

D3. Area of a circle with radius 7? +

Answer: 49π
Area = πr² = π × 7² = 49π.

D4. What is the GCF of 36 and 48? +

Answer: 12
36 = 2² × 3²; 48 = 2⁴ × 3. GCF = 2² × 3 = 12.

D5. A triangle has angles 40° and 75°. What is the third angle? +

Answer: 65°
Triangle angles sum to 180°: 180 − 40 − 75 = 65°.

D6. (x+3)(x−3) = ? +

Answer: x² − 9
Difference of squares: (a+b)(a−b) = a² − b². Here a=x, b=3, so result is x² − 9.

D7. What is the median of: 3, 7, 1, 9, 5? +

Answer: 5
Sort first: 1, 3, 5, 7, 9. The middle value (3rd of 5) is 5.

D8. A square has area 64. What is its perimeter? +

Answer: 32
Area = s² = 64 → s = 8. Perimeter = 4s = 4 × 8 = 32.

D9. What is 2/3 + 3/4? +

Answer: 17/12
Common denominator 12: 2/3 = 8/12; 3/4 = 9/12. Sum = 17/12.

D10. If n is even, is n² odd or even? +

Answer: Even
Even × Even = Even. n² = n × n. If n is even, n² is always even (e.g., 4² = 16, 6² = 36).

Common Mistake Pattern	What Goes Wrong	Fix
Average speed trap	Averaging rates instead of using harmonic mean	Use 2r₁r₂/(r₁+r₂) for equal distances
Distributing exponents	(a+b)² = a²+b² (WRONG)	(a+b)² = a²+2ab+b²
Units digit patterns	Not recognizing cyclical patterns	Memorize cycles for 2,3,7,8 (length 4)
DS contamination	Using info from both statements for one	Physically cover statement you're not testing
Percent change base	Using new value instead of original as base	Percent change = (New−Old)/Old × 100
√x² = x	Forgetting the absolute value	√x² = \|x\|; could be positive or negative x

Raw Score Estimate	Scaled Score	Percentile (approx)
40–45 correct	85+	96th+
33–39 correct	75–84	75th–95th
25–32 correct	60–74	45th–74th
15–24 correct	45–59	20th–44th
Below 15	Below 45	Below 20th

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