01
Pre-Algebra
Variables, equations, inequalities, ratios, and number properties
★ KEY MEMORY · "ISOLATE the variable — do the SAME thing to BOTH sides"
Solving Equations
Q1
Equation
Step 2 — Divide both sides by 3: x = ?
Memory: ISOLATE → undo addition first, then division.
If 3x + 7 = 22, what is the value of x?
How to solve
Step 1 — Subtract 7 from both sides: 3x = 15Step 2 — Divide both sides by 3: x = ?
Memory: ISOLATE → undo addition first, then division.
Correct answer: D (x = 5)
3x + 7 = 22 → subtract 7 → 3x = 15 → divide by 3 → x = 5.
Trap: Many students forget to subtract 7 first and divide 22 by 3, getting x ≈ 7.3. Always undo addition/subtraction before multiplication/division.
3x + 7 = 22 → subtract 7 → 3x = 15 → divide by 3 → x = 5.
Trap: Many students forget to subtract 7 first and divide 22 by 3, getting x ≈ 7.3. Always undo addition/subtraction before multiplication/division.
Q2
Tricky ⚠
Solve for x: -2x + 5 = 13
Correct answer: A (x = −4)
−2x + 5 = 13 → subtract 5 → −2x = 8 → divide by −2 → x = −4.
Trap: Dividing by a negative number flips the sign. x = 4 is wrong because −2 × 4 = −8, not 8.
−2x + 5 = 13 → subtract 5 → −2x = 8 → divide by −2 → x = −4.
Trap: Dividing by a negative number flips the sign. x = 4 is wrong because −2 × 4 = −8, not 8.
Order of Operations (PEMDAS)
★ KEY MEMORY · "PEMDAS — Please Excuse My Dear Aunt Sally"
Q3
PEMDAS
Parentheses first: (5−1) = 4
Exponents: 22 = 4
Multiply: 3 × 4 = 12
Then add/subtract left to right.
Evaluate: 4 + 3 × 22 − (5 − 1)
Order of Operations
P → E → M/D → A/SParentheses first: (5−1) = 4
Exponents: 22 = 4
Multiply: 3 × 4 = 12
Then add/subtract left to right.
Correct answer: C (12)
4 + 3 × 2² − (5−1) = 4 + 3 × 4 − 4 = 4 + 12 − 4 = 12.
Trap: Adding 4 + 3 = 7 first (ignoring PEMDAS) gives 7 × 4 − 4 = 24, which is wrong.
4 + 3 × 2² − (5−1) = 4 + 3 × 4 − 4 = 4 + 12 − 4 = 12.
Trap: Adding 4 + 3 = 7 first (ignoring PEMDAS) gives 7 × 4 − 4 = 24, which is wrong.
Inequalities
★ KEY MEMORY · "FLIP the sign when multiplying / dividing by NEGATIVE"
Q4
Tricky ⚠
Which values of x satisfy -3x < 12?
Correct answer: B (x > −4)
Divide both sides by −3 → sign FLIPS → x > −4.
This is the #1 most missed inequality rule. Negative division = flip the sign. Always!
Divide both sides by −3 → sign FLIPS → x > −4.
This is the #1 most missed inequality rule. Negative division = flip the sign. Always!
Ratios & Proportions
★ KEY MEMORY · "CROSS-MULTIPLY to solve proportions: a/b = c/d → ad = bc"
Q5
Proportion
A recipe uses 2 cups of flour for every 3 cups of sugar. If you use 12 cups of sugar, how many cups of flour do you need?
Correct answer: C (8 cups)
Set up: 2/3 = x/12 → cross multiply → 3x = 24 → x = 8.
Trap: Adding 12 − 3 = 9, then 2 + 9 = 11 (wrong). Always use cross-multiplication for proportions.
Set up: 2/3 = x/12 → cross multiply → 3x = 24 → x = 8.
Trap: Adding 12 − 3 = 9, then 2 + 9 = 11 (wrong). Always use cross-multiplication for proportions.
Percent Problems
★ KEY MEMORY · "IS over OF = PERCENT over 100"
Q6
Percent
Method B: Sale price = 75% of $40 = 0.75 × 40 = ?
A shirt costs $40. It is on sale for 25% off. What is the sale price?
Two methods
Method A: Find 25% of $40 = 0.25 × 40 = $10 discount → $40 − $10 = ?Method B: Sale price = 75% of $40 = 0.75 × 40 = ?
Correct answer: B ($30)
25% of $40 = $10. $40 − $10 = $30.
Trap: $10 is the discount, not the sale price! The question asks for the final price.
25% of $40 = $10. $40 − $10 = $30.
Trap: $10 is the discount, not the sale price! The question asks for the final price.
Distributive Property
★ KEY MEMORY · "a(b + c) = ab + ac — DISTRIBUTE to EVERY term inside"
Q7
Distribution
Simplify: 3(2x − 4) + 5x
Correct answer: D (11x − 12)
3(2x − 4) = 6x − 12. Then 6x − 12 + 5x = 11x − 12.
Trap B: 3 × (−4) = −12, not −4. You must multiply 3 by BOTH terms inside the parentheses.
3(2x − 4) = 6x − 12. Then 6x − 12 + 5x = 11x − 12.
Trap B: 3 × (−4) = −12, not −4. You must multiply 3 by BOTH terms inside the parentheses.
Word Problem — Two Variables
Q8
Tricky ⚠
Total: x + 2x = 36 → 3x = 36 → x = 12
So Tom has 2x = ?
Tom has twice as many stickers as Lisa. Together they have 36 stickers. How many does Tom have?
Set up the equation
Let Lisa = x → Tom = 2xTotal: x + 2x = 36 → 3x = 36 → x = 12
So Tom has 2x = ?
Correct answer: B (24)
x + 2x = 36 → x = 12 (Lisa). Tom = 2 × 12 = 24.
Trap A: 12 is Lisa's count, not Tom's. Read carefully — the question asks for Tom.
x + 2x = 36 → x = 12 (Lisa). Tom = 2 × 12 = 24.
Trap A: 12 is Lisa's count, not Tom's. Read carefully — the question asks for Tom.
Exponents
★ KEY MEMORY · "x⁰ = 1 (always!) · x¹ = x · negative exponent = flip to fraction"
Q9
Exponents
So 2³ × 2² = 23+2 = 2?
What is the value of 23 × 22?
Exponent Rule
When multiplying same base: am × an = am+nSo 2³ × 2² = 23+2 = 2?
Correct answer: C (2⁵ = 32)
Same base → ADD exponents: 2³ × 2² = 2^(3+2) = 2⁵ = 32.
Trap A: You ADD exponents (not multiply) → 2^6 = 64 is wrong. 3+2=5, not 6.
Same base → ADD exponents: 2³ × 2² = 2^(3+2) = 2⁵ = 32.
Trap A: You ADD exponents (not multiply) → 2^6 = 64 is wrong. 3+2=5, not 6.
Number Line & Absolute Value
★ KEY MEMORY · "|x| = DISTANCE from zero — always positive or zero"
Q10
Tricky ⚠
Evaluate: |−8| + |3 − 10|
Correct answer: C (15)
|−8| = 8. |3 − 10| = |−7| = 7. Total = 8 + 7 = 15.
Trap: Absolute value is never negative. |−7| = 7, not −7.
|−8| = 8. |3 − 10| = |−7| = 7. Total = 8 + 7 = 15.
Trap: Absolute value is never negative. |−7| = 7, not −7.
02
Geometry
Area, perimeter, angles, triangles, circles, and coordinate geometry
Area & Perimeter
★ KEY MEMORY · "AREA = space inside · PERIMETER = fence around outside"
Q11
Area
Perimeter = 2l + 2w = 2(l + w)
A rectangle has a length of 9 cm and a width of 4 cm. What is the area?
Formulas to remember
Area of rectangle = length × width = l × wPerimeter = 2l + 2w = 2(l + w)
Correct answer: B (36 cm²)
Area = 9 × 4 = 36 cm². Note: area always uses cm² (squared units).
Trap A: 26 cm is the perimeter (2×9 + 2×4). Trap D: 26 cm² is wrong units for the perimeter value.
Area = 9 × 4 = 36 cm². Note: area always uses cm² (squared units).
Trap A: 26 cm is the perimeter (2×9 + 2×4). Trap D: 26 cm² is wrong units for the perimeter value.
Triangle Area
★ KEY MEMORY · "Area of Triangle = ½ × base × height (the height must be PERPENDICULAR)"
Q12
Triangle
A triangle has a base of 10 m and a height of 6 m. What is its area?
Correct answer: B (30 m²)
Area = ½ × 10 × 6 = ½ × 60 = 30 m².
Trap A: Forgetting the ½. Students often compute 10 × 6 = 60 and stop. Always divide by 2 for triangles!
Area = ½ × 10 × 6 = ½ × 60 = 30 m².
Trap A: Forgetting the ½. Students often compute 10 × 6 = 60 and stop. Always divide by 2 for triangles!
Pythagorean Theorem
★ KEY MEMORY · "a² + b² = c² — c is always the HYPOTENUSE (longest side, opposite right angle)"
Q13
Pythagorean
3² + 4² = c²
9 + 16 = c²
c = √25 = ?
A right triangle has legs of length 3 and 4. What is the length of the hypotenuse?
Pythagorean Theorem
a² + b² = c²3² + 4² = c²
9 + 16 = c²
c = √25 = ?
Correct answer: D (5)
3² + 4² = 9 + 16 = 25. √25 = 5. This is the famous "3-4-5 Pythagorean triple!"
Trap A: 3 + 4 = 7 (just adding the legs — wrong! You must square then add).
3² + 4² = 9 + 16 = 25. √25 = 5. This is the famous "3-4-5 Pythagorean triple!"
Trap A: 3 + 4 = 7 (just adding the legs — wrong! You must square then add).
Circle — Circumference & Area
★ KEY MEMORY · "Circumference = 2πr · Area = πr² · π ≈ 3.14"
Q14
Tricky ⚠
A circle has a diameter of 10 cm. What is its area? (Use π ≈ 3.14)
Correct answer: B (78.5 cm²)
Diameter = 10, so radius r = 5. Area = π × 5² = 3.14 × 25 = 78.5 cm².
Trap A: Using diameter instead of radius → π × 10² = 314. Always halve the diameter to get r first!
Diameter = 10, so radius r = 5. Area = π × 5² = 3.14 × 25 = 78.5 cm².
Trap A: Using diameter instead of radius → π × 10² = 314. Always halve the diameter to get r first!
Angle Relationships
★ KEY MEMORY · "Straight line = 180° · Full turn = 360° · Right angle = 90°"
Q15
Angles
Complementary = two angles that add to 90°
Trick: Supplement starts with S, Straight starts with S → 180°
Two angles are supplementary. One angle is 65°. What is the other?
Angle vocabulary
Supplementary = two angles that add to 180°Complementary = two angles that add to 90°
Trick: Supplement starts with S, Straight starts with S → 180°
Correct answer: C (115°)
Supplementary angles sum to 180°. 180 − 65 = 115°.
Trap A: 90 − 65 = 25 → that's the complement, not the supplement. Remember: S for Supplementary = Straight = 180°.
Supplementary angles sum to 180°. 180 − 65 = 115°.
Trap A: 90 − 65 = 25 → that's the complement, not the supplement. Remember: S for Supplementary = Straight = 180°.
Volume of a Rectangular Prism
★ KEY MEMORY · "Volume = length × width × height = l × w × h (cubic units!)"
Q16
Volume
A box measures 5 cm × 4 cm × 3 cm. What is its volume?
Correct answer: A (60 cm³)
V = 5 × 4 × 3 = 60 cm³. Volume always uses cubic units (cm³).
Trap D: 5 × 4 × 6? No — it's 5 × 4 × 3 = 60. Double-check each dimension.
V = 5 × 4 × 3 = 60 cm³. Volume always uses cubic units (cm³).
Trap D: 5 × 4 × 6? No — it's 5 × 4 × 3 = 60. Double-check each dimension.
Coordinate Geometry
★ KEY MEMORY · "Distance Formula: d = √[(x₂−x₁)² + (y₂−y₁)²] · Midpoint: ((x₁+x₂)/2, (y₁+y₂)/2)"
Q17
Coordinates
What is the midpoint of the segment connecting (2, 6) and (8, 4)?
Correct answer: C (5, 5)
Midpoint x = (2+8)/2 = 5. Midpoint y = (6+4)/2 = 5. → (5, 5).
Trap A: (10, 10) is the sum, not the average. Always divide by 2 for the midpoint!
Midpoint x = (2+8)/2 = 5. Midpoint y = (6+4)/2 = 5. → (5, 5).
Trap A: (10, 10) is the sum, not the average. Always divide by 2 for the midpoint!
Triangle Angles
★ KEY MEMORY · "All angles in a triangle add up to EXACTLY 180°"
Q18
Tricky ⚠
OR: Find interior angle 3 first → 180 − 45 − 75 = 60°
Then exterior angle = 180 − 60 = ?
A triangle has angles of 45° and 75°. What is the exterior angle at the third vertex?
Exterior Angle Theorem
Exterior angle = sum of the two non-adjacent interior anglesOR: Find interior angle 3 first → 180 − 45 − 75 = 60°
Then exterior angle = 180 − 60 = ?
Correct answer: B (120°)
Interior angle = 180 − 45 − 75 = 60°. Exterior angle = 180 − 60 = 120°.
Or: Exterior = 45 + 75 = 120° (Exterior Angle Theorem shortcut!).
Trap A: 60° is the interior angle. The exterior angle is its supplement on a straight line.
Interior angle = 180 − 45 − 75 = 60°. Exterior angle = 180 − 60 = 120°.
Or: Exterior = 45 + 75 = 120° (Exterior Angle Theorem shortcut!).
Trap A: 60° is the interior angle. The exterior angle is its supplement on a straight line.
Scale & Similar Figures
★ KEY MEMORY · "Similar figures have EQUAL angles and PROPORTIONAL sides"
Q19
Similar Figures
Then area ratio = a² : b²
So ratio = 2² : 5² = 4 : 25
If small area = 8, large area = ?
Two similar triangles have sides in ratio 2 : 5. The smaller triangle has area 8 cm². What is the area of the larger triangle?
Area ratio rule
If sides ratio = a : bThen area ratio = a² : b²
So ratio = 2² : 5² = 4 : 25
If small area = 8, large area = ?
Correct answer: D (50 cm²)
Area ratio = 4 : 25. If small = 8, then 8/4 = 2 (unit value). Large = 25 × 2 = 50 cm².
Trap A: Just multiplying 8 × (5/2) = 20 — wrong! Areas scale by the square of the side ratio, not the side ratio itself.
Area ratio = 4 : 25. If small = 8, then 8/4 = 2 (unit value). Large = 25 × 2 = 50 cm².
Trap A: Just multiplying 8 × (5/2) = 20 — wrong! Areas scale by the square of the side ratio, not the side ratio itself.
Surface Area
★ KEY MEMORY · "Surface Area = sum of ALL faces · cube has 6 equal square faces"
Q20
Tricky ⚠
A cube has a side length of 4 cm. What is its total surface area?
Correct answer: C (96 cm²)
Each face = 4² = 16 cm². A cube has 6 faces → 6 × 16 = 96 cm².
Trap A: 16 is just one face. Trap B: 64 is the volume (4³). Surface area ≠ volume!
Each face = 4² = 16 cm². A cube has 6 faces → 6 × 16 = 96 cm².
Trap A: 16 is just one face. Trap B: 64 is the volume (4³). Surface area ≠ volume!