Math Mastery · Pre-Algebra & Geometry

Pre-Algebra

Questions 1–10 Algebra

Q 01 Order of Operations

⚡ Memory Key

PEMDAS — Parentheses · Exponents · Multiply/Divide · Add/Subtract (left→right)

Evaluate: 3 + 6 × (5 + 4) ÷ 3 − 7

Explanation

✓ Correct Answer: A — 14

Step 1: Parentheses first → 5 + 4 = 9
Step 2: Left to right for × and ÷ → 6 × 9 = 54, then 54 ÷ 3 = 18
Step 3: Add/Subtract → 3 + 18 − 7 = 14
Trap: Many students do 3 + 6 = 9 first — wrong! Multiplication comes before addition.

Q 02 Negative Numbers

⚡ Memory Key

SAME signs → ADD · DIFFERENT signs → SUBTRACT (keep the sign of the bigger absolute value)

What is −8 + (−5) − (−3)?

Explanation

✓ Correct Answer: B — −10

Rewrite: −8 + (−5) − (−3) = −8 − 5 + 3
Subtracting a negative = adding a positive!
−8 − 5 = −13, then −13 + 3 = −10
Trap: Forgetting that −(−3) = +3 is the most common mistake here.

Q 03 Fractions

⚡ Memory Key

KCF — Keep · Change · Flip (for division of fractions)

Simplify: 34 ÷ 916

Explanation

✓ Correct Answer: C — 4/3

KCF: Keep 3/4, Change ÷ to ×, Flip 9/16 → 16/9
(3/4) × (16/9) = 48/36 = 4/3
Cancel first: 3 and 9 → 1/3; 4 and 16 → 4/1
Result: 4/3

Q 04 Solving Equations

⚡ Memory Key

ISOLATE x — Do the same operation to BOTH sides to keep balance

Solve for x: 4x − 7 = 2x + 11

Explanation

✓ Correct Answer: D — x = 9

Move x terms left: 4x − 2x = 11 + 7
2x = 18 → x = 9
Check: 4(9) − 7 = 29 and 2(9) + 11 = 29 ✓
Trap: Forgetting to move the constant to the right side too.

Q 05 Percentages

⚡ Memory Key

IS/OF = %/100 — "IS" is the part, "OF" is the whole

A shirt costs $40. It's on sale for 35% off. What is the sale price?

Explanation

✓ Correct Answer: A — $26

Discount = 35% of $40 = 0.35 × 40 = $14
Sale price = $40 − $14 = $26
Shortcut: You pay 65% → 0.65 × 40 = $26
Trap: Answer B is just the discount, not the sale price!

Q 06 Ratios & Proportions

⚡ Memory Key

CROSS-MULTIPLY — a/b = c/d → a×d = b×c

A car travels 150 miles in 3 hours. At the same rate, how far will it travel in 5 hours?

Explanation

✓ Correct Answer: B — 250 miles

Rate = 150 ÷ 3 = 50 mph
Distance = 50 × 5 = 250 miles
Or: 150/3 = x/5 → 3x = 750 → x = 250

Q 07 Exponents

⚡ Memory Key

SAME BASE → ADD exponents (×) · SUBTRACT exponents (÷)

Simplify: 2³ × 2⁴ ÷ 2²

Explanation

✓ Correct Answer: C — 2⁵ = 32

Multiply: 2³ × 2⁴ = 2^(3+4) = 2⁷
Divide: 2⁷ ÷ 2² = 2^(7−2) = 2⁵
2⁵ = 32
Trap: Never multiply the bases! 2 stays as 2, only exponents change.

Q 08 Inequalities

⚡ Memory Key

FLIP the sign when multiplying or dividing by a NEGATIVE number

Solve: −3x + 6 > 15. Which value of x is a solution?

Explanation

✓ Correct Answer: A — x = −5

−3x + 6 > 15 → −3x > 9
Divide by −3 → FLIP the sign! → x < −3
Only x = −5 satisfies x < −3
Trap: NOT flipping the inequality sign when dividing by −3 is the #1 error.

Q 09 Distributive Property

⚡ Memory Key

DISTRIBUTE — a(b + c) = ab + ac. Multiply the outside by EVERY term inside.

Expand and simplify: 2(3x − 4) + 5(x + 2)

Explanation

✓ Correct Answer: D — 11x + 2

2(3x − 4) = 6x − 8
5(x + 2) = 5x + 10
Combine: 6x + 5x = 11x, −8 + 10 = 2
Answer: 11x + 2
Trap: Forgetting to distribute the negative: 2(3x − 4) ≠ 6x − 4

Q 10 Word Problems · Two Variables

⚡ Memory Key

NAME → EQUATION — Assign a variable to each unknown, then write the equation

A class has 32 students. There are 4 more girls than boys. How many girls are in the class?

Explanation

✓ Correct Answer: C — 18 girls

Let boys = b, girls = b + 4
b + (b + 4) = 32 → 2b + 4 = 32 → 2b = 28 → b = 14
Girls = 14 + 4 = 18
Check: 14 + 18 = 32 ✓

Geometry

Questions 11–20 Geometry

Q 11 Pythagorean Theorem

⚡ Memory Key

a² + b² = c² — c is always the HYPOTENUSE (longest side, opposite right angle)

A right triangle has legs of length 6 and 8. What is the length of the hypotenuse?

Explanation

✓ Correct Answer: B — 10

6² + 8² = c² → 36 + 64 = 100 → c = √100 = 10
This is a 3-4-5 triple × 2 = 6-8-10. Memorize common triples: 3-4-5, 5-12-13, 8-15-17.

Q 12 Area of Triangle

⚡ Memory Key

A = ½ × base × height — height must be PERPENDICULAR to the base

A triangle has a base of 14 cm and a height of 9 cm. What is its area?

Explanation

✓ Correct Answer: C — 63 cm²

A = ½ × 14 × 9 = ½ × 126 = 63 cm²
Trap: Answer A = 126 is the rectangle area (forgetting the ½). Always halve for a triangle!

Q 13 Circle — Circumference & Area

⚡ Memory Key

C = 2πr · A = πr² — Circumference uses r once, Area uses r SQUARED

A circle has a diameter of 10 cm. What is its area? (Use π ≈ 3.14)

Explanation

✓ Correct Answer: A — 78.5 cm²

Diameter = 10, so radius = 10 ÷ 2 = 5
A = π × 5² = 3.14 × 25 = 78.5 cm²
Trap: Using diameter (10) instead of radius (5) → gives 314, which is answer C!

Q 14 Angles — Supplementary & Complementary

⚡ Memory Key

C comes before S · 90 comes before 180 — Complementary = 90°, Supplementary = 180°

Two angles are supplementary. One angle is 3 times the other. Find both angles.

Explanation

✓ Correct Answer: B — 45° and 135°

Supplementary = sum of 180°
Let angle = x → other = 3x
x + 3x = 180 → 4x = 180 → x = 45°
Angles: 45° and 135°. Check: 45 + 135 = 180° ✓

Q 15 Perimeter of Rectangle

⚡ Memory Key

P = 2(l + w) — Add length and width FIRST, then multiply by 2

A garden is rectangular with a perimeter of 56 m. The length is 5 m more than the width. What is the width?

Explanation

✓ Correct Answer: D — 11.5 m

Let width = w, length = w + 5
2(w + w + 5) = 56 → 2(2w + 5) = 56
4w + 10 = 56 → 4w = 46 → w = 11.5 m
Check: length = 16.5, 2(11.5 + 16.5) = 2(28) = 56 ✓

Q 16 Volume of Rectangular Prism

⚡ Memory Key

V = l × w × h — Volume fills 3D space, always use cubic units (cm³, m³)

A box is 8 cm long, 5 cm wide, and 4 cm tall. What is its volume?

Explanation

✓ Correct Answer: C — 160 cm³

V = 8 × 5 × 4 = 160 cm³
Trap: Answer A = 8 × 5 × 2 (forgetting to use actual height). Always multiply ALL three dimensions.

Q 17 Parallel Lines & Transversals

⚡ Memory Key

CO-INTERIOR = 180° · ALTERNATE = EQUAL · CORRESPONDING = EQUAL

Two parallel lines are cut by a transversal. One angle is 65°. What is the measure of its co-interior (same-side interior) angle?

Explanation

✓ Correct Answer: D — 115°

Co-interior angles (also called consecutive interior angles) are supplementary.
65° + x = 180° → x = 115°
Trap: Many confuse this with alternate interior angles (which ARE equal = 65°).

Q 18 Interior Angles of Polygon

⚡ Memory Key

Sum = (n − 2) × 180° — n is the number of sides

What is the sum of interior angles of a hexagon (6-sided polygon)?

Explanation

✓ Correct Answer: B — 720°

(6 − 2) × 180° = 4 × 180° = 720°
Reference: Triangle = 180°, Quad = 360°, Pentagon = 540°, Hexagon = 720°
Each additional side adds 180°.

Q 19 Scale & Similarity

⚡ Memory Key

SIMILAR = same shape, different size → corresponding sides are PROPORTIONAL

Two similar triangles have sides in the ratio 3 : 5. The smaller triangle has a perimeter of 24 cm. What is the perimeter of the larger triangle?

Explanation

✓ Correct Answer: A — 40 cm

Perimeters scale by the same ratio as sides.
24/x = 3/5 → 3x = 120 → x = 40 cm
Or: 24 × (5/3) = 40 cm

Q 20 Coordinate Geometry — Distance

⚡ Memory Key

d = √[(x₂−x₁)² + (y₂−y₁)²] — It's Pythagorean theorem on the coordinate plane!

Find the distance between points A(1, 2) and B(4, 6).

Explanation

✓ Correct Answer: C — 5

d = √[(4−1)² + (6−2)²]
= √[3² + 4²] = √[9 + 16] = √25 = 5
It's a 3-4-5 right triangle on the coordinate plane!

Pre-Algebra

Geometry

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