Math Mastery — Pre-Algebra & Geometry

Pre-Algebra

10 Questions

Order of Operations

Easy

Evaluate: 3 + 4 × 2² − 6 ÷ 3

⚠️ Students often add before multiplying — be careful!

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Memory Point

PEMDAS: Parentheses → Exponents → Multiply/Divide → Add/Subtract

💡 Solution Breakdown

Follow PEMDAS step by step:
3 + 4 × 2² − 6 ÷ 3
Step 1 — Exponents: 2² = 4
→ 3 + 4 × 4 − 6 ÷ 3
Step 2 — Multiply & Divide (left to right): 4×4 = 16, 6÷3 = 2
→ 3 + 16 − 2
Step 3 — Add & Subtract: 3 + 16 − 2 = 17 ✓

Integers & Negative Numbers

Easy

Which expression has the greatest value?
−3 × (−4) | −12 ÷ (−2) | −5 × 3 | 20 ÷ (−4)

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Memory Point

SAME signs → Positive result
DIFFERENT signs → Negative result
"Same = Smile (+), Different = Danger (−)"

💡 Compare All Values

Compute each: A = 12, B = 6, C = −15, D = −5
Ordering: 12 > 6 > −5 > −15
The greatest is A: −3 × (−4) = 12.
Key rule: negative × negative = positive!

Solving One-Step Equations

Easy

Solve for x: 5x − 8 = 22

⚠️ Don't forget to move the constant FIRST before dividing!

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Memory Point

ISOLATE x:
① Move constants → ② Divide by coefficient
"Undo in reverse PEMDAS order"

💡 Step-by-Step

5x − 8 = 22
Add 8 to both sides: 5x = 30
Divide both sides by 5: x = 6
Check: 5(6) − 8 = 30 − 8 = 22 ✓

Fractions & Ratios

Medium

A recipe uses ¾ cup of sugar for 12 cookies. How many cups of sugar are needed for 36 cookies?

⚠️ Many students multiply by 36 directly — first find the scale factor!

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Memory Point

SCALE FACTOR = new ÷ original
Then multiply: original amount × scale factor
"Find the multiplier first!"

💡 Ratio Method

Scale factor: 36 ÷ 12 = 3
Sugar needed: ¾ × 3 = 9/4 = 2¼ cups
Or use proportion: ¾/12 = x/36 → x = ¾ × 3 = 2.25

Percentages

Easy

A shirt originally costs $80. It is on sale for 30% off. What is the sale price?

⚠️ 30% off ≠ final price is 30% of original! Subtract the discount.

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Memory Point

DISCOUNT: Sale Price = Original × (1 − rate)
30% off → multiply by 0.70
"Keep 70%, lose 30%"

💡 Two Methods

Method 1: Discount = 80 × 0.30 = $24 → Sale = 80 − 24 = $56
Method 2: Sale Price = 80 × (1 − 0.30) = 80 × 0.70 = $56
Common mistake: Choosing $24 (which is the discount, not the price!)

Inequalities

Medium

Solve: −2x + 5 > 11. Which represents the solution?

⚠️ The #1 mistake: forgetting to FLIP the inequality when dividing by a negative!

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Memory Point

FLIP the sign when:
× or ÷ by a NEGATIVE number
"Negative flips the sign — always!"

💡 Don't Forget to Flip!

−2x + 5 > 11
Subtract 5: −2x > 6
Divide by −2 (FLIP the sign!): x < −3
If you chose A (x > −3), you forgot to flip — very common error!

Exponents & Scientific Notation

Medium

Simplify: (2³ × 2⁴) ÷ 2⁵

⚠️ Students often multiply the exponents — you ADD when multiplying same bases!

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Memory Point

SAME BASE rules:
× → ADD exponents (aᵐ × aⁿ = aᵐ⁺ⁿ)
÷ → SUBTRACT exponents (aᵐ ÷ aⁿ = aᵐ⁻ⁿ)

💡 Exponent Laws

2³ × 2⁴ = 2^(3+4) = 2⁷
2⁷ ÷ 2⁵ = 2^(7−5) = 2² = 4
Answer: 4. Note: Never multiply the exponents here (that would give 2¹²).

Variables & Expressions

Easy

If a = −2 and b = 5, what is the value of a² − 3b + 1?

⚠️ (−2)² = 4, NOT −4. Squaring removes the negative!

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Memory Point

SUBSTITUTION: Replace variable, keep parentheses!
(−2)² = (−2)(−2) = +4
"Negative squared = POSITIVE"

💡 Substitute Carefully

a² − 3b + 1 = (−2)² − 3(5) + 1
= 4 − 15 + 1
= −10
Careful: (−2)² = 4 (positive!), not −4.

Word Problems — Linear Equations

Medium

A taxi charges $2.50 base fare plus $1.75 per mile. Maria paid $15.75. How many miles did she ride?

⚠️ Set up the equation first — don't just guess and check!

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Memory Point

WORD → EQUATION:
Total = base + (rate × amount)
"Identify: fixed cost + variable cost = total"

💡 Set Up the Equation

Let m = miles.
2.50 + 1.75m = 15.75
1.75m = 13.25
m = 13.25 ÷ 1.75 ≈ 7.57 miles
Closest answer: ~7.6 miles (C)

Graphing & Slope

Medium

A line passes through (1, 3) and (4, 9). What is the equation of the line?

⚠️ Students mix up rise/run — slope is Δy ÷ Δx, NOT Δx ÷ Δy!

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Memory Point

SLOPE = rise ÷ run = (y₂−y₁) ÷ (x₂−x₁)
y = mx + b → plug in a point to find b
"Rise over Run"

💡 Find Slope, Then Intercept

Slope: m = (9−3)÷(4−1) = 6÷3 = 2
Use point (1,3): 3 = 2(1) + b → b = 1
Equation: y = 2x + 1
Check with (4,9): 2(4)+1 = 9 ✓

Geometry

10 Questions

Pythagorean Theorem

Easy

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

⚠️ The hypotenuse is always the longest side — opposite the right angle!

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Memory Point

PYTHAGOREAN THEOREM: a² + b² = c²
c = hypotenuse (longest side)
Common triples: 3-4-5, 5-12-13, 6-8-10

💡 Apply the Theorem

c² = 6² + 8² = 36 + 64 = 100
c = √100 = 10 cm
This is the famous 6-8-10 triple (double of 3-4-5). Memorize it!

Area of Composite Figures

Medium

A figure is made of a rectangle (6×4) with a semicircle on top (diameter = 6). What is the total area? (Use π ≈ 3.14)

⚠️ Students use full circle area instead of HALF — it's a semicircle!

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Memory Point

COMPOSITE = add each part separately
Semicircle Area = ½ × π × r²
"Break it down, add it up"

💡 Split the Shape

Rectangle: 6 × 4 = 24 cm²
Semicircle radius = 3: ½ × 3.14 × 3² = ½ × 28.26 = 14.13 cm²
Total: 24 + 14.13 = 38.13 cm²

Angles — Parallel Lines

Easy

Two parallel lines are cut by a transversal. One angle measures 65°. What is the measure of its co-interior (same-side interior) angle?

⚠️ Co-interior angles are SUPPLEMENTARY (sum to 180°), NOT equal!

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Memory Point

PARALLEL LINE angle pairs:
Alternate interior → EQUAL ("Z angles")
Co-interior/Same-side → SUPPLEMENTARY (= 180°)
Corresponding → EQUAL ("F angles")

💡 Co-Interior = Supplementary

Co-interior angles add up to 180°:
65° + x = 180°
x = 115°
Common mistake: Choosing 65° (alternate interior would be equal, but co-interior are NOT equal).

Volume of 3D Shapes

Easy

A cylinder has a radius of 5 cm and height of 12 cm. What is its volume? (Use π ≈ 3.14)

⚠️ Square the RADIUS first, then multiply — not the diameter!

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Memory Point

CYLINDER Volume = π × r² × h
Radius = diameter ÷ 2
"π r² is the circle area — then × height"

💡 Formula Application

V = π × r² × h
= 3.14 × 5² × 12
= 3.14 × 25 × 12
= 3.14 × 300 = 942 cm³
Error check: If you used diameter (10) instead of radius (5), you'd get 3,768 — way too big!

Triangle Properties

Medium

In a triangle, two angles measure 47° and 83°. The exterior angle at the third vertex is:

⚠️ Exterior angle = sum of the two NON-adjacent interior angles!

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Memory Point

EXTERIOR ANGLE THEOREM:
Exterior angle = sum of 2 remote interior angles
Interior angles of triangle always = 180°

💡 Two Approaches

Method 1 (Exterior Angle Theorem): 47° + 83° = 130°
Method 2 (find third angle first):
Third interior angle = 180° − 47° − 83° = 50°
Exterior = 180° − 50° = 130°
Both give 130°!

Circles — Arc & Sector

Medium

A circle has a radius of 9 cm. A sector has a central angle of 120°. What is the area of this sector? (Use π ≈ 3.14)

⚠️ Students use the full circle area — multiply by the FRACTION of 360°!

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Memory Point

SECTOR Area = (angle ÷ 360) × π × r²
"Sector is a FRACTION of the full circle"
120° = ⅓ of 360°

💡 Fraction of Circle

Full circle area: π × 9² = 3.14 × 81 = 254.34 cm²
Fraction: 120 ÷ 360 = ⅓
Sector area: ⅓ × 254.34 = 84.78 cm²

Similar Triangles

Medium

Triangle ABC ~ Triangle DEF. If AB = 8, BC = 12, and DE = 6, what is EF?

⚠️ Match CORRESPONDING sides — order matters in similar triangles!

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Memory Point

SIMILAR TRIANGLES: corresponding sides proportional
AB/DE = BC/EF = AC/DF
"Set up the proportion — cross multiply to solve"

💡 Set Up the Proportion

Scale factor: DE ÷ AB = 6 ÷ 8 = ¾
So: EF = BC × ¾ = 12 × ¾ = 9
Or: AB/DE = BC/EF → 8/6 = 12/EF → EF = (12×6)/8 = 9

Surface Area

Medium

What is the total surface area of a rectangular prism with length 5 cm, width 3 cm, and height 4 cm?

⚠️ There are 3 PAIRS of faces — don't count just 3 faces!

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Memory Point

SURFACE AREA = 2(lw + lh + wh)
Think: 3 pairs of rectangles × 2
"Box has 6 faces = 3 pairs"

💡 All 6 Faces

SA = 2(lw + lh + wh)
= 2(5×3 + 5×4 + 3×4)
= 2(15 + 20 + 12)
= 2(47) = 94 cm²
Common error: Forgetting the ×2 → gets 47 cm² (answer B — only half the surface).

Coordinate Geometry

Easy

What is the distance between points A(1, 2) and B(7, 10)?

⚠️ This IS the Pythagorean theorem — Δx and Δy are the legs!

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Memory Point

DISTANCE = √[(x₂−x₁)² + (y₂−y₁)²]
= √(Δx² + Δy²)
"It's Pythagorean theorem in disguise!"

💡 Distance Formula

Δx = 7−1 = 6, Δy = 10−2 = 8
d = √(6² + 8²) = √(36 + 64) = √100 = 10
Recognize the 6-8-10 Pythagorean triple again!

Transformations

Medium

Point P(3, −2) is reflected over the y-axis. Then the result is rotated 180° about the origin. What are the final coordinates?

⚠️ Do transformations IN ORDER — reflect first, THEN rotate the new point!

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Memory Point

REFLECTIONS: y-axis: (x,y) → (−x, y)
x-axis: (x,y) → (x, −y)
ROTATION 180°: (x,y) → (−x, −y)
"Order matters! Apply each rule step by step"

💡 Apply Rules In Order

Step 1 — Reflect P(3,−2) over y-axis: (3,−2) → (−3,−2)
Step 2 — Rotate (−3,−2) by 180°: (−3,−2) → (3, 2)
Final answer: (3, 2)
Key: 180° rotation flips BOTH signs: (x,y) → (−x, −y)

Pre-Algebra &Geometry

💡 Solution Breakdown

💡 Compare All Values

💡 Step-by-Step

💡 Ratio Method

💡 Two Methods

💡 Don't Forget to Flip!

💡 Exponent Laws

💡 Substitute Carefully

💡 Set Up the Equation

💡 Find Slope, Then Intercept

💡 Apply the Theorem

💡 Split the Shape

💡 Co-Interior = Supplementary

💡 Formula Application

💡 Two Approaches

💡 Fraction of Circle

💡 Set Up the Proportion

💡 All 6 Faces

💡 Distance Formula

💡 Apply Rules In Order

Pre-Algebra &
Geometry