Math Mastery — Pre-Algebra & Geometry

1

Order of Operations · PEMDAS

Evaluate: 3 + 4 × 2 − (6 ÷ 3)

⚡ Please Excuse My Dear Aunt Sally → Parentheses · Exponents · Multiply/Divide · Add/Subtract

📘 Solution

Step 1 — Parentheses first: (6 ÷ 3) = 2
Step 2 — Multiply: 4 × 2 = 8
Step 3 — Left to right: 3 + 8 − 2 = 9

Trap: many students add 3 + 4 first, getting 28. Always multiply before adding!

2

Integer Operations · Negatives

What is the value of −3 × (−4) + (−5)?

⚡ neg × neg = pos | neg × pos = neg → Same signs: +, Different signs: −

📘 Solution

Step 1 — Multiply: (−3) × (−4) = +12 (negative × negative = positive)
Step 2 — Add: 12 + (−5) = 12 − 5 = 7

Trap: Students often forget that (−) × (−) = (+). Keep the sign rules handy!

3

Fractions · Adding Unlike Denominators

Simplify: 23 + 56

⚡ LCD → Convert → Add → Simplify → Find Least Common Denominator first

📘 Solution

LCD of 3 and 6 is 6.
Convert: 2/3 = 4/6
Add: 4/6 + 5/6 = 9/6
Simplify: 9/6 = 1½

Trap: Adding denominators directly (3 + 6 = 9) is the #1 fraction mistake!

4

Solving One-Step Equations

If 5x − 3 = 22, what is the value of x?

⚡ ISOLATE → INVERSE OPERATIONS → Undo addition/subtraction first, then multiplication/division

📘 Solution

5x − 3 = 22
Add 3 to both sides: 5x = 25
Divide by 5: x = 5
Check: 5(5) − 3 = 25 − 3 = 22 ✓

Always CHECK your answer by substituting back into the original equation!

5

Ratios & Proportions

A car travels 180 miles in 3 hours. At the same rate, how many miles does it travel in 5 hours?

⚡ UNIT RATE × TIME = DISTANCE → Find rate per 1 unit first, then scale up

📘 Solution

Unit rate: 180 ÷ 3 = 60 miles per hour
In 5 hours: 60 × 5 = 300 miles
(Or use proportion: 180/3 = x/5 → 3x = 900 → x = 300)

Setting up the proportion correctly is key — keep the same unit in the same position!

6

Percent · Increase & Decrease

A shirt costs $40. It is on sale for 25% off. What is the sale price?

⚡ Sale Price = Original × (1 − rate) → 25% off means you pay 75% of original

📘 Solution

Discount amount: 40 × 0.25 = $10
Sale price: 40 − 10 = $30
Shortcut: 40 × 0.75 = $30

Trap: students answer $10 (the discount amount), not the final price!

7

Exponents · Powers & Roots

Which of the following is equal to 2³ × 2⁴?

⚡ SAME BASE → ADD exponents → aᵐ × aⁿ = aᵐ⁺ⁿ

📘 Solution

Same base (2), so add exponents: 2³ × 2⁴ = 2³⁺⁴ = 2⁷ = 128

Trap: students multiply exponents (3 × 4 = 12). That rule is for (2³)⁴, not 2³ × 2⁴!

8

Inequalities · Graphing on a Number Line

Solve: −2x + 4 < 10. Which values make this true?

⚡ FLIP the sign when ÷ or × by NEGATIVE → Dividing both sides by −2: flip < to >

📘 Solution

−2x + 4 < 10
Subtract 4: −2x < 6
Divide by −2 and flip the inequality: x > −3

This is the most common mistake in inequalities — forgetting to flip the sign!

9

Algebraic Expressions · Distributive Property

Expand and simplify: 3(x + 4) − 2(x − 1)

⚡ DISTRIBUTE → COMBINE like terms → Multiply each term inside parentheses by the factor outside

📘 Solution

3(x + 4) = 3x + 12
−2(x − 1) = −2x + 2 ← careful with the negative!
Combine: (3x − 2x) + (12 + 2) = x + 14

Trap: −2(x − 1) is often wrongly written as −2x − 2. Distributing a negative flips both signs!

10

Word Problem · Setting Up an Equation

Maria has twice as many stickers as Jake. Together they have 42 stickers. How many stickers does Maria have?

⚡ Let x = smaller → define others in terms of x → Write equation from "together" or "total"

📘 Solution

Let Jake = x, Maria = 2x
x + 2x = 42 → 3x = 42 → x = 14 (Jake)
Maria = 2(14) = 28

Trap: students often give Jake's amount (14). Re-read the question — it asks for Maria!

11

Angles · Complementary & Supplementary

Two angles are supplementary. One angle measures 73°. What is the measure of the other angle?

⚡ Complementary = 90° | Supplementary = 180° → "S" for Supplementary, "S" for Straight line (180°)

📘 Solution

Supplementary means they sum to 180°.
180° − 73° = 107°

Trap: subtracting from 90° gives 17° (complementary, not supplementary)!

12

Triangles · Interior Angle Sum

A triangle has angles of 47° and 68°. What is the measure of the third angle?

⚡ Triangle Angle Sum = 180° → Always. No exceptions. Third = 180 − (sum of other two)

📘 Solution

47° + 68° = 115°
Third angle: 180° − 115° = 65°

Always check: do all three angles add to 180°? 47 + 68 + 65 = 180 ✓

13

Pythagorean Theorem

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

⚡ a² + b² = c² (c = hypotenuse) → Memorize 3-4-5 and 6-8-10 Pythagorean triples!

📘 Solution

a² + b² = c²
6² + 8² = 36 + 64 = 100
c = √100 = 10

Note: √100 = 10, so A and D are actually the same value. Answer A is the simplified form.

14

Area of a Triangle

A triangle has a base of 10 cm and a height of 7 cm. What is its area?

⚡ A = ½ × base × height → Don't forget the ½ — triangle is half a rectangle!

📘 Solution

A = ½ × b × h = ½ × 10 × 7 = ½ × 70 = 35 cm²

Trap: forgetting the ½ gives 70 (choice A). A triangle always has half the area of a rectangle with the same base and height.

15

Circles · Circumference

A circle has a diameter of 14 cm. What is its circumference? (Use π ≈ 3.14)

⚡ C = πd OR C = 2πr → Diameter = 2 × radius. Don't double-count!

📘 Solution

C = πd = 3.14 × 14 = 43.96 cm

Trap: using radius (7) instead of diameter gives 21.98 (choice A). Also, circumference is in cm, not cm² — D has wrong units (that formula is for area)!

16

Area of a Circle

What is the area of a circle with a radius of 5 m? (Use π ≈ 3.14)

⚡ A = πr² ← square the RADIUS, not the diameter → Always check: are you given r or d?

📘 Solution

A = πr² = 3.14 × 5² = 3.14 × 25 = 78.5 m²

Trap: squaring the diameter (10² = 100 → 314) is very common. Remember r = 5, not 10!

17

Parallel Lines & Transversals · Alternate Interior Angles

Two parallel lines are cut by a transversal. An angle on one side measures 115°. What is the measure of its alternate interior angle?

⚡ Alternate Interior = EQUAL | Co-interior = 180° → "Z" shape = alternate = equal. "C" shape = co-interior = supplementary.

📘 Solution

Alternate interior angles are equal when lines are parallel.
Answer: 115°

Trap: co-interior angles (same-side interior) = 180° − 115° = 65°. Alternate interior angles are equal, not supplementary!

18

Volume of a Rectangular Prism

A box has a length of 8 cm, width of 5 cm, and height of 4 cm. What is its volume?

⚡ V = l × w × h (All three dimensions!) → Volume is 3D → multiply ALL three measurements

📘 Solution

V = l × w × h = 8 × 5 × 4 = 160 cm³

Trap: forgetting one dimension (8 × 5 = 40 or 8 × 4 = 32) gives wrong answers. Always multiply all three!

19

Congruence & Similarity · Scale Factor

Two similar triangles have corresponding sides of 4 cm and 10 cm. If the smaller triangle has a perimeter of 18 cm, what is the perimeter of the larger triangle?

⚡ Similar → sides PROPORTIONAL → perimeters proportional too → Scale factor applies to ALL lengths (sides, perimeter)

📘 Solution

Scale factor: 10 ÷ 4 = 2.5
Larger perimeter: 18 × 2.5 = 45 cm

Note: Scale factor applies to AREA too — but for area, you square the scale factor. For length/perimeter, just multiply by scale factor.

20

Coordinate Geometry · Distance Formula

What is the distance between points A(1, 2) and B(4, 6) on a coordinate plane?

⚡ d = √[(x₂−x₁)² + (y₂−y₁)²] → It's the Pythagorean theorem in disguise!

📘 Solution

Δx = 4 − 1 = 3, Δy = 6 − 2 = 4
d = √(3² + 4²) = √(9 + 16) = √25 = 5

Recognize this? It's the 3-4-5 Pythagorean triple! Note: √25 = 5, so B and D are the same value. 5 is the simplified answer.

Pre-Algebra

Geometry

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