Multiply first: \(4 \times 2 = 8\). Then left to right: \(3 + 8 - 1 = \mathbf{10}\). The trap is adding 3+4 first — always multiply before adding!
02
Pre-Algebra
Negative Numbers
What is \((-3) \times (-5)\)?
🔑SIGN RULE — Same signs → Positive · Different signs → Negative
Pattern
\((-2)(-4) = +8\) · \((-2)(+4) = -8\)
💡 Explanation
Both signs are negative (same) → result is positive. \(3 \times 5 = 15\), so \((-3)(-5) = +15\). Many students write −15 because they see two negatives — remember, two negatives cancel each other!
03
Pre-Algebra
Solving One-Step Equations
Solve for \(x\): \(x - 7 = 12\)
🔑INVERSE OP — Do the opposite operation to both sides to isolate \(x\)
Example
\(x - 3 = 9 \Rightarrow x = 9 + 3 = 12\)
💡 Explanation
Add 7 to both sides: \(x - 7 + 7 = 12 + 7\), so \(x = \mathbf{19}\). Check: \(19 - 7 = 12\) ✓. A common mistake is subtracting 7 and getting 5.
04
Pre-Algebra
Fractions — Multiplication
\(\dfrac{2}{3} \times \dfrac{9}{4} = \; ?\)
🔑CROSS-CANCEL — Simplify diagonally before multiplying to keep numbers small
If 3 pencils cost $1.50, how much do 7 pencils cost?
🔑UNIT RATE — Find cost per 1 item, then multiply
Strategy
4 apples = $2 → 1 apple = $0.50 → 9 apples = $4.50
💡 Explanation
Unit rate: \(\$1.50 \div 3 = \$0.50\) per pencil. For 7 pencils: \(7 \times \$0.50 = \mathbf{\$3.50}\). Cross-multiply also works: \(\frac{1.50}{3} = \frac{x}{7} \Rightarrow x = \frac{7 \times 1.50}{3} = 3.50\).
06
Pre-Algebra
Exponents
Simplify: \(2^3 \times 2^4\)
🔑SAME BASE → ADD — \(a^m \times a^n = a^{m+n}\)
Example
\(5^2 \times 5^3 = 5^{2+3} = 5^5\)
💡 Explanation
Same base (2), so add the exponents: \(2^3 \times 2^4 = 2^{3+4} = \mathbf{2^7} = 128\). Do NOT multiply the exponents (that's the power rule: \((2^3)^4 = 2^{12}\)).
07
Pre-Algebra
Percent
What is 30% of 80?
🔑OF = × — Convert % to decimal: 30% = 0.30, then multiply
Shortcut
10% of 80 = 8 → 30% = 3 × 8 = 24
💡 Explanation
\(0.30 \times 80 = \mathbf{24}\). Or: 10% of 80 is 8, so 30% = 3 × 8 = 24. Choosing 2.4 means forgetting to multiply by 80 (just moved the decimal).
08
Pre-Algebra
Distributive Property
Expand: \(4(3x - 5)\)
🔑DISTRIBUTE — Multiply the outside number by each term inside
Distribute 4 to each term: \(4 \times 3x = 12x\) and \(4 \times (-5) = -20\). So the answer is \(\mathbf{12x - 20}\). Forgetting to multiply 4 by the −5 gives the wrong answer 12x − 5.
09
Pre-Algebra
Inequalities
Solve: \(-2x > 10\)
🔑FLIP IT — When dividing or multiplying by a negative, reverse the inequality sign
Divide both sides by −2 and flip the sign: \(\frac{-2x}{-2} < \frac{10}{-2}\) → \(x < \mathbf{-5}\). The #1 mistake is forgetting to flip: choosing \(x > -5\).
10
Pre-Algebra
Two-Step Equations
Solve: \(3x + 6 = 21\)
🔑UNDO IN REVERSE — Subtract/add first, then multiply/divide
Step 1: Subtract 6 from both sides → \(3x = 15\). Step 2: Divide by 3 → \(x = \mathbf{5}\). Check: \(3(5)+6 = 15+6 = 21\) ✓
Geometry
Questions 11–20
11
Geometry
Pythagorean Theorem
A right triangle has legs \(a = 6\) and \(b = 8\). Find the hypotenuse \(c\).
🔑PYTHAGORAS — \(a^2 + b^2 = c^2\) · Hypotenuse = longest side (opposite right angle)
Famous Triple
3-4-5: \(3^2+4^2 = 9+16 = 25 = 5^2\) ✓ 6-8-10 is a scaled version!
💡 Explanation
\(6^2 + 8^2 = 36 + 64 = 100 = c^2\), so \(c = \sqrt{100} = \mathbf{10}\). This is the 3-4-5 triple scaled by 2. Many students forget to take the square root and answer 100.
12
Geometry
Area of a Triangle
A triangle has base \(b = 10\) cm and height \(h = 7\) cm. What is its area?
🔑HALF BASE-HEIGHT — \(A = \dfrac{1}{2} \times b \times h\) · Height must be perpendicular to base
Example
Base = 6, Height = 4 → \(A = \frac{1}{2}(6)(4) = 12\text{ cm}^2\)
💡 Explanation
\(A = \frac{1}{2} \times 10 \times 7 = \frac{70}{2} = \mathbf{35 \text{ cm}^2}\). Forgetting the \(\frac{1}{2}\) gives 70 — the most common error here!
13
Geometry
Angles — Triangle Sum
Two angles of a triangle are 55° and 80°. What is the third angle?
🔑TRIANGLE SUM = 180° — Always. No exceptions.
Formula
Third angle = \(180° - \text{(sum of other two)}\)
A circle has radius \(r = 5\). Find its circumference. (Use \(\pi \approx 3.14\))
🔑C = 2πr (or πd) · Don't mix up circumference (perimeter) with area (\(\pi r^2\))
Quick check
r = 1 → C ≈ 6.28 (just over 6) · r = 10 → C ≈ 62.8
💡 Explanation
\(C = 2\pi r = 2 \times 3.14 \times 5 = \mathbf{31.4}\). Also correct as \(10\pi\). The trap answer 78.5 is the area (\(\pi r^2 = 3.14 \times 25\)), not the circumference!
15
Geometry
Complementary & Supplementary Angles
Angle A and Angle B are supplementary. If Angle A = 113°, find Angle B.
🔑S for Straight (180°) — Supplementary = 180° · C for Corner (90°) — Complementary = 90°
Memory trick
"S" is a bigger letter → Supplementary (180°) is the bigger sum
💡 Explanation
Supplementary means \(A + B = 180°\). So \(B = 180° - 113° = \mathbf{67°}\). If you got 23°, you used 90° instead of 180° — that's complementary.
16
Geometry
Volume of a Rectangular Prism
A box is 4 cm long, 3 cm wide, and 5 cm tall. What is its volume?
🔑V = l × w × h — Multiply ALL three dimensions · Answer in cubic units (cm³)
Versus
Surface Area = sum of all 6 face areas · Volume = space inside
💡 Explanation
\(V = 4 \times 3 \times 5 = \mathbf{60 \text{ cm}^3}\). The answer 94 cm³ is the surface area (\(2(lw+lh+wh) = 2(12+20+15) = 94\)), not the volume.
17
Geometry
Parallel Lines & Transversals
Two parallel lines are cut by a transversal. One angle is 70°. What is the measure of its alternate interior angle?
🔑Z-ANGLE — Alternate interior angles are EQUAL (they form a Z shape)
Three pairs to know
Corresponding = equal (F-shape) · Alternate interior = equal (Z-shape) · Co-interior = 180° (C-shape)
💡 Explanation
Alternate interior angles between parallel lines are always equal. So the answer is \(\mathbf{70°}\). Co-interior (same-side interior) angles are supplementary (= 180°), which is where 110° comes from — a different pair!
18
Geometry
Area of a Circle
A circle has diameter \(d = 10\). What is its area? (Use \(\pi \approx 3.14\))
🔑DIAMETER ÷ 2 = RADIUS — Area uses radius: \(A = \pi r^2\). Don't plug in diameter!
Common trap
d = 10 → r = 5 (NOT 10) → \(A = \pi(5)^2 = 25\pi\)
💡 Explanation
Diameter = 10, so radius \(r = 5\). \(A = \pi r^2 = 3.14 \times 25 = \mathbf{78.5}\). Using diameter instead of radius gives \(3.14 \times 100 = 314\) — the #1 mistake!
19
Geometry
Coordinate Geometry — Distance
Find the distance between points \(A(1, 2)\) and \(B(4, 6)\).
🔑DISTANCE = PYTHAGORAS — \(d = \sqrt{(\Delta x)^2 + (\Delta y)^2}\) where Δ means "change in"
\(\Delta x = 3,\; \Delta y = 4\). \(d = \sqrt{3^2 + 4^2} = \sqrt{9+16} = \sqrt{25} = \mathbf{5}\). This is the 3-4-5 right triangle! Forgetting the square root gives 25.
20
Geometry
Similar Triangles
Triangle ABC is similar to Triangle DEF. Side AB = 6, side DE = 9, and side BC = 8. Find side EF.
🔑SCALE FACTOR — Corresponding sides are proportional: \(\dfrac{AB}{DE} = \dfrac{BC}{EF}\)
Strategy
Find ratio first: \(\frac{6}{9} = \frac{2}{3}\), so each side of DEF = \(\frac{3}{2}\) × ABC side
💡 Explanation
Scale factor: \(\frac{DE}{AB} = \frac{9}{6} = \frac{3}{2}\). So \(EF = BC \times \frac{3}{2} = 8 \times \frac{3}{2} = \mathbf{12}\). Cross-multiply: \(\frac{6}{9} = \frac{8}{EF} \Rightarrow 6 \cdot EF = 72 \Rightarrow EF = 12\).