πŸ¦• Dino Rex's Pre-Algebra Adventure

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πŸ“– Concepts & Key Formulas
πŸ”’
Unit 1Number Systems & Integers
πŸ“Œ Key Concept
Integers include β€¦βˆ’3, βˆ’2, βˆ’1, 0, 1, 2, 3… Absolute value |x| = distance from 0. Adding integers with the same sign: add magnitudes, keep sign. Different signs: subtract smaller from larger, take sign of the larger absolute value.
|a + b| ≀ |a| + |b|
Memorise: |βˆ’7| = 7 Β· (βˆ’3) Γ— (βˆ’4) = +12 Β· (βˆ’3) Γ— (+4) = βˆ’12
✏️ Example
Calculate: βˆ’8 + 3 = ?
Answer: βˆ’5 (larger absolute value is 8, sign is βˆ’)
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Unit 2Fractions & Decimals
πŸ“Œ Key Concept
To add/subtract fractions: find LCD, convert, then operate. To multiply fractions: multiply straight across. To divide: multiply by the reciprocal (flip the second fraction).
a/b Γ· c/d = a/b Γ— d/c
Memorise: LCD Β· KFC rule (Keep, Flip, Change)
✏️ Example
2/3 + 1/4 = ? LCD = 12 β†’ 8/12 + 3/12
Answer: 11/12
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Unit 3Ratios & Proportions
πŸ“Œ Key Concept
A ratio compares two quantities. A proportion states two ratios are equal. Cross-multiply to solve: if a/b = c/d, then ad = bc.
a/b = c/d ⟹ a Γ— d = b Γ— c
Cross-multiplication property
✏️ Example
x/5 = 12/20 β†’ 20x = 60
Answer: x = 3
%
Unit 4Percentages
πŸ“Œ Key Concept
Percent means "per hundred." To find P% of W: multiply W Γ— (P/100). Percent change = (New βˆ’ Old)/Old Γ— 100. Tip: convert percent to decimal by dividing by 100.
Percent Change = ((New βˆ’ Old) / Old) Γ— 100%
Memorise: is/of = %/100
✏️ Example
What is 30% of 80?
Answer: 80 Γ— 0.30 = 24
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Unit 5Expressions & Variables
πŸ“Œ Key Concept
A variable is a letter representing an unknown. An expression is a combination of variables, numbers, and operations (no equal sign). Evaluate by substituting values. Simplify by combining like terms.
3x + 5x = 8x Β· 4a βˆ’ a = 3a
Like terms have identical variable parts
✏️ Example
Evaluate 2xΒ² βˆ’ 3x + 1 when x = 2
Answer: 2(4) βˆ’ 3(2) + 1 = 8 βˆ’ 6 + 1 = 3
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Unit 6Equations
πŸ“Œ Key Concept
An equation has an equal sign. Solve by performing inverse operations to isolate the variable. Whatever you do to one side, do to the other. Two-step equations: first undo addition/subtraction, then multiplication/division.
2x + 5 = 13 β†’ 2x = 8 β†’ x = 4
Inverse operations keep balance
✏️ Example
Solve: 3x βˆ’ 7 = 14
Answer: 3x = 21 β†’ x = 7
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Unit 7Inequalities
πŸ“Œ Key Concept
Solve like equations BUT: when multiplying or dividing by a NEGATIVE number, FLIP the inequality sign! Graph on number line: open circle for < or >, closed circle for ≀ or β‰₯.
βˆ’2x < 8 β†’ x > βˆ’4 (sign flipped!)
⚠️ Flip when Γ· or Γ— by negative!
✏️ Example
Solve: 3x + 4 ≀ 13
Answer: 3x ≀ 9 β†’ x ≀ 3
πŸ“
Unit 8Coordinate Plane & Functions
πŸ“Œ Key Concept
Points are written (x, y). Slope = rise/run = (yβ‚‚ βˆ’ y₁)/(xβ‚‚ βˆ’ x₁). Slope-intercept form: y = mx + b where m = slope and b = y-intercept. Quadrants: I(+,+), II(βˆ’,+), III(βˆ’,βˆ’), IV(+,βˆ’).
slope m = (yβ‚‚ βˆ’ y₁) / (xβ‚‚ βˆ’ x₁)
y = mx + b Β· Memorise: rise over run
✏️ Example
Slope between (1, 3) and (4, 9)?
Answer: (9βˆ’3)/(4βˆ’1) = 6/3 = 2
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Unit 9Exponents & Powers
πŸ“Œ Key Concept
xⁿ means x multiplied by itself n times. Product rule: xᡃ Γ— xᡇ = xᡃ⁺ᡇ. Quotient rule: xᡃ Γ· xᡇ = xᡃ⁻ᡇ. Power of a power: (xᡃ)ᡇ = xᡃᡇ. Any number to the 0 power = 1. Negative exponent: x⁻ⁿ = 1/xⁿ.
x⁰ = 1 Β· x⁻ⁿ = 1/xⁿ Β· xᡃ·xᡇ = xᡃ⁺ᡇ
Laws of Exponents
✏️ Example
Simplify: 2Β³ Γ— 2⁴
Answer: 2³⁺⁴ = 2⁷ = 128
πŸ“
Unit 10Geometry: Area & Perimeter
πŸ“Œ Key Concept
Area is the space inside a shape (square units). Perimeter is the total distance around. Triangle area = Β½ Γ— base Γ— height. Circle: Area = Ο€rΒ², Circumference = 2Ο€r. Use Ο€ β‰ˆ 3.14 unless told otherwise.
Rectangle: A = lΓ—w Β· Triangle: A = Β½bh Β· Circle: A = Ο€rΒ²
Perimeter = sum of all sides Β· C = 2Ο€r
✏️ Example
Area of triangle with base 10 and height 6?
Answer: Β½ Γ— 10 Γ— 6 = 30 sq units
πŸ“ Practice Questions
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Questions Correct
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πŸ“‹ Answer Key & Full Explanations
Q1 β€” Number Systems
βœ“ Answer: A β€” βˆ’5, βˆ’1, 0, 2, 3
On a number line, numbers increase from left to right. More negative = smaller. So the order is βˆ’5 (smallest), βˆ’1, 0, 2, 3 (largest). Option B is greatest to least (reverse). Options C and D break the negative ordering rule.
Q2 β€” Integer Operations
βœ“ Answer: A β€” 9
Follow order of operations (PEMDAS): multiply and divide first, then add.
  • Step 1: (βˆ’4) Γ— (βˆ’3) = +12 (negative Γ— negative = positive)
  • Step 2: (βˆ’6) Γ· 2 = βˆ’3 (negative Γ· positive = negative)
  • Step 3: 12 + (βˆ’3) = 12 βˆ’ 3 = 9
Q3 β€” Fractions
βœ“ Answer: A β€” 7/12
Find LCD of 4 and 6: LCD = 12.
  • 3/4 = 9/12 (multiply top and bottom by 3)
  • 1/6 = 2/12 (multiply top and bottom by 2)
  • 9/12 βˆ’ 2/12 = 7/12
Q4 β€” Fractions (Division)
βœ“ Answer: A β€” 10/3
Convert mixed number and apply KFC (Keep, Flip, Change):
  • 2Β½ = 5/2 (convert to improper fraction: 2Γ—2+1=5)
  • 5/2 Γ· 3/4 = 5/2 Γ— 4/3 (flip 3/4 to 4/3)
  • = (5Γ—4)/(2Γ—3) = 20/6 = 10/3
Q5 β€” Ratios & Proportions
βœ“ Answer: B β€” $15.00
Set up a proportion: 4/6 = 10/x.
  • Cross multiply: 4x = 6 Γ— 10 = 60
  • x = 60 Γ· 4 = $15.00
Q6 β€” Percentages
βœ“ Answer: B β€” $60
Find the discount amount, then subtract:
  • Discount = 25% of $80 = 0.25 Γ— 80 = $20
  • Sale price = $80 βˆ’ $20 = $60
Q7 β€” Percent Change
βœ“ Answer: C β€” 30%
Use percent change formula: (New βˆ’ Old) / Old Γ— 100.
  • Change = 650 βˆ’ 500 = 150
  • Percent change = 150/500 Γ— 100 = 0.30 Γ— 100 = 30%
Q8 β€” Simplifying Expressions
βœ“ Answer: A β€” 3x + 10y
Combine like terms separately:
  • x-terms: 5x βˆ’ 2x = 3x
  • y-terms: 3y + 7y = 10y
  • Result: 3x + 10y
Q9 β€” Evaluating Expressions
βœ“ Answer: A β€” 16
Substitute a = 2, b = βˆ’1, c = 5 into 4a βˆ’ 3b + c:
  • 4(2) = 8
  • βˆ’3(βˆ’1) = +3 (negative Γ— negative = positive)
  • c = 5
  • Total: 8 + 3 + 5 = 16
Q10 β€” Equations
βœ“ Answer: B β€” x = 6
Solve 5x βˆ’ 9 = 21 using inverse operations:
  • Step 1: Add 9 to both sides β†’ 5x = 30
  • Step 2: Divide both sides by 5 β†’ x = 6
  • Check: 5(6) βˆ’ 9 = 30 βˆ’ 9 = 21 βœ“
Q11 β€” Writing Equations
βœ“ Answer: A β€” 3n βˆ’ 8 = 19
Translate word by word:
  • "A number" = n
  • "tripled" = Γ—3 β†’ 3n
  • "8 is subtracted" β†’ 3n βˆ’ 8
  • "giving 19" β†’ = 19
  • Equation: 3n βˆ’ 8 = 19 β†’ n = 9
Q12 β€” Inequalities
βœ“ Answer: A β€” x < βˆ’4
Solve βˆ’3x + 6 > 18:
  • Step 1: Subtract 6 from both sides β†’ βˆ’3x > 12
  • Step 2: Divide by βˆ’3 (FLIP the sign!) β†’ x < βˆ’4
  • ⚠️ Always flip inequality when dividing by a negative number!
Q13 β€” Slope
βœ“ Answer: A β€” 2
Use slope formula m = (yβ‚‚ βˆ’ y₁)/(xβ‚‚ βˆ’ x₁):
  • (x₁,y₁) = (2,5), (xβ‚‚,yβ‚‚) = (6,13)
  • m = (13 βˆ’ 5)/(6 βˆ’ 2) = 8/4 = 2
Q14 β€” Slope-Intercept Form
βœ“ Answer: A β€” (0, 5)
In y = mx + b: m is the slope, b is the y-intercept.
  • y = βˆ’2x + 5: slope m = βˆ’2, y-intercept b = 5
  • The y-intercept is where x = 0: point is (0, 5)
  • Note: (5, 0) would be the x-intercept β€” don't confuse!
Q15 β€” Exponents (Power of a Power)
βœ“ Answer: A β€” 729
Apply the power of a power rule: (xᡃ)ᡇ = xᡃˣᡇ
  • (3Β²)Β³ = 3^(2Γ—3) = 3⁢
  • 3⁢ = 3Γ—3Γ—3Γ—3Γ—3Γ—3 = 729
Q16 β€” Negative Exponents
βœ“ Answer: A β€” 1/25
Negative exponent rule: x⁻ⁿ = 1/xⁿ
  • 5⁻² = 1/5Β² = 1/25
  • Key: Negative exponents do NOT make the result negative! They create fractions.
Q17 β€” Rectangle Perimeter
βœ“ Answer: A β€” 11 cm
Perimeter of rectangle: P = 2l + 2w
  • 36 = 2l + 2(7)
  • 36 = 2l + 14
  • 2l = 22 β†’ l = 11 cm
  • Check: 2(11) + 2(7) = 22 + 14 = 36 βœ“
Q18 β€” Circle Area
βœ“ Answer: A β€” 78.5
Area of circle: A = Ο€rΒ²
  • r = 5, so rΒ² = 25
  • A = 3.14 Γ— 25 = 78.5 square units
  • Note: 31.4 would be the circumference: 2Ο€r = 2Γ—3.14Γ—5 = 31.4
Q19 β€” Distributive Property
βœ“ Answer: A β€” 11x βˆ’ 12
Distribute first, then combine like terms:
  • 3(2x βˆ’ 4) = 3Γ—2x βˆ’ 3Γ—4 = 6x βˆ’ 12
  • Add 5x: 6x βˆ’ 12 + 5x
  • Combine x-terms: 6x + 5x = 11x
  • Result: 11x βˆ’ 12
Q20 β€” Direct Variation
βœ“ Answer: A β€” 40
Direct variation: y = kx, where k is the constant of variation.
  • Find k: 15 = k Γ— 3 β†’ k = 5
  • When x = 8: y = 5 Γ— 8 = 40
  • The ratio y/x always equals k: 15/3 = 5 and 40/8 = 5 βœ“