Pre-Algebra
Integers · Fractions · Equations · Ratios · Exponents
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Memory Key PEMDAS: Parentheses → Exponents → Multiply/Divide → Add/Subtract (left to right)
Quick Example
\(2 + 3 \times 4 = 2 + 12 = 14\) (NOT 20 — multiply first!)
Evaluate: \(3 + 4 \times (2^2 - 1) \div 3\)
Step-by-Step Solution
Step 1 — Parentheses: \(2^2 - 1 = 4 - 1 = 3\)Step 2 — Multiply & Divide (left to right): \(4 \times 3 \div 3 = 12 \div 3 = 4\)
Step 3 — Add: \(3 + 4 = 7\) ✓
Trap: Many students add 3+4 first and get 28. Always multiply/divide before adding!
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Memory Key "Subtracting a negative = ADDING": \(a - (-b) = a + b\)
Quick Example
\(5 - (-3) = 5 + 3 = 8\) (two negatives → positive)
What is \(-8 - (-13) + (-5)\)?
Step-by-Step Solution
Rewrite: \(-8 + 13 + (-5)\)\(= (-8 - 5) + 13 = -13 + 13 = 0\) ✓
Trap: \(-(-13)\) becomes \(+13\), not still \(-13\). Double negative always flips the sign!
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Memory Key LCD (Least Common Denominator) first — then add numerators only, NEVER add denominators!
Quick Example
\(\dfrac{1}{4} + \dfrac{1}{6} = \dfrac{3}{12} + \dfrac{2}{12} = \dfrac{5}{12}\)
Simplify: \(\dfrac{3}{4} - \dfrac{2}{3} + \dfrac{1}{6}\)
Step-by-Step Solution
LCD of 4, 3, 6 = 12\(\dfrac{9}{12} - \dfrac{8}{12} + \dfrac{2}{12} = \dfrac{9-8+2}{12} = \dfrac{3}{12} = \dfrac{1}{4}\) ✓
Trap: Don't add/subtract denominators (e.g., 4+3+6=13). Find LCD first!
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Memory Key INVERSE OPERATION: whatever is done TO x, undo it on BOTH sides equally.
Quick Example
\(x + 5 = 12 \Rightarrow x = 12 - 5 = 7\)
Solve for \(x\): \(-3x + 7 = -5\)
Step-by-Step Solution
\(-3x + 7 = -5\)Subtract 7: \(-3x = -12\)
Divide by \(-3\): \(x = 4\) ✓
Trap: Dividing by a negative doesn't change direction in an equation (only in inequalities)!
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Memory Key CROSS-MULTIPLY: \(\dfrac{a}{b} = \dfrac{c}{d} \Rightarrow ad = bc\)
Quick Example
\(\dfrac{3}{4} = \dfrac{x}{8} \Rightarrow 3 \times 8 = 4x \Rightarrow x = 6\)
A recipe uses 2 cups of flour for every 3 cookies. How many cups of flour are needed to make 45 cookies?
Step-by-Step Solution
Set up proportion: \(\dfrac{2}{3} = \dfrac{x}{45}\)Cross-multiply: \(3x = 90\)
Divide: \(x = 30\) cups ✓
Trap: Don't multiply 2 × 45 and forget to divide by 3!
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Memory Key IS/OF rule: \(\text{percent} = \dfrac{\text{IS}}{\text{OF}} \times 100\) or \(\text{part} = \text{rate} \times \text{whole}\)
Quick Example
30% of 80 = \(0.30 \times 80 = 24\)
A shirt costs $40. After a 15% discount, what is the sale price?
Step-by-Step Solution
Discount = \(0.15 \times 40 = 6\)Sale price = \(40 - 6 = \$34\) ✓
OR: \(40 \times 0.85 = \$34\) (multiply by 1 − 0.15)
Trap: The question asks for sale price, NOT the discount amount ($6).
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Memory Key Negative exponent = FLIP it: \(a^{-n} = \dfrac{1}{a^n}\). Zero exponent: anything\(^0 = 1\).
Quick Example
\(2^{-3} = \dfrac{1}{2^3} = \dfrac{1}{8}\) ; \(5^0 = 1\)
Simplify: \(2^3 \times 2^{-1} + 3^0\)
Step-by-Step Solution
\(2^3 \times 2^{-1} = 2^{3+(-1)} = 2^2 = 4\) (add exponents when multiplying same base)\(3^0 = 1\)
Answer: \(4 + 1 = 5\) ✓
Trap: \(3^0\) is NOT 0 — any nonzero number to the power 0 is 1!
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Memory Key DISTRIBUTE means multiply outside value to EVERY term inside: \(a(b+c) = ab + ac\)
Quick Example
\(3(x + 5) = 3x + 15\) ; \(-2(x - 4) = -2x + 8\)
Simplify: \(-2(3x - 4) + 5x\)
Step-by-Step Solution
\(-2(3x - 4) = -6x + 8\) (distribute \(-2\) to both terms — note sign flip!)Add \(5x\): \(-6x + 8 + 5x = -x + 8\) ✓
Trap: \(-2 \times (-4) = +8\) (negative × negative = positive). Many students write \(-8\) by mistake!
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Memory Key FLIP the inequality sign when multiplying or dividing by a NEGATIVE number!
Quick Example
\(-2x > 6 \Rightarrow x < -3\) (sign flips!)
Solve: \(-4x + 3 \leq 11\) and identify the correct solution.
Step-by-Step Solution
\(-4x + 3 \leq 11\)Subtract 3: \(-4x \leq 8\)
Divide by \(-4\) → FLIP sign: \(x \geq -2\) ✓
Trap: Forgetting to flip the inequality sign when dividing by \(-4\). This is the #1 most common mistake!
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Memory Key READ → DEFINE variable → WRITE equation → SOLVE → CHECK back in original problem.
Quick Example
"3 more than twice a number is 13" → \(2n + 3 = 13 \Rightarrow n = 5\)
Tom saves $12 each week. He already has $35 saved. After how many full weeks will he have at least $95?
Step-by-Step Solution
Equation: \(12w + 35 \geq 95\)\(12w \geq 60\)
\(w \geq 5\) → minimum 5 weeks ✓
Trap: Don't forget the $35 he already has — always include starting amount in word problems!
Geometry
Angles · Triangles · Circles · Area · Pythagorean Theorem · Volume
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Memory Key C for Complementary = Corner (90°). S for Supplementary = Straight line (180°). "C comes before S → 90° before 180°"
Quick Example
Complement of 35° = \(90 - 35 = 55°\) Supplement of 35° = \(180 - 35 = 145°\)
Two angles are supplementary. One angle is three times the other. Find the smaller angle.
Step-by-Step Solution
Let smaller angle = \(x\). Then larger = \(3x\).Supplementary: \(x + 3x = 180°\)
\(4x = 180 \Rightarrow x = 45°\) ✓
Trap: The question asks for the smaller angle (45°), not the larger one (135°)!
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Memory Key ALL triangles: interior angles always add up to EXACTLY 180°. No exceptions!
Quick Example
Angles 50° + 70° + ? = 180° → missing angle = 60°
A triangle has angles in ratio 1 : 2 : 3. What is the largest angle?
Step-by-Step Solution
Ratio sum = \(1+2+3 = 6\) partsEach part = \(180° \div 6 = 30°\)
Largest angle = \(3 \times 30° = 90°\) ✓ (This is a right triangle!)
Note: Any triangle with angle ratio 1:2:3 is always a right triangle.
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Memory Key \(a^2 + b^2 = c^2\) where \(c\) is ALWAYS the HYPOTENUSE (longest side, opposite right angle).
Quick Example
Legs 3, 4 → hypotenuse: \(\sqrt{3^2 + 4^2} = \sqrt{25} = 5\) (the famous 3-4-5 triple!)
A ladder leans against a wall. The base is 5 m from the wall, and the ladder is 13 m long. How high does it reach?
Step-by-Step Solution
Hypotenuse = ladder = 13 m, one leg = 5 m, find other leg \(h\):\(5^2 + h^2 = 13^2\)
\(25 + h^2 = 169\)
\(h^2 = 144 \Rightarrow h = 12\) m ✓ (5-12-13 Pythagorean triple!)
Trap: Don't add: \(5^2 + 13^2\). The 13 is the hypotenuse, so it goes on the right side of the equation.
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Memory Key SPLIT complex shape into simple ones (rectangles, triangles). ADD or SUBTRACT their areas.
Quick Example
L-shape = big rectangle − small rectangle removed from corner
A rectangle is 10 cm × 6 cm. A square of side 2 cm is cut out from one corner. What is the remaining area?
Step-by-Step Solution
Rectangle area = \(10 \times 6 = 60\) cm²Square cut out = \(2 \times 2 = 4\) cm²
Remaining = \(60 - 4 = 56\) cm² ✓
Trap: Choice A (60 cm²) is the original area before cutting — read the question carefully!
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Memory Key "Cherry Pie Delicious — Apple Pie Round": \(C = \pi d\) & \(A = \pi r^2\). radius = diameter ÷ 2!
Quick Example
Circle radius 5: \(C = 2\pi(5) = 10\pi\) ; \(A = \pi(5)^2 = 25\pi\)
A circle has a diameter of 10 cm. Which value equals its area? (Use \(\pi \approx 3.14\))
Step-by-Step Solution
Diameter = 10, so radius = 5 cmArea = \(\pi r^2 = 3.14 \times 5^2 = 3.14 \times 25 = 78.5\) cm² ✓
Trap: Using diameter (10) instead of radius (5) in the formula gives \(3.14 \times 100 = 314\) — that's choice B and a very common error!
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Memory Key "Alternate = Equal, Co-interior = Add up to 180°". F-angles (corresponding) = equal; Z-angles (alternate) = equal; C-angles (co-interior) = 180°.
Quick Example
If corresponding angle = 70°, then its pair = 70° (same position on parallel lines)
Two parallel lines are cut by a transversal. One co-interior (same-side interior) angle is 112°. What is the other co-interior angle?
Step-by-Step Solution
Co-interior angles are supplementary (add to 180°):\(180° - 112° = 68°\) ✓
Trap: Co-interior angles are NOT equal — that's alternate interior angles. Many students choose 112° by mistake.
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Memory Key Volume = Base Area × Height. For rectangular prism: \(V = l \times w \times h\). Don't confuse with surface area!
Quick Example
Box 3×4×5: \(V = 60\) units³ ; SA = \(2(3×4 + 4×5 + 3×5) = 94\) units²
A fish tank is 50 cm long, 30 cm wide, and 40 cm deep. How many liters of water does it hold? (1 liter = 1,000 cm³)
Step-by-Step Solution
Volume = \(50 \times 30 \times 40 = 60{,}000\) cm³Convert: \(60{,}000 \div 1{,}000 = 60\) liters ✓
Trap: Forgetting to divide by 1,000 to convert cm³ to liters — or dividing by 100 (wrong conversion).
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Memory Key Similar triangles: same SHAPE, different SIZE. Corresponding sides in proportion. \(\dfrac{a_1}{a_2} = \dfrac{b_1}{b_2} = \dfrac{c_1}{c_2}\)
Quick Example
Triangles 3-4-5 and 6-8-? → scale factor 2 → missing side = 10
Two similar triangles have corresponding sides in ratio 3 : 5. The smaller triangle has an area of 27 cm². What is the area of the larger triangle?
Step-by-Step Solution
Side ratio = \(\dfrac{3}{5}\) → Area ratio = \(\left(\dfrac{3}{5}\right)^2 = \dfrac{9}{25}\)\(\dfrac{27}{A} = \dfrac{9}{25} \Rightarrow A = \dfrac{27 \times 25}{9} = 75\) cm² ✓
Trap: Students multiply \(27 \times \dfrac{5}{3} = 45\) (using side ratio, not squared). Area scales by the square of the side ratio!
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Memory Key EXTERIOR angle = SUM of TWO non-adjacent interior angles. (The outside angle "collects" both far-away angles.)
Quick Example
Interior angles 40° and 65° → exterior angle opposite = \(40 + 65 = 105°\)
In a triangle, two interior angles are 48° and 73°. What is the exterior angle at the third vertex?
Step-by-Step Solution
Exterior angle = \(48° + 73° = 121°\) ✓OR: Third interior angle = \(180 - 48 - 73 = 59°\), exterior = \(180 - 59 = 121°\) ✓
Trap: Some students subtract to find the third angle (59°) and stop there — but 59° is the interior angle, not the exterior!
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Memory Key Cylinder SA = 2 circles + rectangle "unrolled": \(SA = 2\pi r^2 + 2\pi r h\). Factor: \(2\pi r(r + h)\)
Quick Example
Cylinder r=3, h=5: \(SA = 2\pi(9) + 2\pi(3)(5) = 18\pi + 30\pi = 48\pi\)
A cylinder has radius 4 cm and height 7 cm. What is the total surface area? (Leave answer in terms of \(\pi\))
Step-by-Step Solution
Two circles: \(2\pi r^2 = 2\pi(16) = 32\pi\)Lateral rectangle: \(2\pi r h = 2\pi(4)(7) = 56\pi\)
Total: \(32\pi + 56\pi = 88\pi\) cm² ✓
Trap: Forgetting the TWO circles (only computing lateral area = \(56\pi\)).