Pre-Algebra
Numbers, Variables & Equations
Core topics: integers, fractions, ratios, expressions, solving equations
Evaluate: 3 + 4 × 2 − 1
Be careful — multiplication comes before addition!
⚡ Memory Key
PEMDAS — Parentheses · Exponents · Multiply · Divide · Add · Subtract
Always do × ÷ BEFORE + −
Always do × ÷ BEFORE + −
💡 Explanation
Step 1: Do multiplication first → 4 × 2 = 8
Step 2: Left to right → 3 + 8 − 1 = 10
Common mistake: doing 3 + 4 = 7 first, then × 2 = 14 (wrong — ignore left-to-right when × is involved with +)
Step 2: Left to right → 3 + 8 − 1 = 10
Common mistake: doing 3 + 4 = 7 first, then × 2 = 14 (wrong — ignore left-to-right when × is involved with +)
What is −3 × (−4) + (−5)?
⚡ Memory Key
SAME signs → Positive | DIFFERENT signs → Negative
(−)(−) = + (+)(−) = −
(−)(−) = + (+)(−) = −
💡 Explanation
Step 1: −3 × (−4) = +12 (same signs = positive)
Step 2: 12 + (−5) = 12 − 5 = 7
Trap: Many students say 12 + 5 = 17 (forgetting the negative on −5)
Step 2: 12 + (−5) = 12 − 5 = 7
Trap: Many students say 12 + 5 = 17 (forgetting the negative on −5)
Simplify: 2/3 + 3/4
⚡ Memory Key
LCD = Least Common Denominator → find the smallest number both denominators divide into evenly.
3 × 4 → LCD = 12 | Then convert both fractions to /12
3 × 4 → LCD = 12 | Then convert both fractions to /12
💡 Explanation
LCD = 12
2/3 = 8/12 3/4 = 9/12
8/12 + 9/12 = 17/12
Trap B: Adding numerators AND denominators (2+3)/(3+4) — NEVER do this!
2/3 = 8/12 3/4 = 9/12
8/12 + 9/12 = 17/12
Trap B: Adding numerators AND denominators (2+3)/(3+4) — NEVER do this!
Solve for x: 2x + 5 = 13
⚡ Memory Key
ISOLATE the variable — undo operations in reverse PEMDAS order.
First undo + or −, then undo × or ÷
First undo + or −, then undo × or ÷
💡 Explanation
2x + 5 = 13
Step 1: Subtract 5 from both sides → 2x = 8
Step 2: Divide both sides by 2 → x = 4
Check: 2(4) + 5 = 8 + 5 = 13 ✓
Step 1: Subtract 5 from both sides → 2x = 8
Step 2: Divide both sides by 2 → x = 4
Check: 2(4) + 5 = 8 + 5 = 13 ✓
If the ratio of boys to girls in a class is 3:5, and there are 15 boys, how many girls are there?
⚡ Memory Key
PROPORTION = two equal ratios → set up a/b = c/d, then CROSS-MULTIPLY
3/5 = 15/x → 3x = 75 → x = 25
3/5 = 15/x → 3x = 75 → x = 25
💡 Explanation
Set up proportion: 3/5 = 15/x
Cross-multiply: 3x = 5 × 15 = 75
x = 75 ÷ 3 = 25 girls
Trap A: Some students multiply 3 × 3 = 9 (just adding the ratio value, not scaling it)
Cross-multiply: 3x = 5 × 15 = 75
x = 75 ÷ 3 = 25 girls
Trap A: Some students multiply 3 × 3 = 9 (just adding the ratio value, not scaling it)
What is 35% of 80?
Percents appear on every test. Master this formula!
⚡ Memory Key
% formula: Part = Percent × Whole
Translate: "of" = × | "is" = = | "what" = variable
35% × 80 = 0.35 × 80
Translate: "of" = × | "is" = = | "what" = variable
35% × 80 = 0.35 × 80
💡 Explanation
35% = 35/100 = 0.35
0.35 × 80 = 28
Or: 10% of 80 = 8, so 30% = 24, 5% = 4 → 24 + 4 = 28 ✓
0.35 × 80 = 28
Or: 10% of 80 = 8, so 30% = 24, 5% = 4 → 24 + 4 = 28 ✓
Expand: 3(2x − 4)
⚡ Memory Key
DISTRIBUTE = multiply the outside number to every term inside.
a(b + c) = ab + ac | Don't forget to distribute the sign!
a(b + c) = ab + ac | Don't forget to distribute the sign!
💡 Explanation
3 × 2x = 6x 3 × (−4) = −12
Result: 6x − 12
Trap B: Only multiplying the first term (3 × 2x = 6x, but leaving −4 unchanged)
Trap D: Forgetting the negative sign → 3 × 4 = +12 (wrong sign)
Result: 6x − 12
Trap B: Only multiplying the first term (3 × 2x = 6x, but leaving −4 unchanged)
Trap D: Forgetting the negative sign → 3 × 4 = +12 (wrong sign)
Which is equal to 2³ × 2²?
⚡ Memory Key
PRODUCT RULE: Same base → ADD the exponents
aᵐ × aⁿ = aᵐ⁺ⁿ | Do NOT multiply the exponents (that's Power Rule)
aᵐ × aⁿ = aᵐ⁺ⁿ | Do NOT multiply the exponents (that's Power Rule)
💡 Explanation
Same base (2) → add exponents: 3 + 2 = 5
So 2³ × 2² = 2⁵ = 32
Trap A: Multiplying exponents 3 × 2 = 6 (that's the Power Rule, for (2³)²)
So 2³ × 2² = 2⁵ = 32
Trap A: Multiplying exponents 3 × 2 = 6 (that's the Power Rule, for (2³)²)
Simplify: 5x + 3y − 2x + y
⚡ Memory Key
LIKE TERMS = same variable AND same exponent.
Only combine terms with the same "letter-power". x and y are different — can't combine!
Only combine terms with the same "letter-power". x and y are different — can't combine!
💡 Explanation
x terms: 5x − 2x = 3x
y terms: 3y + y = 4y
Answer: 3x + 4y
Trap A: Combining x and y together (they're NOT like terms — different variables)
y terms: 3y + y = 4y
Answer: 3x + 4y
Trap A: Combining x and y together (they're NOT like terms — different variables)
Solve: −2x < 8
This is the #1 most-missed inequality question!
⚡ Memory Key
FLIP the inequality sign when you multiply or divide by a NEGATIVE number.
< becomes > | > becomes < | This is the #1 trap!
< becomes > | > becomes < | This is the #1 trap!
💡 Explanation
−2x < 8
Divide both sides by −2 → FLIP the sign!
x > 8 ÷ (−2) = −4
Answer: x > −4
Trap B: Forgetting to flip the < to > when dividing by −2
Divide both sides by −2 → FLIP the sign!
x > 8 ÷ (−2) = −4
Answer: x > −4
Trap B: Forgetting to flip the < to > when dividing by −2
Geometry
Shapes, Angles & Measurement
Core topics: angles, triangles, area, perimeter, circles, coordinate geometry
Two angles are supplementary. One measures 65°. What is the other?
⚡ Memory Key
SUPPlementary = 180° (think: S for Straight line)
COMPlementary = 90° (think: C for Corner)
Sup → 180 | Comp → 90
COMPlementary = 90° (think: C for Corner)
Sup → 180 | Comp → 90
💡 Explanation
Supplementary = sum is 180°
180° − 65° = 115°
Trap A: Using 90° (that's complementary, not supplementary) → 90 − 65 = 35
180° − 65° = 115°
Trap A: Using 90° (that's complementary, not supplementary) → 90 − 65 = 35
A triangle has angles of 45° and 80°. Find the third angle.
⚡ Memory Key
TRIANGLE SUM THEOREM: All three angles of any triangle always add up to 180°.
Missing angle = 180 − (sum of other two)
Missing angle = 180 − (sum of other two)
💡 Explanation
Sum of all angles = 180°
Third angle = 180° − 45° − 80° = 55°
Trap D: 180 − 45 only = 135, then forgetting to subtract 80
Third angle = 180° − 45° − 80° = 55°
Trap D: 180 − 45 only = 135, then forgetting to subtract 80
A triangle has base 10 cm and height 6 cm. What is its area?
⚡ Memory Key
Area of Triangle = ½ × base × height
The height must be PERPENDICULAR to the base (straight up, not the slanted side!)
The height must be PERPENDICULAR to the base (straight up, not the slanted side!)
💡 Explanation
A = ½ × b × h = ½ × 10 × 6 = ½ × 60 = 30 cm²
Trap B: Forgetting the ½ → 10 × 6 = 60 (this is a rectangle, not a triangle!)
Trap B: Forgetting the ½ → 10 × 6 = 60 (this is a rectangle, not a triangle!)
A right triangle has legs of 3 and 4. What is the hypotenuse?
⚡ Memory Key
PYTHAGOREAN THEOREM: a² + b² = c² (only for right triangles!)
c = hypotenuse (longest side, opposite the right angle)
Memorize: 3-4-5 and 5-12-13 are classic Pythagorean triples!
c = hypotenuse (longest side, opposite the right angle)
Memorize: 3-4-5 and 5-12-13 are classic Pythagorean triples!
💡 Explanation
a² + b² = c²
3² + 4² = 9 + 16 = 25
c = √25 = 5
Trap A: Adding legs directly 3 + 4 = 7 (not how Pythagorean theorem works!)
3² + 4² = 9 + 16 = 25
c = √25 = 5
Trap A: Adding legs directly 3 + 4 = 7 (not how Pythagorean theorem works!)
A circle has a radius of 7 cm. What is its circumference? (Use π ≈ 3.14)
⚡ Memory Key
Circumference = 2πr or πd | Area = πr²
r = radius, d = diameter = 2r
Tip: C has 2 letters → C = 2πr uses 2 | Area = πr² uses r²
r = radius, d = diameter = 2r
Tip: C has 2 letters → C = 2πr uses 2 | Area = πr² uses r²
💡 Explanation
C = 2 × π × r = 2 × 3.14 × 7 = 43.96 cm
Trap B: Using area formula πr² = 3.14 × 49 = 153.86 (that's AREA, not circumference)
Trap D: Using C = πr instead of 2πr
Trap B: Using area formula πr² = 3.14 × 49 = 153.86 (that's AREA, not circumference)
Trap D: Using C = πr instead of 2πr
Two parallel lines are cut by a transversal. One of the alternate interior angles is 72°. What is the other alternate interior angle?
⚡ Memory Key
ALTERNATE INTERIOR angles = EQUAL (Z-shape in the lines)
CO-INTERIOR (same-side) = adds up to 180°
CORRESPONDING angles = EQUAL (F-shape)
CO-INTERIOR (same-side) = adds up to 180°
CORRESPONDING angles = EQUAL (F-shape)
💡 Explanation
Alternate interior angles are always equal when lines are parallel.
Answer: 72°
Trap B: Students confuse with co-interior angles (same-side) which = 180° − 72° = 108°
Answer: 72°
Trap B: Students confuse with co-interior angles (same-side) which = 180° − 72° = 108°
A circle has diameter 10 cm. What is its area? (π ≈ 3.14)
Watch out — they gave you the DIAMETER, not the radius!
⚡ Memory Key
DIAMETER → RADIUS FIRST! r = d ÷ 2
If diameter = 10, then radius = 5. THEN use A = πr²
This "diameter trap" appears on almost every test.
If diameter = 10, then radius = 5. THEN use A = πr²
This "diameter trap" appears on almost every test.
💡 Explanation
Diameter = 10 → radius = 5
A = π × r² = 3.14 × 25 = 78.5 cm²
Trap A: Using diameter directly → 3.14 × 10² = 314 (used d instead of r — classic mistake!)
A = π × r² = 3.14 × 25 = 78.5 cm²
Trap A: Using diameter directly → 3.14 × 10² = 314 (used d instead of r — classic mistake!)
A box is 5 cm long, 4 cm wide, and 3 cm tall. What is its volume?
⚡ Memory Key
VOLUME = length × width × height
Area is 2D (cm²) | Volume is 3D (cm³) — always include the cube unit!
Think: l×w = base area, then × h = how tall you stack it
Area is 2D (cm²) | Volume is 3D (cm³) — always include the cube unit!
Think: l×w = base area, then × h = how tall you stack it
💡 Explanation
V = l × w × h = 5 × 4 × 3 = 60 cm³
Trap A: Only multiplying two dimensions (4 × 3 × 2 = 24 — added instead of multiplied)
Trap B: Adding all three dimensions 5 + 4 + 3 = 12 (that's not volume!)
Trap A: Only multiplying two dimensions (4 × 3 × 2 = 24 — added instead of multiplied)
Trap B: Adding all three dimensions 5 + 4 + 3 = 12 (that's not volume!)
Find the midpoint of the segment with endpoints (2, 4) and (8, 10).
⚡ Memory Key
MIDPOINT = average of x-values, average of y-values
M = ((x₁+x₂)/2, (y₁+y₂)/2)
Think: "middle = average" — just like finding the middle of two numbers on a number line
M = ((x₁+x₂)/2, (y₁+y₂)/2)
Think: "middle = average" — just like finding the middle of two numbers on a number line
💡 Explanation
x: (2 + 8)/2 = 10/2 = 5
y: (4 + 10)/2 = 14/2 = 7
Midpoint = (5, 7)
Trap C: Adding but not dividing by 2 → (2+8, 4+10) = (10, 14)
y: (4 + 10)/2 = 14/2 = 7
Midpoint = (5, 7)
Trap C: Adding but not dividing by 2 → (2+8, 4+10) = (10, 14)
Two similar triangles: Triangle A has sides 3, 4, 5. Triangle B is similar with its shortest side = 6. What is the longest side of Triangle B?
⚡ Memory Key
SIMILAR = same shape, different size. All sides scale by the same SCALE FACTOR.
Find the scale factor first: new short side ÷ old short side = 6 ÷ 3 = 2
Then multiply ALL sides by that factor.
Find the scale factor first: new short side ÷ old short side = 6 ÷ 3 = 2
Then multiply ALL sides by that factor.
💡 Explanation
Scale factor = 6 ÷ 3 = 2
Longest side of A = 5
Longest side of B = 5 × 2 = 10
Trap D: Adding scale factor instead of multiplying → 5 + (6−3) = 8, or misidentifying sides
Longest side of A = 5
Longest side of B = 5 × 2 = 10
Trap D: Adding scale factor instead of multiplying → 5 + (6−3) = 8, or misidentifying sides