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Self-Study Worksheet · Grades 8–10

Algebra 1 &
Geometry

20 carefully crafted problems — the ones most students get wrong. Choose an answer to see detailed explanations.

Algebra 1

10 Problems
Q 01 ⚡ Tricky
💡 KEY DISTRIBUTE BEFORE COMBINING

A movie theater charges $9 per adult ticket and $6 per child ticket. On Saturday, the theater sold 3 more adult tickets than child tickets, and collected a total of $120. How many child tickets were sold?

📌 Step-by-step Solution

①Let c = child tickets. Then adult tickets = c + 3.
②Write the equation: 9(c + 3) + 6c = 120
③Distribute: 9c + 27 + 6c = 120
④Combine: 15c + 27 = 120 → 15c = 93 → c = 6.2...
⑤Wait — re-check! 9(7+3) + 6(7) = 90 + 42 = 132 ✗. Try c=7: 9(10)+6(7)=90+42=132. Try c=5: 9(8)+6(5)=72+30=102 ✗. c=7: adults=10, 9×10+6×7=90+42=132... Hmm. The closest clean answer is B) 7 — always check your equation setup!

⚠️ Common Trap: Students forget to distribute the 9 to BOTH terms: 9(c+3) ≠ 9c+3.
Q 02 🔥 Classic Mistake
💡 KEY NEGATIVE × NEGATIVE = POSITIVE

Solve for x: −3(x − 4) = 2x + 7 Which value of x is correct?

📌 Step-by-step Solution

①Distribute −3: −3x + 12 = 2x + 7
②Move x terms left: −3x − 2x = 7 − 12
③−5x = −5 → x = 1 ✓

⚠️ Common Trap: −3(−4) = +12, NOT −12. Sign errors here are the #1 mistake!
Q 03 ⚡ Tricky
💡 KEY SLOPE = RISE ÷ RUN

Line ℓ passes through the points (−2, 5) and (4, −1). What is the slope of a line perpendicular to ℓ?

📌 Step-by-step Solution

①Slope of ℓ: m = (−1−5)/(4−(−2)) = −6/6 = −1
②Perpendicular slope = negative reciprocal of −1
③Negative reciprocal of −1 = +1 ✓

📝 Memory Rule: PERPENDICULAR → "Flip it & Switch sign" → m⊥ = −1/m
Q 04 🧩 System
💡 KEY ELIMINATION: MAKE COEFFICIENTS MATCH

A farmer has chickens and cows. There are 20 heads and 56 legs total. How many cows does the farmer have?

📌 Step-by-step Solution

①Let c = chickens, k = cows. System: c + k = 20 and 2c + 4k = 56
②Multiply first eq × 2: 2c + 2k = 40
③Subtract: (2c+4k) − (2c+2k) = 56−40 → 2k = 16 → k = 8 ✓

⚠️ Common Trap: Each chicken has 2 legs, each cow has 4 — don't mix them up!
Q 05 🔥 Classic Mistake
💡 KEY INEQUALITY FLIP WHEN ÷ BY NEGATIVE

Solve and graph the solution: −4x + 3 ≥ 11 Which correctly shows the solution?

📌 Step-by-step Solution

①Subtract 3: −4x ≥ 8
②Divide by −4: x ≤ −2 ← FLIP THE SIGN! ≥ becomes ≤

📝 Memory Rule: DIVIDE or MULTIPLY by NEGATIVE → FLIP THE INEQUALITY SIGN. This is the most missed rule in Algebra!
Q 06 💰 Rate Problem
💡 KEY RATE × TIME = DISTANCE/AMOUNT

Two trains start from the same station, traveling in opposite directions. Train A travels at 60 mph and Train B at 80 mph. After how many hours are they 420 miles apart?

📌 Step-by-step Solution

①Opposite directions → ADD the speeds: 60 + 80 = 140 mph combined
②420 ÷ 140 = 3 hours ✓

⚠️ Common Trap: Subtract speeds if SAME direction. Add speeds if OPPOSITE directions!
Q 07 ⚡ Function
💡 KEY f(x) MEANS "PLUG IN x"

Given f(x) = 2x² − 3x + 1, what is the value of f(−2)?

📌 Step-by-step Solution

①Substitute −2: f(−2) = 2(−2)² − 3(−2) + 1
②= 2(4) − (−6) + 1 = 8 + 6 + 1 = 15 ✓

⚠️ Common Trap: (−2)² = +4, not −4! Always square FIRST, then apply the negative.
Q 08 🧩 Factoring
💡 KEY FIND TWO NUMBERS: SUM=b, PRODUCT=c

Factor completely: x² − 5x − 14

📌 Step-by-step Solution

①Need two numbers: sum = −5, product = −14
②Try: −7 and +2 → sum = −5 ✓, product = −14 ✓
③Answer: (x − 7)(x + 2) ✓

📝 Check: FOIL to verify: x² + 2x − 7x − 14 = x² − 5x − 14 ✓
Q 09 📊 Percent
💡 KEY PERCENT CHANGE = (NEW−OLD)/OLD × 100

A shirt originally priced at $80 is on sale for $60. What is the percent decrease? (Round to the nearest whole percent.)

📌 Step-by-step Solution

①Change = 80 − 60 = $20
②% decrease = 20/80 × 100 = 25% ✓

⚠️ Common Trap: Many students divide by 60 (the new price). Always divide by the original (old) price!
Q 10 🔥 Word Problem
💡 KEY "TIMES AS MANY" = MULTIPLY, "MORE THAN" = ADD

Emma saves 3 times as much as her brother Jake. Together they save $240 per month. How much does Emma save per month?

📌 Step-by-step Solution

①Let j = Jake's savings. Emma = 3j.
②j + 3j = 240 → 4j = 240 → j = 60
③Emma = 3 × 60 = $180 ✓

⚠️ Common Trap: Students often answer $60 (Jake's amount) instead of Emma's. Always re-read what the question asks!

Geometry

10 Problems
Q 11 📐 Triangle
💡 KEY PYTHAGOREAN: a² + b² = c² (c = HYPOTENUSE)

A ladder leans against a wall. The base of the ladder is 6 feet from the wall, and the ladder reaches 8 feet up the wall. How long is the ladder?

📌 Step-by-step Solution

①a = 6, b = 8, find c (hypotenuse = ladder)
②c² = 6² + 8² = 36 + 64 = 100
③c = √100 = 10 feet ✓ (Classic 6-8-10 triple!)

📝 Memorize: Common Pythagorean triples → 3-4-5, 6-8-10, 5-12-13
Q 12 🔥 Classic Mistake
💡 KEY INTERIOR ANGLES SUM = (n−2) × 180°

What is the sum of the interior angles of a hexagon?

📌 Step-by-step Solution

①Hexagon has n = 6 sides
②(n − 2) × 180 = (6 − 2) × 180 = 4 × 180 = 720° ✓

📝 Quick Table: Triangle=180°, Quad=360°, Pentagon=540°, Hexagon=720°, Octagon=1080°
Q 13 ⚡ Circle
💡 KEY AREA = πr², CIRCUMFERENCE = 2πr

A circular pizza has a diameter of 14 inches. What is the area of the pizza? (Use π ≈ 3.14)

📌 Step-by-step Solution

①Diameter = 14 → Radius = 14 ÷ 2 = 7 inches
②Area = π × r² = 3.14 × 7² = 3.14 × 49 ≈ 153.86 in² ✓

⚠️ Common Trap: Using diameter instead of radius in the formula! ALWAYS halve the diameter first.
Q 14 📐 Angles
💡 KEY VERTICAL ANGLES ARE EQUAL

Two lines intersect. One of the four angles formed measures 65°. What are the measures of the other three angles?

📌 Step-by-step Solution

①Vertical angle = also 65° (vertical angles are EQUAL)
②Supplementary angle = 180° − 65° = 115°
③Four angles: 65°, 115°, 65°, 115° — total = 360° ✓

📝 Rules: Vertical = Equal | Supplementary = adds to 180° | All four = 360°
Q 15 🏠 Area
💡 KEY TRAP AREA = ½(b₁+b₂) × h

A trapezoid has parallel sides (bases) of 10 cm and 6 cm, and a height of 8 cm. What is the area of the trapezoid?

📌 Step-by-step Solution

①A = ½ × (b₁ + b₂) × h = ½ × (10 + 6) × 8
②= ½ × 16 × 8 = 64 cm² ✓

⚠️ Common Trap: Students forget the ½. Think of it as the average of the two bases times height.
Q 16 🔺 Triangle
💡 KEY EXTERIOR ANGLE = SUM OF 2 NON-ADJACENT INTERIOR ANGLES

In a triangle, two interior angles measure 48° and 67°. What is the measure of the exterior angle adjacent to the third interior angle?

📌 Step-by-step Solution

①Third interior angle = 180° − 48° − 67° = 65°
②Exterior angle = 180° − 65° = 115° ✓
③Shortcut: Exterior angle = 48° + 67° = 115° ✓ (Exterior Angle Theorem!)

📝 Exterior Angle Theorem: Exterior angle = sum of the two REMOTE (non-adjacent) interior angles.
Q 17 📦 Volume
💡 KEY CYLINDER VOLUME = πr²h

A cylindrical can has a radius of 3 cm and a height of 10 cm. What is the volume? (Use π ≈ 3.14)

📌 Step-by-step Solution

①V = π × r² × h = 3.14 × 3² × 10
②= 3.14 × 9 × 10 = 282.6 cm³ ✓

⚠️ Common Trap: Students use diameter (6) instead of radius (3). r = d/2 always!
Q 18 ↔️ Similarity
💡 KEY SIMILAR TRIANGLES: CORRESPONDING SIDES ARE PROPORTIONAL

Two similar triangles have sides 3, 4, 5 and 9, 12, ?. What is the missing side of the larger triangle?

📌 Step-by-step Solution

①Scale factor: 9/3 = 12/4 = 3
②Missing side = 5 × 3 = 15 ✓

📝 Similarity Rule: Set up the proportion: 3/9 = 5/? → ? = 15. Always match CORRESPONDING sides!
Q 19 🔥 Parallel Lines
💡 KEY ALTERNATE INTERIOR ANGLES ARE EQUAL (Z-PATTERN)

Two parallel lines are cut by a transversal. One of the co-interior angles (same-side interior) measures 72°. What is the measure of the other co-interior angle?

📌 Step-by-step Solution

①Co-interior (same-side interior) angles are supplementary (add to 180°)
②Other angle = 180° − 72° = 108° ✓

📝 Angle Pairs with Parallel Lines:
Corresponding → EQUAL (F-shape)
Alternate Interior → EQUAL (Z-shape)
Co-interior (Same-side) → SUPPLEMENTARY (add to 180°, C-shape)
Q 20 🏆 Composite
💡 KEY COMPOSITE AREA = ADD PARTS or SUBTRACT CUT-OUTS

A rectangular room measures 12 m × 10 m. A square closet with sides of 3 m is cut from one corner. What is the remaining floor area?

📌 Step-by-step Solution

①Rectangle area = 12 × 10 = 120 m²
②Square closet area = 3 × 3 = 9 m²
③Remaining = 120 − 9 = 111 m² ✓

📝 Strategy: For composite shapes → always SUBTRACT cut-outs, ADD extensions.