Algebra 1 · 10 Problems
Variables, Equations & Functions
Core topics: solving equations, word problems, slope, systems, inequalities
⚡
Quick Memory Points
Before you start — lock these keywords into your brain:
ISOLATE
BALANCE
INVERSE
DISTRIBUTE
SUBSTITUTE
SLOPE = rise/run
RATE × TIME = DISTANCE
Age Word Problem
Emma is 3 times as old as her younger brother. In 4 years, she will be twice her brother's age. How old is Emma now?
💡 Key: Set up TWO equations. Let b = brother's age now.
Now: Emma = 3b
In 4 years: (3b + 4) = 2(b + 4)
In 4 years: (3b + 4) = 2(b + 4)
Step-by-step solution:
Let b = brother's age now, so Emma = 3b
In 4 years: 3b + 4 = 2(b + 4)
3b + 4 = 2b + 8
b = 4 → Emma = 3 × 4 = 12 years old ✓
Common mistake: Forgetting to add 4 to BOTH sides of the equation.
In 4 years: 3b + 4 = 2(b + 4)
3b + 4 = 2b + 8
b = 4 → Emma = 3 × 4 = 12 years old ✓
Common mistake: Forgetting to add 4 to BOTH sides of the equation.
Coin Problem
A jar contains only dimes (10¢) and quarters (25¢). There are 18 coins totaling $2.85. How many quarters are there?
💡 Key words: TOTAL COINS + TOTAL VALUE → two equations, two unknowns.
d + q = 18
0.10d + 0.25q = 2.85
0.10d + 0.25q = 2.85
Step-by-step:
From d + q = 18 → d = 18 − q
Sub into value equation: 0.10(18 − q) + 0.25q = 2.85
1.80 − 0.10q + 0.25q = 2.85
0.15q = 1.05 → q = 7... wait, let's recheck:
0.15q = 1.05 → q = 7. But with 7 quarters: 7×0.25=1.75, 11×0.10=1.10 → total=2.85 ✓
Hmm — let's recount. q = 7 → answer is A? No, check: 0.15q = 1.05 → q = 7.
Correct: 9 quarters. Let me redo: 10(18−q) + 25q = 285 → 180 + 15q = 285 → 15q = 105 → q = 7.
Wait — q = 7 is correct. But listed answer is C=9. Let me verify C: 9×25 + 9×10 = 225+90=315≠285. A is correct: 7×25+11×10=175+110=285 ✓
Answer is A: 7 quarters. Recheck by substitution always!
Sub into value equation: 0.10(18 − q) + 0.25q = 2.85
1.80 − 0.10q + 0.25q = 2.85
0.15q = 1.05 → q = 7... wait, let's recheck:
0.15q = 1.05 → q = 7. But with 7 quarters: 7×0.25=1.75, 11×0.10=1.10 → total=2.85 ✓
Hmm — let's recount. q = 7 → answer is A? No, check: 0.15q = 1.05 → q = 7.
Correct: 9 quarters. Let me redo: 10(18−q) + 25q = 285 → 180 + 15q = 285 → 15q = 105 → q = 7.
Wait — q = 7 is correct. But listed answer is C=9. Let me verify C: 9×25 + 9×10 = 225+90=315≠285. A is correct: 7×25+11×10=175+110=285 ✓
Answer is A: 7 quarters. Recheck by substitution always!
Mixture Problem
A chemist mixes a 20% acid solution with a 50% acid solution to make 60 mL of a 30% acid solution. How many mL of the 20% solution are needed?
💡 MIXTURE KEY: (concentration × volume) + (concentration × volume) = final concentration × total volume
0.20x + 0.50(60 − x) = 0.30 × 60
Step-by-step:
0.20x + 0.50(60 − x) = 18
0.20x + 30 − 0.50x = 18
−0.30x = −12
x = 40 mL of the 20% solution ✓
Check: 0.20(40) + 0.50(20) = 8 + 10 = 18 = 0.30(60) ✓
0.20x + 30 − 0.50x = 18
−0.30x = −12
x = 40 mL of the 20% solution ✓
Check: 0.20(40) + 0.50(20) = 8 + 10 = 18 = 0.30(60) ✓
Distance–Rate–Time
Two trains leave the same station traveling in opposite directions. Train A travels at 60 mph and Train B at 80 mph. After how many hours will they be 420 miles apart?
💡 OPPOSITE DIRECTION: Add speeds! Total distance = (speed₁ + speed₂) × time
(60 + 80) × t = 420
140t = 420
140t = 420
Solution:
t = 420 ÷ 140 = 3 hours ✓
Tricky trap: Students often subtract speeds (80−60) instead of adding them when going in OPPOSITE directions.
Tricky trap: Students often subtract speeds (80−60) instead of adding them when going in OPPOSITE directions.
Slope & Linear Equation
A line passes through (2, 5) and (6, 13). Which equation correctly represents this line?
💡 SLOPE FORMULA: m = (y₂ − y₁) ÷ (x₂ − x₁), then use y = mx + b
m = (13 − 5) ÷ (6 − 2) = 8 ÷ 4 = 2
Solution:
m = 2 ✓ Now find b using point (2, 5):
5 = 2(2) + b → 5 = 4 + b → b = 1
Equation: y = 2x + 1 ✓
Check with (6,13): 2(6)+1 = 13 ✓
5 = 2(2) + b → 5 = 4 + b → b = 1
Equation: y = 2x + 1 ✓
Check with (6,13): 2(6)+1 = 13 ✓
Work Rate Problem
Printer A can print a report in 4 hours. Printer B can print the same report in 6 hours. If both printers work together, how long will it take to finish the report?
💡 WORK RATE KEY: Rate = 1/time. Together: 1/A + 1/B = 1/T
¹⁄₄ + ¹⁄₆ = ¹⁄T
Solution:
1/4 + 1/6 = 3/12 + 2/12 = 5/12
T = 12/5 = 2.4 hours = 2 hours 24 minutes ✓
Common trap: Just adding 4+6=10, then dividing by 2 gives 5 hours — WRONG. Always use reciprocals!
T = 12/5 = 2.4 hours = 2 hours 24 minutes ✓
Common trap: Just adding 4+6=10, then dividing by 2 gives 5 hours — WRONG. Always use reciprocals!
Inequality Word Problem
Marcus wants to buy notebooks at $2.50 each and has $20. He must keep at least $4.00 for bus fare. What is the maximum number of notebooks he can buy?
💡 AT LEAST: means ≥. Remaining money must be ≥ $4
2.50n ≤ 20 − 4
Solution:
2.50n ≤ 16
n ≤ 6.4
Maximum whole number = 6 notebooks ✓
Trap: Round DOWN for "maximum whole units" — never round up inequalities!
n ≤ 6.4
Maximum whole number = 6 notebooks ✓
Trap: Round DOWN for "maximum whole units" — never round up inequalities!
Percent Increase
A jacket originally costs $80. It is marked up 25%, then the marked-up price is discounted 20%. What is the final price?
💡 MULTIPLY — don't add/subtract percent! Two steps, two separate multipliers.
$80 × 1.25 × 0.80 = ?
Solution:
After markup: 80 × 1.25 = $100
After discount: 100 × 0.80 = $80.00 ✓
Surprise: +25% then −20% returns to original! 1.25 × 0.80 = 1.00. But this is a coincidence — it doesn't always happen!
After discount: 100 × 0.80 = $80.00 ✓
Surprise: +25% then −20% returns to original! 1.25 × 0.80 = 1.00. But this is a coincidence — it doesn't always happen!
Systems of Equations
A farm has chickens and cows. There are 30 animals total. The total number of legs is 84. How many cows are on the farm?
💡 KEY FACT: Chickens = 2 legs, Cows = 4 legs. Write TWO equations.
c + k = 30
4c + 2k = 84
4c + 2k = 84
Solution:
From eq 1: k = 30 − c
Sub: 4c + 2(30 − c) = 84
4c + 60 − 2c = 84
2c = 24 → c = 12 cows ✓
Check: 12 cows + 18 chickens = 30, legs: 48+36=84 ✓
Sub: 4c + 2(30 − c) = 84
4c + 60 − 2c = 84
2c = 24 → c = 12 cows ✓
Check: 12 cows + 18 chickens = 30, legs: 48+36=84 ✓
Function & Pattern
A parking lot charges $3 for the first hour and $1.50 for each additional hour. If someone paid $12.00 total, how many hours did they park?
💡 SETUP: First hour is different — isolate it. Remaining cost = total − first hour cost.
3 + 1.50(h − 1) = 12
Solution:
3 + 1.50(h − 1) = 12
1.50(h − 1) = 9
h − 1 = 6
h = 7 hours ✓
1.50(h − 1) = 9
h − 1 = 6
h = 7 hours ✓
Geometry · 10 Problems
Shapes, Angles & Proofs
Core topics: triangles, circles, area/perimeter, angles, Pythagorean theorem, similarity
🔷
Quick Memory Points
Geometry = visual + formula. Memorize these first:
PYTHAGOREAN: a²+b²=c²
TRIANGLE: 180°
SUPPLEMENTARY: 180°
COMPLEMENTARY: 90°
CIRCLE AREA: πr²
SIMILAR: PROPORTIONAL
VERTICAL ANGLES: EQUAL
Triangle Angles
In a triangle, two angles measure 47° and 68°. What is the measure of the third angle?
💡 TRIANGLE SUM: All three angles always add up to 180°. No exceptions.
∠A + ∠B + ∠C = 180°
Solution:
Third angle = 180° − 47° − 68° = 65° ✓
Pythagorean Theorem
A ladder 13 feet long leans against a wall. The base of the ladder is 5 feet from the wall. How high up the wall does the ladder reach?
💡 PYTHAGOREAN: a² + b² = c². Here c = ladder (hypotenuse), a = base distance.
5² + b² = 13²
b² = 169 − 25 = 144
b² = 169 − 25 = 144
Solution:
b² = 169 − 25 = 144
b = √144 = 12 feet ✓
5-12-13 is a Pythagorean triple — memorize it!
b = √144 = 12 feet ✓
5-12-13 is a Pythagorean triple — memorize it!
Area of a Composite Shape
An "L-shaped" floor consists of a 10 m × 8 m rectangle with a 3 m × 4 m rectangle cut out from one corner. What is the area of the floor?
💡 COMPOSITE: Big shape − small shape. Don't forget to subtract!
Area = (10 × 8) − (3 × 4)
Solution:
Full rectangle: 10 × 8 = 80 m²
Cut-out: 3 × 4 = 12 m²
L-shape area = 80 − 12 = 68 m² ✓
Cut-out: 3 × 4 = 12 m²
L-shape area = 80 − 12 = 68 m² ✓
Parallel Lines & Transversal
Two parallel lines are cut by a transversal. One of the co-interior angles (same-side interior) measures 110°. What is the measure of the other co-interior angle?
💡 CO-INTERIOR (same-side): They are SUPPLEMENTARY → add up to 180°. NOT equal!
Co-interior angles: α + β = 180°
Solution:
180° − 110° = 70° ✓
Trap: Many students confuse alternate interior angles (EQUAL) with co-interior angles (SUPPLEMENTARY = 180°).
Trap: Many students confuse alternate interior angles (EQUAL) with co-interior angles (SUPPLEMENTARY = 180°).
Circle Area & Circumference
A circular pizza has a diameter of 14 inches. What is its area? (Use π ≈ 3.14)
💡 DIAMETER → RADIUS: r = d/2 = 7. Then Area = πr². Don't use diameter directly!
A = π × r² = 3.14 × 7²
Solution:
r = 14/2 = 7 inches
A = 3.14 × 7² = 3.14 × 49 = 153.86 in² ✓
Trap: Option D uses diameter instead of radius (3.14 × 14² = 615.44) — WRONG!
A = 3.14 × 7² = 3.14 × 49 = 153.86 in² ✓
Trap: Option D uses diameter instead of radius (3.14 × 14² = 615.44) — WRONG!
Similar Triangles
Two similar triangles have corresponding sides in ratio 3:5. The area of the smaller triangle is 27 cm². What is the area of the larger triangle?
💡 AREA RATIO = (side ratio)² — not the same as the side ratio!
Area ratio = (3/5)² = 9/25
Solution:
Side ratio = 3:5, so Area ratio = 9:25
27 / 9 = 3, so larger area = 3 × 25 = 75 cm² ✓
Classic mistake: Multiplying 27 × (5/3) = 45 — this uses side ratio, not area ratio!
27 / 9 = 3, so larger area = 3 × 25 = 75 cm² ✓
Classic mistake: Multiplying 27 × (5/3) = 45 — this uses side ratio, not area ratio!
Volume of a Rectangular Prism
A rectangular box is 8 cm long, 5 cm wide, and 3 cm tall. What is its volume?
💡 VOLUME = length × width × height — all three dimensions multiplied.
V = l × w × h = 8 × 5 × 3
Solution:
V = 8 × 5 × 3 = 120 cm³ ✓
Exterior Angle Theorem
An exterior angle of a triangle measures 125°. One of the non-adjacent interior angles is 72°. What is the other non-adjacent interior angle?
💡 EXTERIOR ANGLE THEOREM: Exterior angle = sum of the two NON-adjacent interior angles.
Exterior ∠ = ∠A + ∠B
125° = 72° + ∠B
125° = 72° + ∠B
Solution:
∠B = 125° − 72° = 53° ✓
Trap: Students often subtract the exterior angle from 180° first. Don't! Use the theorem directly.
Trap: Students often subtract the exterior angle from 180° first. Don't! Use the theorem directly.
Perimeter of a Triangle (Pythagorean)
A right triangle has legs of 9 cm and 12 cm. What is the perimeter of the triangle?
💡 PERIMETER needs ALL 3 sides. Find hypotenuse first using a² + b² = c².
c² = 9² + 12² = 81 + 144 = 225
c = 15 cm
Perimeter = 9 + 12 + 15
c = 15 cm
Perimeter = 9 + 12 + 15
Solution:
Hypotenuse: √(81 + 144) = √225 = 15 cm
Perimeter = 9 + 12 + 15 = 36 cm ✓
9-12-15 = 3×(3-4-5) Pythagorean triple!
Perimeter = 9 + 12 + 15 = 36 cm ✓
9-12-15 = 3×(3-4-5) Pythagorean triple!
Arc Length
A sector of a circle has a central angle of 60° and a radius of 12 cm. What is the arc length? (Use π ≈ 3.14)
💡 ARC LENGTH = (angle/360°) × 2πr — the fraction of the full circle's circumference.
Arc = (60/360) × 2 × 3.14 × 12
Solution:
Arc = (60/360) × 2 × 3.14 × 12
= (1/6) × 75.36
= 12.56 cm ✓
Trap: Option B forgets to multiply by 1/6 and uses just 2πr.
= (1/6) × 75.36
= 12.56 cm ✓
Trap: Option B forgets to multiply by 1/6 and uses just 2πr.