Self-Study Worksheet

Math Mastery
Algebra & Geometry

Core concepts, tricky questions, and memory anchors for confident problem-solving.

20 Questions 2 Units Multiple Choice
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Unit 1

Algebra 1 — Word Problems

Variables, equations, inequalities, and linear relationships. The most commonly missed question types.

01
Linear Equations · Setting Up
Maya has $45. She earns $8 per hour babysitting. She wants to buy a jacket that costs $93. Which equation correctly represents the number of hours, h, she needs to work?
⚡ KEY: "has" → start value · "earns" → rate × variable · "wants" → goal
A \(45h + 8 = 93\)
B \(8h - 45 = 93\)
C \(8h + 45 = 93\)
D \(8 + 45h = 93\)
✗ Incorrect. Maya already has $45. Each hour earns $8, so after h hours she has 8h + 45. Set that equal to 93: 8h + 45 = 93. Answer: C.
02
Two-Step Equations · Tricky Wording
Three times a number decreased by 7 equals 20. What is the number?
⚠ Watch: "decreased by" means subtract FROM the expression, not from 7.
⚡ TRANSLATE: "three times x decreased by 7" → 3x − 7 (NOT 7 − 3x)
A \(x = 4\)
B \(x = 9\)
C \(x = 13\)
D \(x = \frac{27}{3}\)
✗ Incorrect. Equation: \(3x - 7 = 20\). Add 7: \(3x = 27\). Divide by 3: \(x = 9\). Note that D also equals 9, but B is the simplified correct answer. Answer: B.
03
Inequalities · Real-World Context
A roller coaster requires riders to be at least 48 inches tall. Leo is 42 inches tall. How many more inches, x, does Leo need to grow to ride? Which inequality represents this situation?
⚡ "at least" → ≥ · "at most" → ≤ · "more than" → > · "less than" → <
A \(42 + x > 48\)
B \(42 + x \geq 48\)
C \(x > 6\)
D \(42 - x \geq 48\)
✗ Incorrect. Leo needs his current height plus growth to meet OR exceed 48 in. "At least" means ≥, so: \(42 + x \geq 48\). Option C is the solved form but misses the setup. Answer: B.
04
Rate & Distance · Classic Trap
Two trains leave the same station at the same time, traveling in opposite directions. Train A travels at 60 mph, Train B at 40 mph. After how many hours are they 300 miles apart?
⚡ OPPOSITE directions → ADD speeds: (rate₁ + rate₂) × time = total distance
A \(3 \text{ hours}\)
B \(5 \text{ hours}\)
C \(7.5 \text{ hours}\)
D \(2.5 \text{ hours}\)
✗ Incorrect. Combined speed = 60 + 40 = 100 mph. \(100t = 300\), so \(t = 3\) hours. Common mistake: dividing 300 by only one speed. Answer: A.
05
Consecutive Integers · Number Tricks
The sum of three consecutive even integers is 78. What is the largest of the three integers?
⚠ "Consecutive even" means they differ by 2, not 1!
⚡ CONSECUTIVE EVEN: x, x+2, x+4 · CONSECUTIVE ODD: same pattern x, x+2, x+4
A \(24\)
B \(26\)
C \(28\)
D \(30\)
✗ Incorrect. Let integers be \(x, x+2, x+4\). Sum: \(3x + 6 = 78\), so \(3x = 72\), \(x = 24\). The three integers are 24, 26, 28. Largest = 28. Answer: C.
06
Mixture Problems · Percent Confusion
A store sells two types of nuts. Type A costs $4/lb, Type B costs $7/lb. You want to make 15 lbs of a blend worth $5/lb. How many pounds of Type A are needed?
⚡ MIXTURE: (price₁)(qty₁) + (price₂)(qty₂) = (blended price)(total qty)
A \(5 \text{ lbs}\)
B \(8 \text{ lbs}\)
C \(10 \text{ lbs}\)
D \(12 \text{ lbs}\)
✗ Incorrect. Let \(x\) = lbs of Type A, then \(15-x\) = lbs of Type B. \(4x + 7(15-x) = 5(15)\) → \(4x + 105 - 7x = 75\) → \(-3x = -30\) → \(x = 10\). Answer: C.
07
Slope-Intercept · Reading Graphs
A phone plan charges a $20 monthly fee plus $0.05 per text. Which equation gives the total monthly cost \(C\) for sending \(t\) texts? Which value represents the y-intercept and what does it mean?
⚡ y = mx + b · m = rate (per unit) · b = starting value (when x=0)
A \(C = 20t + 0.05\); b = 0.05
B \(C = 0.05t + 20\); b = 20, fixed fee
C \(C = 0.05t + 20\); b = 0.05, per-text cost
D \(C = 20t\); no y-intercept
✗ Incorrect. The equation is \(C = 0.05t + 20\). The y-intercept is 20, which is the fixed monthly fee — the cost even with zero texts. Answer: B.
08
Systems of Equations · Age Problem
Emma is 3 times as old as her brother Jake. In 4 years, she will be twice as old as Jake. How old is Jake now?
⚠ "In 4 years" applies to BOTH people's ages.
⚡ "In n years" → add n to BOTH sides: (Emma+n) = 2(Jake+n)
A \(4 \text{ years old}\)
B \(6 \text{ years old}\)
C \(8 \text{ years old}\)
D \(12 \text{ years old}\)
✗ Incorrect. Now: \(E = 3J\). In 4 years: \(3J+4 = 2(J+4)\) → \(3J+4 = 2J+8\) → \(J=4\). Jake is 4 years old now. Answer: A.
09
Proportion · Unit Rate
A car travels 150 miles in 2.5 hours. At the same rate, how far does it travel in 4 hours?
⚡ UNIT RATE first: miles ÷ hours = mph → then multiply by new time
A \(200 \text{ miles}\)
B \(220 \text{ miles}\)
C \(240 \text{ miles}\)
D \(260 \text{ miles}\)
✗ Incorrect. Unit rate = \(\frac{150}{2.5} = 60\) mph. Distance in 4 hours = \(60 \times 4 = 240\) miles. Answer: C.
10
Percent Change · The Sneaky One
A shirt costs $80. It goes on sale for 25% off. Then, the sale price is increased by 25%. What is the final price?
⚠ Most students say $80 — they are wrong!
⚡ −25% then +25% ≠ same price! Percentages apply to DIFFERENT bases
A \(\$80.00\)
B \(\$75.00\)
C \(\$82.50\)
D \(\$70.00\)
✗ Incorrect. Sale price: \(80 \times 0.75 = \$60\). Then +25%: \(60 \times 1.25 = \$75\). The two 25%s act on different amounts! Answer: B.
Unit 2

Geometry — Core Concepts

Angles, triangles, area, volume, and the Pythagorean theorem. High-frequency exam questions.

11
Triangle Angles · Interior Sum
In triangle ABC, angle A = 55° and angle B = 78°. What is the measure of angle C?
⚡ TRIANGLE SUM THEOREM: ∠A + ∠B + ∠C = 180° (always, no exceptions)
A \(37°\)
B \(43°\)
C \(47°\)
D \(52°\)
✗ Incorrect. \(55 + 78 + C = 180\) → \(133 + C = 180\) → \(C = 47°\). Answer: C.
12
Pythagorean Theorem · Missing Side
A right triangle has legs of length 5 and 12. What is the length of the hypotenuse?
⚡ a² + b² = c² · Hypotenuse is ALWAYS c (the longest side, opposite right angle)
A \(11\)
B \(13\)
C \(15\)
D \(17\)
✗ Incorrect. \(5^2 + 12^2 = 25 + 144 = 169\). \(\sqrt{169} = 13\). This is the famous 5-12-13 triple! Answer: B.
13
Area of a Triangle · Tricky Height
A triangle has a base of 14 cm and a height of 9 cm. What is its area?
⚠ Many students forget the ½ — do NOT use base × height alone.
⚡ A = ½ × base × height · Think: triangle is HALF of a rectangle
A \(126 \text{ cm}^2\)
B \(63 \text{ cm}^2\)
C \(46 \text{ cm}^2\)
D \(94.5 \text{ cm}^2\)
✗ Incorrect. \(A = \frac{1}{2} \times 14 \times 9 = \frac{126}{2} = 63 \text{ cm}^2\). Option A is the wrong answer students get by forgetting the ½. Answer: B.
14
Circle Area · Radius vs. Diameter
A circle has a diameter of 10 cm. What is its area? Use \(\pi \approx 3.14\).
⚠ The problem gives diameter — you must halve it first!
⚡ A = πr² · r = d ÷ 2 · ALWAYS halve diameter before squaring
A \(314 \text{ cm}^2\)
B \(31.4 \text{ cm}^2\)
C \(78.5 \text{ cm}^2\)
D \(157 \text{ cm}^2\)
✗ Incorrect. Radius = \(\frac{10}{2} = 5\) cm. \(A = 3.14 \times 5^2 = 3.14 \times 25 = 78.5 \text{ cm}^2\). Option A uses diameter as radius — the classic mistake. Answer: C.
15
Supplementary & Complementary Angles
Angle X and angle Y are supplementary. Angle X is 38° more than angle Y. What is angle X?
⚠ Supplementary = sum is 180°. Complementary = sum is 90°.
⚡ S-U-pplementary = 1-8-0° · C-O-mplementary = 9-0° (C comes before S, 90 before 180)
A \(71°\)
B \(109°\)
C \(64°\)
D \(116°\)
✗ Incorrect. Let \(Y = y\), then \(X = y + 38\). Supplementary: \(y + (y+38) = 180\) → \(2y = 142\) → \(y = 71°\). So \(X = 71 + 38 = 109°\). Answer: B.
16
Volume of a Rectangular Prism
A fish tank is 40 cm long, 25 cm wide, and 30 cm tall. If the tank is filled 80% full, how many cubic centimeters of water does it hold?
⚡ V = l × w × h · then multiply by percent as decimal: 80% → × 0.80
A \(24{,}000 \text{ cm}^3\)
B \(30{,}000 \text{ cm}^3\)
C \(25{,}000 \text{ cm}^3\)
D \(27{,}500 \text{ cm}^3\)
✗ Incorrect. Full volume = \(40 \times 25 \times 30 = 30{,}000 \text{ cm}^3\). 80% full: \(30{,}000 \times 0.8 = 24{,}000 \text{ cm}^3\). Answer: A.
17
Parallel Lines & Transversals · Angle Pairs
Two parallel lines are cut by a transversal. One angle measures 65°. What is the measure of the co-interior (same-side interior) angle?
⚠ Co-interior angles are supplementary, NOT equal!
⚡ Alternate angles = EQUAL · Co-interior angles = SUPPLEMENTARY (add to 180°)
A \(65°\)
B \(90°\)
C \(115°\)
D \(125°\)
✗ Incorrect. Co-interior (same-side interior) angles are supplementary: \(65 + x = 180\) → \(x = 115°\). Students often confuse this with alternate angles (which are equal). Answer: C.
18
Perimeter of Composite Shapes
An L-shaped figure is formed by removing a 3 × 4 rectangle from the corner of a 7 × 8 rectangle. What is the perimeter of the L-shape?
⚠ Perimeter = outside boundary only — the cut-out creates NEW outer edges.
⚡ COMPOSITE PERIMETER: trace ONLY outside edges · cut adds edges, doesn't remove outer ones
A \(28 \text{ units}\)
B \(30 \text{ units}\)
C \(34 \text{ units}\)
D \(38 \text{ units}\)
✗ Incorrect. Original rectangle perimeter = \(2(7+8)=30\). Removing a corner 3×4 piece: two sides disappear (3 and 4) but two new sides appear (3 and 4). Net change = 0. Perimeter = 30 units. Answer: B.
19
Similar Triangles · Scale Factor
Triangle PQR is similar to triangle XYZ. In triangle PQR, the sides are 6, 8, and 10. The shortest side of triangle XYZ is 9. What is the longest side of triangle XYZ?
⚡ SIMILAR: all sides share same RATIO (scale factor) · find k first: k = new/old
A \(12\)
B \(13\)
C \(15\)
D \(18\)
✗ Incorrect. Scale factor: \(k = \frac{9}{6} = 1.5\). Apply to longest side: \(10 \times 1.5 = 15\). Answer: C.
20
Exterior Angle Theorem · The Sneakiest One
In a triangle, two interior angles are 42° and 67°. What is the measure of the exterior angle at the third vertex?
⚠ You do NOT need to find the interior angle first!
⚡ EXTERIOR ANGLE = sum of the TWO non-adjacent interior angles (shortcut!)
A \(71°\)
B \(109°\)
C \(118°\)
D \(131°\)
✗ Incorrect. Exterior Angle Theorem: exterior angle = 42 + 67 = 109°. (Alternatively: interior angle = 180 − 42 − 67 = 71°, exterior = 180 − 71 = 109°.) Answer: B.