Math Competition Prep — Pre-Algebra & Geometry

Section 01 — Pre-Algebra

Word Problems

Ratios, equations, rates, percent, LCM/GCF — the core of pre-algebra competition problems.

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Ratios Tricky

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KEY WORD: "RATIO → PART = UNIT" — find 1 unit first, then scale. If a:b and you know one part, divide to find 1 unit, multiply to find total.

A bag contains red and blue marbles in the ratio $3:5$. If there are 18 red marbles, how many total marbles are in the bag?

✦ Step-by-Step Solution

①Ratio red : blue = 3 : 5. Red = 3 parts = 18 marbles.

②1 part = 18 ÷ 3 = 6 marbles.

③Total parts = 3 + 5 = 8. Total marbles = 8 × 6 = 48.

⚠Common trap: students calculate only blue (30) or forget to add both parts. Always find total parts = a + b.

LCM Key Concept

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LCM = Least Common Multiple. Use prime factorization: take the HIGHEST power of each prime. LCM(a,b,c) = multiply all unique primes at their max powers.

Three buses leave a station at the same time. Bus A returns every 4 hours, Bus B every 6 hours, and Bus C every 9 hours. After how many hours will all three buses be at the station together again for the first time?

✦ Step-by-Step Solution

①Prime factorize: 4 = 2², 6 = 2×3, 9 = 3².

②Take highest powers: 2² and 3².

③LCM = 2² × 3² = 4 × 9 = 36 hours.

⚠Trap: choosing 72 (just multiplying all three), or 18 (forgetting 4's factor of 4). Always use prime factorization — don't just multiply.

Distance & Rate Tricky

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D = R × T (Distance = Rate × Time). When two objects move in the same direction, relative speed = |r₁ − r₂|. Gap = relative speed × time.

A train travels at 60 mph and a car travels at 45 mph in the same direction, both starting from the same point at the same time. How many miles apart are they after 2.5 hours?

✦ Step-by-Step Solution

①Relative speed (same direction) = 60 − 45 = 15 mph.

②Gap = 15 × 2.5 = 37.5 miles.

③Alternatively: Train travels 60 × 2.5 = 150 mi. Car travels 45 × 2.5 = 112.5 mi. Difference = 150 − 112.5 = 37.5 miles. ✓

⚠Trap: D adds speeds (wrong — that's opposite directions). C subtracts wrong values. D is total distance of both, not the gap.

Equations Key Concept

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ISOLATE the variable: undo operations in reverse order (PEMDAS backwards). Fractions? Multiply both sides by the denominator first.

$\dfrac{3}{4}$ of a number, decreased by 8, equals 13. What is the number?

✦ Step-by-Step Solution

①Write the equation: $\frac{3}{4}n - 8 = 13$.

②Add 8 to both sides: $\frac{3}{4}n = 21$.

③Multiply both sides by $\frac{4}{3}$: $n = 21 \times \frac{4}{3} = \frac{84}{3} = \mathbf{28}$.

⚠Trap: Dividing 21 by 3 alone gives 7, then forgetting to multiply by 4. Always flip the fraction when moving it to the other side.

Percent Tricky

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PERCENT of WHAT: "P% of W = A" → W = A ÷ (P/100) = A × (100/P). The word "of" means multiply; "what" is the unknown base.

40% of a certain number is 26. What is 75% of that same number?

✦ Step-by-Step Solution

①0.40 × n = 26 → n = 26 ÷ 0.40 = 65.

②75% of 65 = 0.75 × 65 = 48.75.

⚠Trap: A (39) = 60% of 65, not 75%. D (65) is the number itself, not 75% of it. This is a two-step problem — don't stop at finding the number.

Continued · 6 – 10

Consecutive Integers Key Concept

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CONSECUTIVE INTEGERS: label them n, n+1, n+2, … Sum of k consecutive integers always has a shortcut: middle value × k. Even consecutive: n, n+2, n+4, …

The sum of four consecutive integers is 94. What is the largest of the four integers?

✦ Step-by-Step Solution

①Let the integers be $n, n+1, n+2, n+3$.

②Sum: $4n + 6 = 94$ → $4n = 88$ → $n = 22$.

③The four integers are 22, 23, 24, 25. Largest = 25. ✓ Check: 22+23+24+25 = 94 ✓

⚠Trap: Answering 22 = the smallest, not the largest. Always re-read — the question asks for the LARGEST.

Work Rate Tricky Classic

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WORK RATE FORMULA: Rate = 1/Time. Together: 1/T_A + 1/T_B = 1/T_together. Think of rates as "fraction of job per hour" — ADD the fractions, then flip.

Pipe A can fill a tank in 3 hours and Pipe B can fill the same tank in 6 hours. If both pipes are opened simultaneously, how many hours does it take to fill the tank?

✦ Step-by-Step Solution

①A's rate = $\frac{1}{3}$ tank/hr. B's rate = $\frac{1}{6}$ tank/hr.

②Combined rate = $\frac{1}{3} + \frac{1}{6} = \frac{2}{6} + \frac{1}{6} = \frac{3}{6} = \frac{1}{2}$ tank/hr.

③Time = $\frac{1}{\text{rate}} = \frac{1}{1/2} = \mathbf{2 \text{ hours}}$.

⚠Trap: averaging (4.5 hrs) or adding times (9 hrs). Never add times — ADD rates, then flip to get total time.

Absolute Value Tricky

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|x| = k means TWO equations: x = k OR x = −k. Always solve BOTH cases. Don't forget the negative case — that's where most students lose points.

Solve: $|3x - 7| = 14$. What is the positive solution for $x$?

✦ Step-by-Step Solution

①Case 1: $3x - 7 = 14$ → $3x = 21$ → $x = \mathbf{7}$. ✓ (positive)

②Case 2: $3x - 7 = -14$ → $3x = -7$ → $x = -\frac{7}{3}$. (negative)

③The positive solution is $\mathbf{x = 7}$. Check: |3(7)−7| = |21−7| = |14| = 14 ✓

⚠Trap: only solving Case 1, or mishandling the negative case. Competition problems often ask for the sum or product of both solutions.

Percent Change Tricky Classic

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DOUBLE PERCENT TRAP: +20% then −20% ≠ 0%. Always apply percent changes to the NEW value each time. The net change = 1.2 × 0.8 = 0.96, not 1.00.

The price of a jacket increases by 20%, then later decreases by 20%. The final price compared to the original is:

✦ Step-by-Step Solution

①Start with $100. After +20%: 100 × 1.20 = $120.

②After −20%: 120 × 0.80 = $96.

③Net change = 96 − 100 = −4. So 4% less than original.

④Formula: (1 + r)(1 − r) = 1 − r². Here r = 0.2, so 1 − 0.04 = 0.96, which is 4% less. ✓

⚠The most common wrong answer is A — "they cancel out." This only works for addition and subtraction, NOT multiplication (percentages).

Combined Work Tricky Classic

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TOGETHER FORMULA shortcut: If A takes 'a' days and B takes 'b' days, together = ab/(a+b). Memorize this — it saves time in competitions.

Alice can complete a project in 8 days. Bob can complete the same project in 12 days. If they work together, how many days will it take? Express as a fraction if needed.

✦ Step-by-Step Solution

①Alice's rate = $\frac{1}{8}$/day. Bob's rate = $\frac{1}{12}$/day.

②Combined = $\frac{1}{8} + \frac{1}{12} = \frac{3}{24} + \frac{2}{24} = \frac{5}{24}$/day.

③Time = $\frac{24}{5} = \mathbf{4.8}$ days.

④Shortcut check: $\frac{8 \times 12}{8+12} = \frac{96}{20} = \frac{24}{5}$ ✓

⚠Trap: A=5 (wrong addition of rates), C=10 (average of 8 and 12). The answer 24/5 is less than either individual time — always sanity-check this.

Section 02 — Geometry

Shapes, Angles & Space

Triangles, circles, angles, similarity, volume — the geometric concepts tested in every major competition.

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Triangle Area Key Concept

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TRIANGLE AREA = ½ × base × height. The height must be PERPENDICULAR to the base — it is NOT always a side of the triangle. Watch for right angle marks.

A triangle has a base of 14 cm and a height of 9 cm. A rectangle has the same area as this triangle. If the rectangle's length is 7 cm, what is its width?

✦ Step-by-Step Solution

①Triangle area = $\frac{1}{2} \times 14 \times 9 = 7 \times 9 = \mathbf{63}$ cm².

②Rectangle area = length × width = 63 cm².

③Width = 63 ÷ 7 = 9 cm.

⚠Trap: D (63) is the area, not the width. A (7) = the length given, not the width. Always complete ALL steps of a multi-part problem.

Circles Key Concept

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CIRCLE CHEAT SHEET: C = 2πr = πd. A = πr². Diameter = 2r. Leave answers in terms of π when asked — it's more precise and competition-standard.

A circle has a diameter of 14 cm. What is the exact circumference and area of this circle? (Leave answers in terms of $\pi$.)

✦ Step-by-Step Solution

①Diameter = 14, so radius r = 7 cm.

②Circumference = $2\pi r = 2\pi(7) = \mathbf{14\pi}$ cm.

③Area = $\pi r^2 = \pi(7)^2 = \mathbf{49\pi}$ cm².

⚠Trap B: C = 7π uses radius instead of diameter in the formula. C uses r=14. D uses diameter in C formula correctly but gets C=28π (uses diameter as if it's radius in C=2πr → wrong). Be careful: C = πd = π(14) = 14π, or C = 2π(7) = 14π. Same result.

Pythagorean Theorem Classic

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PYTHAGOREAN TRIPLES to memorize: 3-4-5, 5-12-13, 8-15-17, 7-24-25. These scale up (×2, ×3 …). Recognizing them saves huge time in competitions.

A right triangle has legs of length 5 and 12. A square is drawn on the hypotenuse. What is the area of that square?

✦ Step-by-Step Solution

①Hypotenuse $c = \sqrt{5^2 + 12^2} = \sqrt{25 + 144} = \sqrt{169} = \mathbf{13}$.

②Area of square on hypotenuse = $c^2 = 13^2 = \mathbf{169}$.

③Shortcut: The area of the square on the hypotenuse = $a^2 + b^2 = 25 + 144 = 169$. No need to find c first!

⚠This is the 5-12-13 Pythagorean triple. Trap A = 5²+12²−something. B = 12². Always memorize the classic triples.

Complementary Angles Tricky

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ANGLE VOCABULARY: Complementary = sum is 90°. Supplementary = sum is 180°. Vertical angles = equal. Remember: "C" in Complementary → 90 (Corner). "S" in Supplementary → 180 (Straight).

Two angles are complementary. One angle measures $3x°$ and the other measures $(x + 10)°$. What is the measure of the larger angle?

✦ Step-by-Step Solution

①Complementary: $3x + (x+10) = 90$ → $4x + 10 = 90$ → $4x = 80$ → $x = 20$.

②Angle 1 = $3(20) = 60°$. Angle 2 = $20 + 10 = 30°$.

③The larger angle = 60°. Check: 60 + 30 = 90 ✓

⚠Trap A: x = 20 is not an angle. Trap B: 30° is the smaller angle. Always re-read which angle is asked for.

Rectangle Key Concept

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PERIMETER SHORTCUT: 2(l + w) = P → l + w = P/2. Set up the relationship between l and w from the word problem, substitute, solve. Always draw and label!

The perimeter of a rectangle is 54 cm. The length is 3 cm more than twice the width. What is the area of the rectangle?

✦ Step-by-Step Solution

①Let width = $w$. Then length = $2w + 3$.

②Perimeter: $2(w + 2w + 3) = 54$ → $2(3w + 3) = 54$ → $3w + 3 = 27$ → $3w = 24$ → $w = 8$ cm.

③Length = $2(8) + 3 = 19$ cm. Area = $19 \times 8 = \mathbf{152}$ cm².

④Check perimeter: 2(19 + 8) = 2(27) = 54 ✓

⚠Trap A: using perimeter formula wrong. B: using w=8 and l=16 (forgetting +3). Always write the equation for l in terms of w carefully.

Continued · 16 – 20

Volume — Cylinder Key Concept

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3D VOLUME CHEAT SHEET: Cylinder = πr²h. Cone = ⅓πr²h. Sphere = (4/3)πr³. Prism = Base Area × h. Pattern: "Pointy shapes" (cone, pyramid) = ⅓ × flat version.

A cylindrical can has a radius of 3 cm and a height of 10 cm. What is its volume? (Leave in terms of $\pi$.)

✦ Step-by-Step Solution

①Volume of cylinder = $\pi r^2 h$.

②$V = \pi (3)^2 (10) = \pi \times 9 \times 10 = \mathbf{90\pi}$ cm³.

⚠Trap A: using r instead of r² (π×3×10 = 30π). Trap B: forgetting to square (2×3×10 = 60). Trap D: using diameter (6)² = 36, not radius² = 9. Always square the radius.

Exterior Angles Tricky

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EXTERIOR ANGLE THEOREM: An exterior angle of a triangle = the SUM of the two non-adjacent (remote) interior angles. This is one of the most-tested geometry theorems.

In a triangle, an exterior angle measures 110°. One of the two non-adjacent interior angles is 65°. What is the measure of the third interior angle of the triangle?

✦ Step-by-Step Solution

①Exterior angle = sum of two remote interior angles: $110 = 65 + x$ → $x = \mathbf{45°}$.

②Verify: The adjacent interior angle = 180 − 110 = 70°. All three interior angles: 65 + 45 + 70 = 180° ✓

⚠Trap C: 70° is the interior angle adjacent to the exterior angle (supplement), NOT the third interior angle asked for. Trap B: 110−65=45, not 55 — careful with arithmetic.

Similar Triangles Classic

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SIMILAR TRIANGLES: Corresponding sides are PROPORTIONAL. Find the scale factor = (new side)/(original side), then multiply ALL other sides by that same scale factor.

Two triangles are similar. The first triangle has sides 6, 8, and 10. The second triangle has a shortest side of 9. What is the longest side of the second triangle?

✦ Step-by-Step Solution

①First triangle sides: 6, 8, 10 (note: 3-4-5 triple × 2). Shortest = 6.

②Scale factor = new shortest ÷ original shortest = 9 ÷ 6 = 1.5.

③Longest side of second triangle = 10 × 1.5 = 15.

④Full second triangle: 9, 12, 15 (= 3-4-5 triple × 3). ✓

⚠Trap A=12 is the middle side, not the longest. Trap D=16 adds 6 instead of multiplying by 1.5. Always use multiplication, never addition, for similar figures.

Sector Area Tricky

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SECTOR = "PIZZA SLICE": Area = (θ/360) × πr². Arc length = (θ/360) × 2πr. The fraction θ/360 tells you "what fraction of the full circle." Always put the angle over 360.

A circle has radius 6 cm. A sector of this circle has a central angle of 60°. What is the exact area of the sector?

✦ Step-by-Step Solution

①Full circle area = $\pi (6)^2 = 36\pi$ cm².

②Sector fraction = $\frac{60}{360} = \frac{1}{6}$.

③Sector area = $\frac{1}{6} \times 36\pi = \mathbf{6\pi}$ cm².

④Quick check: 60° = 1/6 of 360°. Makes sense — the sector is 1/6 of the whole pizza. ✓

⚠Trap A: using radius (6) not radius² (36) in the area formula. Trap C: using 60/360 with wrong circle area. Always compute full circle area FIRST, then take the fraction.

Rectangle Diagonal Classic Tricky

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DIAGONAL of RECTANGLE: The diagonal splits the rectangle into two right triangles. Use the Pythagorean theorem: d = √(l² + w²). Memorize 6-8-10 as 3-4-5 scaled by 2.

A rectangle has sides of 6 cm and 8 cm. A diagonal divides it into two triangles. What is the perimeter of one of those triangles?

✦ Step-by-Step Solution

①Diagonal = $\sqrt{6^2 + 8^2} = \sqrt{36+64} = \sqrt{100} = 10$ cm. (This is the 3-4-5 triple × 2!)

②Each triangle has sides: 6, 8, and 10 (the diagonal).

③Perimeter of one triangle = 6 + 8 + 10 = 24 cm.

⚠Trap D=10: that's only the diagonal, not the perimeter. Trap B=28: incorrectly adds 6+8+10+4. Trap C=30: using the full rectangle perimeter. The triangle has only THREE sides: 6, 8, and 10.

Pre-Algebra &Geometry Mastery

Pre-Algebra &
Geometry Mastery