Pre-Algebra
Number, Expression & Equation
Q 1 – 10
⚡ Quick Memory Points
- Order of Operations → PEMDAS (Parentheses · Exponents · Multiply/Divide · Add/Subtract)
- Negative × Negative = Positive → "Two wrongs make a right"
- Distributive Law → a(b + c) = ab + ac
- Proportion → cross-multiply to solve
- Percent → part ÷ whole × 100
1
Order of Operations · PEMDAS
Evaluate: 3 + 4 × 2 − (6 ÷ 3)
KEY: Multiply BEFORE you add. Parentheses FIRST.
✦ Worked Example
2 + 3 × 4 − (8 ÷ 4)
Step 1 – Parentheses: (8 ÷ 4) = 2
Step 2 – Multiply: 3 × 4 = 12
Step 3 – Left to right: 2 + 12 − 2 = 12
Step 1 – Parentheses: (8 ÷ 4) = 2
Step 2 – Multiply: 3 × 4 = 12
Step 3 – Left to right: 2 + 12 − 2 = 12
2
Integer Operations · Negatives
Simplify: −3 × (−4) + (−5)
KEY: neg × neg = pos | pos + neg = subtract
✦ Worked Example
−2 × (−6) + (−3)
Step 1: −2 × −6 = +12
Step 2: 12 + (−3) = 12 − 3 = 9
Step 1: −2 × −6 = +12
Step 2: 12 + (−3) = 12 − 3 = 9
3
Distributive Property
Which expression equals −2(x − 5)?
KEY: Distribute the negative sign to EVERY term inside!
✦ Worked Example
3(x − 4) = 3·x − 3·4 = 3x − 12
−2(x + 3) = −2·x + (−2)·3 = −2x − 6
−2(x + 3) = −2·x + (−2)·3 = −2x − 6
4
Solving One-Step Equations
Solve for x: 3x = −18
KEY: Divide both sides by the coefficient
✦ Worked Example
5x = −35 → divide both sides by 5
x = −35 ÷ 5 = −7
x = −35 ÷ 5 = −7
5
Two-Step Equations
Solve for n: 2n + 7 = 19
KEY: Undo ADD/SUBTRACT first, then MULTIPLY/DIVIDE
✦ Worked Example
4n + 3 = 23
Step 1 – subtract 3: 4n = 20
Step 2 – divide by 4: n = 5
Step 1 – subtract 3: 4n = 20
Step 2 – divide by 4: n = 5
6
Ratio & Proportion
A recipe uses 3 cups of flour for every 2 cups of sugar. How many cups of sugar are needed for 9 cups of flour?
KEY: Set up ratio → cross-multiply → solve
✦ Worked Example
5 apples cost $2. 15 apples cost x.
5/2 = 15/x → 5x = 30 → x = $6
5/2 = 15/x → 5x = 30 → x = $6
7
Percentages
A shirt costs $40. It is on sale for 25% off. What is the sale price?
KEY: Discount = % × original. Sale price = original − discount
✦ Worked Example
$60 item, 10% off:
Discount = 0.10 × 60 = $6
Sale price = 60 − 6 = $54
Discount = 0.10 × 60 = $6
Sale price = 60 − 6 = $54
8
Exponents
What is the value of 23 + 32?
KEY: 2³ = 2×2×2 = 8 | 3² = 3×3 = 9
✦ Worked Example
42 + 23
= 16 + 8 = 24
= 16 + 8 = 24
9
Combining Like Terms
Simplify: 5x − 3 + 2x + 8
KEY: Group x-terms together, group numbers together
✦ Worked Example
4a + 2 − a + 5
= (4a − a) + (2 + 5) = 3a + 7
= (4a − a) + (2 + 5) = 3a + 7
10
Word Problem · Equations
Maya has $5 more than twice the amount Jake has. Together they have $41. How much does Jake have?
KEY: Let j = Jake. Maya = 2j + 5. Write: j + (2j+5) = 41
✦ Set Up
j + 2j + 5 = 41
3j + 5 = 41
3j = 36
j = 12
3j + 5 = 41
3j = 36
j = 12
Geometry
Shape, Angle & Measurement
Q 11 – 20
⚡ Quick Memory Points
- Triangle angles → always add to 180°
- Straight line → 180° | Full turn → 360°
- Pythagorean Theorem → a² + b² = c² (c = hypotenuse)
- Area of triangle → ½ × base × height
- Circle formulas → C = 2πr | A = πr²
- Complementary = 90° | Supplementary = 180°
11
Triangle Angle Sum
A triangle has angles of 47° and 83°. What is the third angle?
KEY: Sum of triangle angles = 180°. Third = 180 − (a + b)
✦ Worked Example
Angles: 55° and 75°
Third = 180 − (55 + 75) = 180 − 130 = 50°
Third = 180 − (55 + 75) = 180 − 130 = 50°
12
Supplementary & Complementary Angles
Angle A and Angle B are supplementary. If Angle A = 68°, what is Angle B?
KEY: Supplementary = 180° | Complementary = 90°
✦ Worked Example
Supplementary: A + B = 180°
A = 55° → B = 180 − 55 = 125°
A = 55° → B = 180 − 55 = 125°
13
Area of a Triangle
A triangle has a base of 10 cm and a height of 6 cm. What is its area?
KEY: Area = ½ × base × height (NOT base × height)
✦ Worked Example
base = 8, height = 5
Area = ½ × 8 × 5 = 20 cm²
Area = ½ × 8 × 5 = 20 cm²
14
Pythagorean Theorem
A right triangle has legs of 6 cm and 8 cm. What is the length of the hypotenuse?
KEY: a² + b² = c² → c = √(a²+b²)
✦ Worked Example
legs = 3 and 4
c² = 3² + 4² = 9 + 16 = 25
c = √25 = 5
c² = 3² + 4² = 9 + 16 = 25
c = √25 = 5
15
Perimeter of Rectangle
A rectangle has a length of 9 m and a width of 4 m. What is its perimeter?
KEY: Perimeter = 2 × (length + width) ← don't forget the 2!
✦ Worked Example
length=7, width=3
P = 2 × (7 + 3) = 2 × 10 = 20 m
P = 2 × (7 + 3) = 2 × 10 = 20 m
16
Circle · Circumference
A circle has a radius of 7 cm. What is its circumference? (Use π ≈ 3.14)
KEY: C = 2πr (radius!) | C = πd (diameter!)
✦ Worked Example
radius = 5
C = 2 × 3.14 × 5 = 31.4 cm
C = 2 × 3.14 × 5 = 31.4 cm
17
Vertical Angles
Two straight lines intersect. One angle formed is 65°. What is the measure of the angle directly opposite it?
KEY: Vertical angles = equal. Adjacent angles on a line = supplementary (180°)
✦ Worked Example
18
Area of Circle
A circular pizza has a diameter of 12 inches. What is the area of the pizza? (Use π ≈ 3.14)
KEY: Diameter given → radius = diameter ÷ 2 FIRST, then A = πr²
✦ Worked Example
diameter = 10 → radius = 5
A = 3.14 × 5² = 3.14 × 25 = 78.5 in²
A = 3.14 × 5² = 3.14 × 25 = 78.5 in²
19
Volume of Rectangular Prism
A box has a length of 5 cm, width of 4 cm, and height of 3 cm. What is its volume?
KEY: Volume = length × width × height (l × w × h)
✦ Worked Example
l=4, w=3, h=2
V = 4 × 3 × 2 = 24 cm³
V = 4 × 3 × 2 = 24 cm³
20
Similar Triangles · Scale Factor
Two similar triangles have corresponding sides in a ratio of 2 : 5. The smaller triangle has a side of 8 cm. What is the corresponding side of the larger triangle?
KEY: Similar = same shape, different size. Set up proportion: small/large = 2/5
✦ Worked Example
Ratio 3:7, small side = 6
6/x = 3/7 → 3x = 42 → x = 14
6/x = 3/7 → 3x = 42 → x = 14