UNIT 1
Number & Operations
Integers · Fractions · Decimals
Key Concepts & Formulas to Memorize
Integers
Whole numbers and their negatives: …, −3, −2, −1, 0, 1, 2, 3, …
Rules: (+)(+) = (+) | (−)(−) = (+) | (+)(−) = (−)
Rules: (+)(+) = (+) | (−)(−) = (+) | (+)(−) = (−)
Fractions
a/b where b ≠ 0. To add/subtract: use a common denominator.
To multiply: (a/b) × (c/d) = ac/bd. To divide: multiply by the reciprocal.
To multiply: (a/b) × (c/d) = ac/bd. To divide: multiply by the reciprocal.
Decimals
Move decimal point: ×10 shifts right, ×0.1 shifts left.
To compare decimals, align decimal points.
To compare decimals, align decimal points.
LCM (Least Common Multiple) used for common denominators
Worked Examples
Example
Compute −3 + (−5)
= −(3 + 5) = −8
Example
Compute 3/4 + 1/6
LCD = 12 → 9/12 + 2/12 = 11/12
Practice Problems
Q1
Evaluate: −8 + 15 − (−4)
Q2
Simplify: (2/3) ÷ (4/9)
UNIT 2
Ratios & Proportions
Rates · Unit Rate · Proportions
Key Concepts & Formulas to Memorize
Ratio
A comparison of two quantities. Written as a:b or a/b.
Proportion
Two equal ratios: a/b = c/d → ad = bc (cross-multiply)
Unit Rate
Rate with denominator 1. Example: 60 miles per hour.
Cross-multiplication: a/b = c/d ⟹ a × d = b × c
Worked Examples
Example
If 5 apples cost $3.00, find cost of 8 apples.
5/3 = 8/x → x = 24/5 = $4.80
Example
Simplify ratio 36 : 48
GCF(36, 48) = 12 → ratio = 3 : 4
Practice Problems
Q3
A car travels 150 miles in 3 hours. How far does it travel in 5 hours?
Q4
Solve the proportion: x / 9 = 4 / 6
UNIT 3
Percentages
Percent · Discount · Tax · Percent Change
Key Concepts & Formulas to Memorize
Percent Formula
Part = Percent% × Whole
Percent Change
% Change = [(New − Old) / Old] × 100
Discount / Tax
Discount = Rate × Original Price; Tax is added to original price.
Part / Whole = Percent / 100
Worked Examples
Example
What is 35% of 80?
0.35 × 80 = 28
Example
Price drops from $50 to $40. What is the % decrease?
[(50 − 40) / 50] × 100 = 20%
Practice Problems
Q5
A jacket costs $120. It is on sale for 25% off. What is the sale price?
Q6
A population grew from 800 to 1000. What is the percent increase?
UNIT 4
Expressions & Equations
Variables · Like Terms · Solving Equations
Key Concepts & Formulas to Memorize
Variable
A letter representing an unknown value. Example: x, y, n.
Like Terms
Terms with the same variable and exponent. Example: 3x + 5x = 8x.
Solving
Use inverse operations to isolate the variable. Keep both sides balanced.
ax + b = c → x = (c − b) / a
Worked Examples
Example
Simplify: 4x + 3 − 2x + 7
= (4x − 2x) + (3 + 7) = 2x + 10
Example
Solve: 3x + 5 = 20
3x = 15 → x = 5
Practice Problems
Q7
Simplify: 5y − 3 + 2y + 8
Q8
Solve: 2x − 7 = 13
UNIT 5
Inequalities
Solving & Graphing Inequalities
Key Concepts & Formulas to Memorize
Symbols
< less than | > greater than | ≤ at most | ≥ at least
Solving
Same as equations EXCEPT: flip the inequality sign when multiplying or dividing by a negative number.
Graphing
Open circle for < or > | Closed circle for ≤ or ≥
If −a < b then a > −b (flip sign when × or ÷ by negative)
Worked Examples
Example
Solve: 2x + 3 > 11
2x > 8 → x > 4
Example
Solve: −3x ≤ 12
Divide by −3 (flip!): x ≥ −4
Practice Problems
Q9
Solve and describe the solution: 4x − 1 < 15
Q10
Solve: −2x + 6 ≥ 14
UNIT 6
Exponents & Square Roots
Powers · Laws of Exponents · Radicals
Key Concepts & Formulas to Memorize
Exponent Rules
xm · xn = xm+n |
xm / xn = xm−n |
(xm)n = xmn
Zero & Negative
x0 = 1 | x−n = 1/xn
Square Roots
√(a·b) = √a · √b | Simplify by finding perfect-square factors.
√(x²) = |x| (always non-negative)
Worked Examples
Example
Simplify: 23 × 24
= 27 = 128
Example
Simplify: √48
= √(16 × 3) = 4√3
Practice Problems
Q11
Simplify: 35 ÷ 32
Q12
Simplify: √72
UNIT 7
Geometry Basics
Perimeter · Area · Volume
Key Concepts & Formulas to Memorize
Perimeter & Area
Rectangle: P = 2(l+w), A = l×w
Triangle: P = a+b+c, A = ½b×h
Circle: C = 2πr, A = πr²
Triangle: P = a+b+c, A = ½b×h
Circle: C = 2πr, A = πr²
Volume
Rectangular prism: V = l×w×h
Cylinder: V = πr²h
Cylinder: V = πr²h
Area of circle = π × r² (use π ≈ 3.14)
Worked Examples
Example
Find area of triangle with base 10 cm, height 6 cm.
A = ½ × 10 × 6 = 30 cm²
Example
Find circumference of circle with radius 7 cm.
C = 2π(7) ≈ 43.96 cm
Practice Problems
Q13
A rectangle has length 12 m and width 5 m. Find its area and perimeter.
Q14
Find the volume of a rectangular prism with l = 8, w = 3, h = 5 (all in cm).
UNIT 8
Coordinate Plane
Plotting Points · Slope · Linear Equations
Key Concepts & Formulas to Memorize
Ordered Pairs
Point (x, y): x is horizontal distance, y is vertical distance from origin.
Slope
m = (y₂ − y₁) / (x₂ − x₁)
Slope-Intercept
y = mx + b where m = slope, b = y-intercept
Slope = rise / run = Δy / Δx
Worked Examples
Example
Find slope between (2, 3) and (6, 11).
m = (11−3) / (6−2) = 8/4 = 2
Example
Write equation with slope 3, y-intercept −1.
y = 3x − 1
Practice Problems
Q15
Find the slope of the line passing through (−1, 4) and (3, −4).
Q16
A line has slope 2 and passes through (0, −5). Write its equation.
UNIT 9
Statistics & Data
Mean · Median · Mode · Range
Key Concepts & Formulas to Memorize
Mean
Sum of all values ÷ number of values.
Median
Middle value when data is ordered. If even number of values, average the two middle values.
Mode & Range
Mode = most frequent value | Range = Max − Min
Mean = (Sum of all data) / (Number of data points)
Worked Examples
Example
Data: 4, 7, 2, 9, 3. Find mean, median, range.
Mean = (4+7+2+9+3)/5 = 5 | Median: 2,3,4,7,9 → 4 | Range = 9−2 = 7
Practice Problems
Q17
Data set: 12, 8, 15, 8, 10, 14. Find the mean, median, mode, and range.
Q18
The mean of five numbers is 12. Four of them are 10, 15, 8, and 14. Find the fifth number.
UNIT 10
Probability
Basic Probability · Outcomes · Complement
Key Concepts & Formulas to Memorize
Probability
P(event) = Favorable outcomes / Total possible outcomes
Range: 0 ≤ P ≤ 1
Range: 0 ≤ P ≤ 1
Complement
P(not A) = 1 − P(A)
Sample Space
All possible outcomes listed systematically.
P(A) = Favorable outcomes / Total outcomes
Worked Examples
Example
A bag has 3 red, 2 blue, 5 green marbles. Find P(red).
P(red) = 3/10
Example
Find P(not red) from the same bag.
P(not red) = 1 − 3/10 = 7/10
Practice Problems
Q19
A standard die (1–6) is rolled. What is the probability of rolling a number greater than 4?
Q20
A spinner has 8 equal sections numbered 1–8. What is the probability of landing on an even number?
Answer Key & Full Solutions
Complete worked solutions for all 20 questions
Q1
−8 + 15 − (−4) = −8 + 15 + 4 = 7 + 4
Answer: 11
Q2
(2/3) ÷ (4/9) = (2/3) × (9/4) = 18/12 = 3/2
Answer: 3/2 (or 1.5)
Q3
150 mi ÷ 3 hr = 50 mph (unit rate) → 50 × 5 = 250
Answer: 250 miles
Q4
x/9 = 4/6 → Cross-multiply: 6x = 36 → x = 36 ÷ 6
Answer: x = 6
Q5
Discount = 25% × $120 = $30 → Sale price = $120 − $30
Answer: $90
Q6
% increase = [(1000 − 800) / 800] × 100 = (200/800) × 100
Answer: 25%
Q7
Combine like terms: (5y + 2y) + (−3 + 8) = 7y + 5
Answer: 7y + 5
Q8
2x = 13 + 7 = 20 → x = 20 ÷ 2
Answer: x = 10
Q9
4x < 15 + 1 = 16 → x < 4 (open circle at 4, shaded to the left)
Answer: x < 4
Q10
−2x ≥ 14 − 6 = 8 → Divide by −2 (flip the sign!) → x ≤ −4
Answer: x ≤ −4
Q11
35 ÷ 32 = 35−2 = 33 = 27
Answer: 27
Q12
√72 = √(36 × 2) = √36 × √2 = 6√2 (perfect square factor: 36)
Answer: 6√2
Q13
Area = 12 × 5 = 60 m² | Perimeter = 2(12 + 5) = 2 × 17 = 34 m
Answer: Area = 60 m², Perimeter = 34 m
Q14
V = l × w × h = 8 × 3 × 5 = 120
Answer: 120 cm³
Q15
m = (−4 − 4) / (3 − (−1)) = −8 / 4
Answer: slope = −2
Q16
m = 2, b = −5 → substitute into y = mx + b
Answer: y = 2x − 5
Q17
Ordered: 8, 8, 10, 12, 14, 15
Mean = (8+8+10+12+14+15) / 6 = 67/6 ≈ 11.17
Median = (10 + 12) / 2 = 11 | Mode = 8 | Range = 15 − 8 = 7
Answer: Mean ≈ 11.2, Median = 11, Mode = 8, Range = 7
Q18
Required sum = 12 × 5 = 60
Sum of known four: 10 + 15 + 8 + 14 = 47
Fifth number = 60 − 47
Answer: 13
Q19
Numbers > 4 on a die: {5, 6} → 2 favorable outcomes out of 6 total
P = 2/6 = 1/3
Answer: 1/3
Q20
Even numbers in 1–8: {2, 4, 6, 8} → 4 favorable outcomes out of 8 total
P = 4/8 = 1/2
Answer: 1/2