Pre-Algebra & Geometry

Pre-Algebra

10 core problems · Variables, equations, ratios, percentages & more

Q 01 Pre-Algebra ⚠ Tricky Variables & Expressions

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Memory Point

SUBSTITUTE → SIMPLIFY → SOLVE
Replace the variable first, then calculate step by step.

📘 Example

If x = 4, find the value of 3x + 5.
→ Substitute: 3(4) + 5 = 12 + 5 = 17

If n = −3, what is the value of 2n² − n + 4?

📖 Explanation

Substitute n = −3:
2(−3)² − (−3) + 4
= 2(9) + 3 + 4
= 18 + 3 + 4 = 25

⚠️ Common mistake: Students often forget that (−3)² = +9, not −9. A negative number squared is always positive!

Q 02 Pre-Algebra Solving Equations

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Memory Point

INVERSE OPERATIONS
Whatever is done to x, undo it — do the same to both sides.

📘 Example

Solve: 3x − 7 = 14
→ Add 7: 3x = 21 → Divide by 3: x = 7

Solve for x: 5x − 3 = 2x + 12

📖 Explanation

5x − 3 = 2x + 12
Subtract 2x from both sides: 3x − 3 = 12
Add 3 to both sides: 3x = 15
Divide by 3: x = 5

Q 03 Pre-Algebra ⚠ Tricky Ratios & Proportions

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Memory Point

CROSS-MULTIPLY → DIVIDE
a/b = c/d → a×d = b×c

📘 Example

Solve: 3/4 = x/20
→ Cross-multiply: 3 × 20 = 4 × x → 60 = 4x → x = 15

A recipe uses 3 cups of flour for every 2 cups of sugar. If you use 9 cups of flour, how many cups of sugar do you need?

📖 Explanation

Set up the proportion: 3/2 = 9/x
Cross-multiply: 3x = 18
Divide: x = 6 cups

Q 04 Pre-Algebra Percentages

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Memory Point

IS / OF = % / 100
"What % of 80 is 20?" → 20/80 × 100

📘 Example

What is 30% of 60?
→ 0.30 × 60 = 18

A jacket originally costs $80. It is on sale for 25% off. What is the sale price?

📖 Explanation

Discount = 25% of $80 = 0.25 × 80 = $20
Sale price = $80 − $20 = $60

Q 05 Pre-Algebra ⚠ Tricky Order of Operations

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Memory Point

PEMDAS = Parentheses · Exponents · Multiply/Divide · Add/Subtract
Always left to right for same-level operations!

📘 Example

Simplify: 4 + 2 × (3²)
→ Exponent first: 3² = 9 → Multiply: 2 × 9 = 18 → Add: 4 + 18 = 22

Evaluate: 3 + 4 × (8 − 5)² ÷ 6

📖 Explanation

Step 1 – Parentheses: (8 − 5) = 3
Step 2 – Exponent: 3² = 9
Step 3 – Multiply: 4 × 9 = 36
Step 4 – Divide: 36 ÷ 6 = 6
Step 5 – Add: 3 + 6 = 9

Q 06 Pre-Algebra Inequalities

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Memory Point

FLIP when DIVIDE/MULTIPLY by NEGATIVE
−2x < 6 → x > −3 (sign flips!)

📘 Example

Solve: −3x < 9
→ Divide by −3 and flip: x > −3

Which value of x is a solution to −2x + 5 > 11?

📖 Explanation

−2x + 5 > 11
Subtract 5: −2x > 6
Divide by −2 (flip the sign!): x < −3
Only x = −4 satisfies x < −3. Answer: x = −4

Q 07 Pre-Algebra ⚠ Tricky Fractions & Operations

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Memory Point

LCD → SAME DENOMINATOR → ADD NUMERATORS
Never add denominators! Only add numerators.

📘 Example

1/3 + 1/4 → LCD = 12 → 4/12 + 3/12 = 7/12

What is the value of ⅔ + ¾ − ½?

📖 Explanation

LCD of 3, 4, 2 = 12
2/3 = 8/12 · 3/4 = 9/12 · 1/2 = 6/12
8/12 + 9/12 − 6/12 = 11/12

Q 08 Pre-Algebra Number Patterns

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Memory Point

FIND THE RULE: ADD? MULTIPLY? SQUARE?
Look at differences between terms first.

📘 Example

Pattern: 2, 5, 8, 11, ___
→ Adding 3 each time → Next: 14

What is the next term in the pattern: 3, 6, 12, 24, ___?

📖 Explanation

Each term is multiplied by 2 (geometric sequence):
3 × 2 = 6 · 6 × 2 = 12 · 12 × 2 = 24 · 24 × 2 = 48

Q 09 Pre-Algebra ⚠ Tricky Word → Equation

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Memory Point

"MORE THAN" = ADD · "TIMES" = MULTIPLY · "IS" = EQUALS
Translate each word into a math symbol.

📘 Example

"5 more than three times a number is 20"
→ 3x + 5 = 20 → x = 5

Tom has twice as many stickers as Emma. Together they have 42 stickers. How many does Tom have?

📖 Explanation

Let Emma = x, Tom = 2x
x + 2x = 42 → 3x = 42 → x = 14
Tom = 2 × 14 = 28

Q 10 Pre-Algebra Integer Operations

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Memory Point

SAME SIGNS → ADD · DIFFERENT SIGNS → SUBTRACT (keep bigger's sign)
−8 + 3 = −5 (8 > 3, so negative wins)

📘 Example

(−5) × (−3) = +15 · (−4) × (+2) = −8

What is the value of (−4) + (−7) × 2 − (−3)?

📖 Explanation

Follow PEMDAS:
(−7) × 2 = −14 first
(−4) + (−14) − (−3)
= −4 − 14 + 3 = −15

Geometry

10 core problems · Angles, area, perimeter, Pythagorean theorem & more

G 01 Geometry Angles

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Memory Point

SUPPLEMENTARY = 180° · COMPLEMENTARY = 90°
Sup → Straight line. Com → Corner (right angle).

📘 Example

Two supplementary angles: one is 65°. The other = 180° − 65° = 115°

Two angles are complementary. One angle measures 37°. What is the measure of the other angle?

📖 Explanation

Complementary angles add up to 90°.
90° − 37° = 53°

G 02 Geometry ⚠ Tricky Area of Triangle

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Memory Point

A = ½ × BASE × HEIGHT
HEIGHT must be perpendicular to the base — not the slanted side!

📘 Example

Triangle: base = 10, height = 6
A = ½ × 10 × 6 = 30

A triangle has a base of 14 cm and a height of 9 cm. What is its area?

📖 Explanation

A = ½ × base × height
A = ½ × 14 × 9 = 7 × 9 = 63 cm²

⚠️ Common mistake: Forgetting to divide by 2! A triangle is exactly half a rectangle.

G 03 Geometry Pythagorean Theorem

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Memory Point

a² + b² = c²
c = HYPOTENUSE (longest side, opposite the right angle)

📘 Example

a = 3, b = 4 → c² = 9 + 16 = 25 → c = 5

A right triangle has legs of length 6 cm and 8 cm. What is the length of the hypotenuse?

📖 Explanation

a² + b² = c²
6² + 8² = 36 + 64 = 100
c = √100 = 10 cm
(3-4-5 triple doubled: 6-8-10)

G 04 Geometry ⚠ Tricky Circle: Circumference

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Memory Point

C = 2πr = πd
RADIUS = half of diameter. Don't mix them up!

📘 Example

Circle with radius 5: C = 2π(5) = 10π ≈ 31.4

A circle has a diameter of 14 cm. What is its circumference? (Use π ≈ 3.14)

📖 Explanation

C = π × d = 3.14 × 14 = 43.96 cm

⚠️ Common mistake: Using C = 2πr and forgetting that r = 7 (half of 14), which gives the same answer but watch your step!

G 05 Geometry Perimeter

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Memory Point

PERIMETER = ADD ALL SIDES
For rectangles: P = 2(length + width)

📘 Example

Rectangle: l = 8, w = 3 → P = 2(8 + 3) = 22

A rectangular garden is 15 m long and 8 m wide. What is the total length of fencing needed to go around it?

📖 Explanation

P = 2(l + w) = 2(15 + 8) = 2(23) = 46 m

G 06 Geometry ⚠ Tricky Triangle Angles

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Memory Point

ALL 3 ANGLES IN A TRIANGLE = 180°
Missing angle = 180° − (sum of the other two)

📘 Example

Triangle angles: 90°, 40°, ? → 180° − 90° − 40° = 50°

In a triangle, two angles measure 48° and 73°. What is the third angle?

📖 Explanation

Sum of all angles = 180°
Third angle = 180° − 48° − 73° = 59°

G 07 Geometry Area of Circle

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Memory Point

A = πr²
Square the RADIUS, then multiply by π. (Not the diameter!)

📘 Example

r = 5 → A = π × 5² = 25π ≈ 78.5

A circular pond has a radius of 7 m. What is its area? (Use π ≈ 3.14)

📖 Explanation

A = π × r² = 3.14 × 7² = 3.14 × 49 = 153.86 m²

G 08 Geometry ⚠ Tricky Volume of Rectangular Prism

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Memory Point

V = LENGTH × WIDTH × HEIGHT
Volume is 3D — multiply all THREE dimensions.

📘 Example

Box: 4 × 3 × 5 = V = 60 cm³

A fish tank is 50 cm long, 30 cm wide, and 40 cm tall. How many cubic centimeters of water can it hold?

📖 Explanation

V = l × w × h = 50 × 30 × 40 = 60,000 cm³

G 09 Geometry Coordinate Geometry

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Memory Point

DISTANCE = √[(x₂−x₁)² + (y₂−y₁)²]
Think Pythagorean theorem on a grid!

📘 Example

Points (0,0) and (3,4):
d = √(3² + 4²) = √(9+16) = √25 = 5

What is the distance between points A(1, 2) and B(5, 5)?

📖 Explanation

d = √[(5−1)² + (5−2)²]
= √[4² + 3²]
= √[16 + 9]
= √25 = 5

G 10 Geometry ⚠ Tricky Similar Figures

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Memory Point

SIMILAR = SAME SHAPE, DIFFERENT SIZE → RATIOS ARE EQUAL
Set up a proportion: side₁/side₂ = other side₁/other side₂

📘 Example

Similar triangles: sides 3 & 4 match to sides 6 & ?
3/6 = 4/x → x = 8

Two similar rectangles: the first has sides 4 cm and 6 cm. The second has a longer side of 9 cm. What is its shorter side?

📖 Explanation

Set up proportion with corresponding sides:
4/6 = x/9
6x = 36 → x = 6 cm