Math Practice — Pre-Algebra & Geometry

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Pre-Algebra

Problems 1 – 10 · Variables, Equations, Ratios, Integers

Order of Operations Tricky!

What is the value of 3 + 4 × 2² − (6 ÷ 3) ?

💡 Memory Key: PEMDAS — Parentheses → Exponents → Multiplication/Division → Addition/Subtraction. Never add before you multiply!

📖 Solution

Step 1 — Exponent first: 2² = 4
Step 2 — Parentheses: 6 ÷ 3 = 2
Step 3 — Multiply: 4 × 4 = 16
Step 4 — Add/Subtract left→right: 3 + 16 − 2 = 17

✅ Answer = 17

Solving One-Step Equations Easy

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

💡 Memory Key: ISOLATE — undo operations in reverse PEMDAS. Add/subtract first, then multiply/divide. Keep both sides balanced!

📖 Solution

5x − 3 = 22
Add 3 to both sides: 5x = 25
Divide by 5: x = 5 ✅

Fractions — Adding Unlike Denominators Tricky!

What is 34 + 56 in simplest form?

💡 Memory Key: LCD — Least Common Denominator. Find the smallest number both denominators divide into. Then convert each fraction before adding.

📖 Solution

LCD of 4 and 6 = 12
Convert: 3/4 = 9/12 and 5/6 = 10/12
Add: 9/12 + 10/12 = 19/12 ✅
(19/12 is already in simplest form — GCF of 19 and 12 is 1)

Ratios & Proportions Watch Out

A recipe uses 2 cups of flour for every 3 cups of sugar. How many cups of flour are needed if you use 12 cups of sugar?

💡 Memory Key: CROSS-MULTIPLY — Set up a/b = c/d, then ad = bc. Make sure units match on each side!

📖 Solution

Set up proportion: 2/3 = x/12
Cross-multiply: 3x = 24
Divide: x = 8 ✅

Integer Operations — Negatives Tricky!

What is (−3) × (−4) + (−6) ÷ 2 ?

💡 Memory Key: SIGN RULES — Neg × Neg = Positive, Pos × Neg = Negative. Remember: two negatives cancel!

📖 Solution

Step 1 — Multiply: (−3) × (−4) = +12 (neg × neg = pos)
Step 2 — Divide: (−6) ÷ 2 = −3 (pos ÷ neg = neg)
Step 3 — Add: 12 + (−3) = 9 ✅

Percent Problems Watch Out

A jacket costs $80. It goes on sale for 25% off. What is the sale price?

💡 Memory Key: PERCENT = PART/WHOLE × 100 — For discount: Sale Price = Original × (1 − rate). Multiply, don't just find the percent!

📖 Solution

Discount amount: 80 × 0.25 = $20
Sale price: 80 − 20 = $60 ✅
OR: 80 × 0.75 = $60 (shortcut!)

Combining Like Terms Tricky!

Simplify: 3x² + 2x − 5 + x² − 4x + 7

💡 Memory Key: LIKE TERMS — Only combine terms with the same variable AND same exponent. 3x² and x² are like. 3x² and 2x are NOT!

📖 Solution

Group like terms:
x² terms: 3x² + x² = 4x²
x terms: 2x − 4x = −2x
Constants: −5 + 7 = +2
Result: 4x² − 2x + 2 ✅

Inequalities Watch Out

Solve: −2x + 5 > 11. Which answer is correct?

💡 Memory Key: FLIP THE SIGN — When you multiply or divide both sides by a NEGATIVE number, the inequality sign flips! (≥ → ≤)

📖 Solution

−2x + 5 > 11
Subtract 5: −2x > 6
Divide by −2 → FLIP the sign! x < −3 ✅
Common mistake: forgetting to flip when dividing by a negative!

Word Problem — Linear Equations Watch Out

Sarah earns $12 per hour. She also received a $30 bonus. She wants to earn at least $150 total. What is the minimum number of hours she must work?

💡 Memory Key: TRANSLATE — Convert words to math: "at least" = ≥, "at most" = ≤. Write the equation first, then solve!

📖 Solution

Equation: 12h + 30 ≥ 150
Subtract 30: 12h ≥ 120
Divide: h ≥ 10
Minimum = 10 hours ✅

Exponents — Negative & Zero Tricky!

What is the value of 2⁰ + 3⁻¹ + 4⁻² ?

💡 Memory Key: ZERO & NEGATIVE EXPONENTS — Any number⁰ = 1. Any number⁻ⁿ = 1 ÷ (numberⁿ). Never say the answer is 0!

📖 Solution

2⁰ = 1 (anything to the zero power = 1)
3⁻¹ = 1/3
4⁻² = 1/4² = 1/16
LCD of 1, 3, 16 = 48:
48/48 + 16/48 + 3/48 = 67/48 ✅
Common mistake (option B): using 4⁻¹ = 1/4 instead of 4⁻² = 1/16

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Geometry

Problems 11 – 20 · Angles, Triangles, Area, Circles, Coordinate Plane

Angle Types — Supplementary Easy

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

💡 Memory Key: C & S — Complementary = 90° (Corner), Supplementary = 180° (Straight line). S has more letters, more degrees!

📖 Solution

Supplementary angles add up to 180°
Other angle = 180° − 73° = 107° ✅

Triangle — Missing Angle Easy

A triangle has angles measuring 45° and 80°. What is the third angle?

💡 Memory Key: TRIANGLE SUM = 180° — The three interior angles of ANY triangle always add up to exactly 180°. Always!

📖 Solution

Triangle angle sum = 180°
Third angle = 180° − 45° − 80° = 55° ✅

Area — Trapezoid Tricky!

A trapezoid has parallel sides of 6 cm and 10 cm, and a height of 5 cm. What is its area?

Area = ½ × (b₁ + b₂) × h

💡 Memory Key: TRAP = AVERAGE BASES × HEIGHT — Add the two parallel bases, divide by 2 (average), then multiply by height. Don't forget the ½!

📖 Solution

Area = ½ × (6 + 10) × 5
= ½ × 16 × 5
= ½ × 80 = 40 cm² ✅

Pythagorean Theorem Watch Out

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

a² + b² = c²

💡 Memory Key: 3-4-5 FAMILY — Memorize Pythagorean triples: 3-4-5, 5-12-13, 6-8-10. These are shortcuts — no calculator needed!

📖 Solution

c² = 6² + 8² = 36 + 64 = 100
c = √100 = 10 ✅
Pro tip: 6-8-10 is a 3-4-5 triple scaled by 2!

Circle — Circumference Tricky!

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

C = πd or C = 2πr

💡 Memory Key: DIAMETER vs RADIUS — d = 2r. If given diameter, use C = πd directly. Don't accidentally halve it and then double it again!

📖 Solution

C = π × d = 3.14 × 14 = 43.96 cm ✅
Common mistake: using radius (7) instead of diameter (14).
That gives 21.98 — exactly half the correct answer!

Vertical Angles Easy

Two lines intersect forming 4 angles. One angle is 65°. What are the measures of the other three angles?

💡 Memory Key: VERTICAL = EQUAL — Vertical angles are across from each other and always equal. Adjacent angles are supplementary (add to 180°).

📖 Solution

Vertical angle = 65° (same as the given angle)
Adjacent angles = 180° − 65° = 115° each
Check: 65 + 115 + 65 + 115 = 360° ✅

Surface Area — Rectangular Prism Tricky!

A rectangular box has length = 5 cm, width = 3 cm, height = 4 cm. What is the total surface area?

SA = 2(lw + lh + wh)

💡 Memory Key: 3 PAIRS — A box has 3 pairs of identical faces: Top/Bottom + Front/Back + Left/Right. Find each pair's area and double it!

📖 Solution

lw = 5×3 = 15
lh = 5×4 = 20
wh = 3×4 = 12
SA = 2(15 + 20 + 12) = 2 × 47 = 94 cm² ✅

Coordinate Plane — Distance Watch Out

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

d = √[(x₂−x₁)² + (y₂−y₁)²]

💡 Memory Key: PYTHAGOREAN IN DISGUISE — Distance formula IS the Pythagorean theorem. The horizontal change is leg a, vertical change is leg b, distance is hypotenuse c!

📖 Solution

Δx = 4 − 1 = 3
Δy = 6 − 2 = 4
d = √(3² + 4²) = √(9 + 16) = √25 = 5 ✅
Another 3-4-5 triple!

Circle — Area Tricky!

A circle has a circumference of 20π cm. What is its area?

💡 Memory Key: FIND r FIRST — C = 2πr → find r, THEN use A = πr². Never use diameter in the area formula — always use radius!

📖 Solution

From circumference: 2πr = 20π → r = 10 cm
Area: A = π × 10² = 100π cm² ✅
Common mistake: using diameter (20) in area formula → gives 400π (4× too large!)

Exterior Angle Theorem Tricky!

An exterior angle of a triangle measures 120°. The two non-adjacent interior angles are x and 2x. Find x.

Exterior Angle = Sum of two non-adjacent interior angles

💡 Memory Key: EAT = SUM — Exterior Angle of Triangle = sum of the two non-adjacent interior angles. Don't use the adjacent angle (supplementary)!

📖 Solution

Exterior angle = sum of two non-adjacent interior angles:
x + 2x = 120°
3x = 120°
x = 40° ✅
Check: interior angles = 40°, 80°, and the adjacent angle = 180°−120° = 60°. Sum = 40+80+60 = 180° ✓

Math PracticePre-Algebra & Geometry

Math Practice
Pre-Algebra & Geometry