Self-Study Worksheet · Algebra 1 & Geometry

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Algebra 1
10 Problems · Word Problems
A · 01 Linear Equations Easy
Key: Isolate → Inverse Operations
Emma has $47 in her piggy bank. She earns $8 per hour babysitting. How many hours must she work to have exactly $103?
⚠️ Common mistake: Forgetting to subtract the money she already has before dividing.
Example First If you have $10 and earn $5/hr, hours to reach $25:
\(10 + 5h = 25 \Rightarrow 5h = 15 \Rightarrow h = 3\)
📘 Explanation Set up: \(47 + 8h = 103\)
Subtract 47 from both sides: \(8h = 56\)
Divide: \(h = 7\) ✓

Trap: Dividing 103 ÷ 8 = 12.875 (forgetting the $47 head start) is the #1 error here.
A · 02 Inequalities Medium
Key: Flip sign when ÷ by NEGATIVE
A store sells notebooks for $3 each. Jake has $20 and wants to buy as many notebooks as possible, but he must keep at least $5 for bus fare. What is the maximum number of notebooks he can buy?
⚠️ Common mistake: Using all $20 without saving $5.
Example First \(3n \leq 20 - 5\) → Money available after saving: \(\$15\) → max notebooks = 5
📘 Explanation Available spending: \(20 - 5 = \$15\)
Inequality: \(3n \leq 15 \Rightarrow n \leq 5\)
Max whole notebooks = 5. Answer: B ✓

Trap: \(20 \div 3 = 6.6\) → 6 notebooks, but this ignores the $5 bus fare constraint.
A · 03 Systems of Equations Medium
Key: Substitution OR Elimination
Two friends, Ava and Ben, together have 34 stickers. Ava has 6 more stickers than Ben. How many stickers does Ben have?
⚠️ Common mistake: Finding Ava's count and reporting it as Ben's.
Example First Two numbers sum to 10, one is 2 more than other: \(a + b = 10,\ a = b+2 \Rightarrow b = 4\)
📘 Explanation Let \(b\) = Ben's stickers, \(a = b + 6\)
\(a + b = 34 \Rightarrow (b+6) + b = 34 \Rightarrow 2b = 28 \Rightarrow b = 14\)
Ben has 14 stickers. Answer: C ✓

Trap: Ava has 20 — that's option D, placed to catch students who answer the wrong person.
A · 04 Proportions & Percent Easy
Key: Part ÷ Whole × 100
A shirt originally costs $40. It is on sale for 25% off. What is the sale price?
⚠️ Common mistake: Reporting the discount amount ($10) instead of the final price.
Example First 20% off $50: discount = \(0.20 \times 50 = \$10\), sale price = \(50 - 10 = \$40\)
📘 Explanation Discount = \(0.25 \times 40 = \$10\)
Sale price = \(40 - 10 = \$30\). Answer: B ✓

Alternatively: \(40 \times 0.75 = \$30\) (multiply by the remaining percentage directly).
A · 05 Rate & Distance Medium
Key: d = r × t
Car A travels at 60 mph. Car B travels at 45 mph in the same direction. Car A leaves 1 hour after Car B. How many hours after Car A departs will Car A catch up to Car B?
⚠️ Common mistake: Forgetting Car B's 1-hour head start.
Example First Head start = \(45 \times 1 = 45\) miles. Close rate = \(60 - 45 = 15\) mph.
📘 Explanation Head start: \(45 \times 1 = 45\) mi
Closing speed: \(60 - 45 = 15\) mph
Time to catch: \(45 \div 15 = 3\) hours. Answer: C ✓
A · 06 Slope & Linear Functions Medium
Key: slope = rise ÷ run = Δy ÷ Δx
A plumber charges a $50 flat fee plus $75 per hour. Write the total cost \(C\) as a function of hours \(h\), then find the cost for 4 hours.
⚠️ Common mistake: Multiplying the flat fee by hours.
Example First \(C = \text{flat fee} + \text{rate} \times h\)
📘 Explanation \(C = 50 + 75h\)
At \(h = 4\): \(C = 50 + 75(4) = 50 + 300 = \$350\). Answer: C ✓

Trap D: \(75 \times 4 \times 50/4 = ...\) — students sometimes multiply everything together.
A · 07 Exponents Hard
Key: Add exponents when multiplying same base
A bacteria population doubles every hour. It starts at 50. How many bacteria are there after 5 hours?
⚠️ Common mistake: Multiplying 50 × 2 × 5 = 500 (linear thinking, not exponential).
Example First Doubles every hour → multiply by 2 each time: \(P = 50 \cdot 2^t\)
📘 Explanation \(P = 50 \cdot 2^5 = 50 \cdot 32 = 1{,}600\). Answer: D ✓

Trap A (500): Using \(50 \times 2 \times 5\) — this is the most common error. Exponential growth is multiplicative, not additive.
A · 08 Quadratic — Factoring Hard
Key: Find two numbers → Product = c, Sum = b
A rectangular garden has an area of 40 m². Its length is 3 m more than its width. Find the width.
⚠️ Common mistake: Accepting negative width as a valid answer.
Example First \(w(w+3) = 40 \Rightarrow w^2 + 3w - 40 = 0\) → factor!
📘 Explanation \(w^2 + 3w - 40 = 0\)
Factor: \((w+8)(w-5) = 0\)
\(w = 5\) or \(w = -8\) → reject negative → width = 5 m. Answer: B ✓
A · 09 Mixture & Ratio Medium
Key: Total amount × Concentration = Pure substance
A chemist has 20 liters of a 30% acid solution and wants to make a 50% acid solution. How many liters of pure acid must be added?
⚠️ Common mistake: Forgetting the added acid also increases total volume.
Example First Acid in + Acid added = Acid in result: \(0.30(20) + x = 0.50(20 + x)\)
📘 Explanation \(6 + x = 0.5(20 + x)\)
\(6 + x = 10 + 0.5x\)
\(0.5x = 4 \Rightarrow x = 8\) liters. Answer: B ✓
A · 10 Functions & Domain Hard
Key: Domain = all valid inputs (x-values)
The function \(f(x) = \dfrac{3}{x - 4}\) models a situation. Which value of \(x\) is NOT in the domain?
⚠️ Common mistake: Setting the numerator to zero instead of the denominator.
Example First \(\frac{5}{x-2}\) is undefined when \(x - 2 = 0 \Rightarrow x = 2\). So 2 is excluded from domain.
📘 Explanation Division by zero is undefined → set denominator = 0:
\(x - 4 = 0 \Rightarrow x = 4\)
Domain is all real numbers except \(x = 4\). Answer: C ✓
Geometry
10 Problems · Core Concepts
G · 01 Angles Easy
Key: Supplementary = 180°, Complementary = 90°
Two angles are supplementary. One angle measures \((3x + 10)°\) and the other measures \((x + 20)°\). Find \(x\).
⚠️ Common mistake: Confusing supplementary (180°) with complementary (90°).
Example First Supplementary: angles sum to 180°: \((2x + 5) + (x + 10) = 180\)
📘 Explanation \((3x+10) + (x+20) = 180\)
\(4x + 30 = 180 \Rightarrow 4x = 150 \Rightarrow x = 37.5\). Answer: B ✓
G · 02 Pythagorean Theorem Easy
Key: a² + b² = c² (c is HYPOTENUSE)
A ladder 13 ft long leans against a wall. The base of the ladder is 5 ft from the wall. How high up the wall does the ladder reach?
⚠️ Common mistake: Adding instead of subtracting: \(\sqrt{13^2 + 5^2}\) (wrong direction).
Example First Hypotenuse = 10, one leg = 6: other leg = \(\sqrt{10^2 - 6^2} = \sqrt{64} = 8\)
📘 Explanation \(5^2 + h^2 = 13^2\)
\(25 + h^2 = 169 \Rightarrow h^2 = 144 \Rightarrow h = 12\) ft. Answer: C ✓

Tip: 5-12-13 is a famous Pythagorean triple worth memorizing!
G · 03 Area — Triangle Easy
Key: A = ½ × base × height
A triangular park has a base of 18 m and a height of 10 m. The city wants to plant grass at $4 per m². What is the total cost?
⚠️ Common mistake: Forgetting the ½ in the triangle area formula.
Example First Base = 6, height = 4: Area = \(\frac{1}{2}(6)(4) = 12 \text{ m}^2\)
📘 Explanation Area = \(\frac{1}{2}(18)(10) = 90 \text{ m}^2\)
Cost = \(90 \times 4 = \$360\). Answer: B ✓

Trap C ($720): forgetting the ½ gives area = 180 m², cost = $720.
G · 04 Circle — Arc & Circumference Medium
Key: Arc = (angle/360) × 2πr
A circle has radius 9 cm. What is the length of an arc that subtends a central angle of 120°? Use \(\pi \approx 3.14\).
⚠️ Common mistake: Using diameter instead of radius in the formula.
Example First Central angle 90° of circle radius 4: arc = \(\frac{90}{360} \cdot 2\pi(4) = \frac{1}{4} \cdot 8\pi \approx 6.28\)
📘 Explanation Arc = \(\frac{120}{360} \times 2\pi(9) = \frac{1}{3} \times 18\pi \approx \frac{1}{3} \times 56.52 \approx 18.84\) cm. Answer: C ✓

Trap D: Forgetting to multiply by ⅓ (not taking the fraction of the circle).
G · 05 Similar Triangles Medium
Key: Corresponding sides are PROPORTIONAL
Triangle ABC is similar to Triangle DEF. The sides of ABC are 4, 6, and 8. The shortest side of DEF is 10. What is the longest side of DEF?
⚠️ Common mistake: Adding the scale factor instead of multiplying.
Example First Scale factor = \(\frac{\text{new shortest}}{\text{old shortest}}\), then multiply all sides.
📘 Explanation Scale factor = \(\frac{10}{4} = 2.5\)
Longest side of DEF = \(8 \times 2.5 = 20\). Answer: C ✓
G · 06 Volume — Cylinder Medium
Key: V = πr²h (radius, NOT diameter!)
A cylindrical water tank has a diameter of 6 ft and a height of 10 ft. What is its volume? (Use \(\pi \approx 3.14\))
⚠️ The most common error: using diameter (6) instead of radius (3) in πr²h.
Example First Diameter = 8, height = 5: r = 4, V = \(\pi(4)^2(5) = 80\pi \approx 251.2\)
📘 Explanation \(r = 6 \div 2 = 3\) ft
\(V = \pi(3)^2(10) = 90\pi \approx 282.6 \text{ ft}^3\). Answer: B ✓

Trap C: Using \(\pi(6)^2(10) = 360\pi \approx 1130.4\) — the diameter-as-radius error.
G · 07 Triangle — Exterior Angle Medium
Key: Exterior angle = Sum of TWO non-adjacent interior angles
In a triangle, two interior angles are 55° and 70°. What is the measure of the exterior angle adjacent to the third interior angle?
⚠️ Common mistake: Subtracting from 180° to find just the interior angle, then stopping.
Example First Interior angles 40°, 60°: third interior = 80°, exterior = 180° − 80° = 100° = 40° + 60° ✓
📘 Explanation Third interior angle = \(180 - 55 - 70 = 55°\)
Exterior angle = \(180 - 55 = 125°\)
OR shortcut: \(55 + 70 = 125°\). Answer: C ✓
G · 08 Coordinate Geometry Medium
Key: Midpoint = average of x's, average of y's
Point M is the midpoint of segment AB. A is at \((2, -4)\) and M is at \((5, 1)\). What are the coordinates of B?
⚠️ Common mistake: Subtracting instead of using the midpoint relationship both ways.
Example First Midpoint formula: \(M = \left(\frac{x_1+x_2}{2},\ \frac{y_1+y_2}{2}\right)\) → solve for unknown endpoint.
📘 Explanation \(\frac{2 + x_2}{2} = 5 \Rightarrow x_2 = 8\)
\(\frac{-4 + y_2}{2} = 1 \Rightarrow y_2 = 6\)
B = \((8, 6)\). Answer: C ✓
G · 09 Surface Area — Rectangular Prism Hard
Key: SA = 2(lw + lh + wh) — SIX faces, THREE pairs
A gift box measures 8 cm × 5 cm × 3 cm. How much wrapping paper (in cm²) is needed to cover it completely?
⚠️ Common mistake: Multiplying all three dimensions together (that's volume, not surface area!).
Example First Box 2×3×4: SA = 2(2·3 + 2·4 + 3·4) = 2(6+8+12) = 52 cm²
📘 Explanation \(SA = 2(8 \times 5 + 8 \times 3 + 5 \times 3) = 2(40 + 24 + 15) = 2(79) = 158 \text{ cm}^2\)
Answer: B ✓

Trap A: \(8 \times 5 \times 3 = 120\) is the volume.
G · 10 Special Right Triangles Hard
Key: 45-45-90 → sides = x, x, x√2
An isosceles right triangle has a hypotenuse of \(10\sqrt{2}\) cm. What is the length of each leg?
⚠️ Common mistake: Dividing by 2 instead of by √2 (or multiplying by √2 instead of dividing).
Example First 45-45-90: if hypotenuse = \(6\sqrt{2}\), then each leg = 6. (hyp = leg × √2)
📘 Explanation In a 45-45-90 triangle: \(\text{hyp} = \text{leg} \times \sqrt{2}\)
\(\text{leg} = \frac{10\sqrt{2}}{\sqrt{2}} = 10\) cm. Answer: B ✓

Memorize: 45-45-90 ratios = \(1 : 1 : \sqrt{2}\)