Table of Contents
Unlike Fractions
Unlike fractions have different denominators. To add or subtract them, we need to find a common denominator first.
Finding the Least Common Denominator (LCD)
To add: 1/3 + 1/4 = ?
Step 1: Find multiples of each denominator:
3: 3, 6, 9, 12, 15...
4: 4, 8, 12, 16...
Step 2: LCD = 12 (smallest common multiple)
Step 3: Convert fractions:
1/3 = 4/12 (multiply by 4/4)
1/4 = 3/12 (multiply by 3/3)
Step 4: Add: 4/12 + 3/12 = 7/12
More Examples
Addition
2/5 + 1/3 = ?
LCD = 15
6/15 + 5/15 = 11/15
Subtraction
3/4 - 1/6 = ?
LCD = 12
9/12 - 2/12 = 7/12
Mixed Number Operations
A mixed number has a whole number and a fraction (like 2½). We can add, subtract, multiply, and divide mixed numbers.
Adding Mixed Numbers
Example: 2¾ + 1½ = ?
Method 1: Add whole numbers and fractions separately
Whole numbers: 2 + 1 = 3
Fractions: 3/4 + 1/2 = 3/4 + 2/4 = 5/4 = 1¼
Total: 3 + 1¼ = 4¼
Converting Mixed to Improper Fractions
2¾ = (2 × 4 + 3) / 4 = 11/4
(whole × denominator + numerator) / denominator
Multiplying Fractions
Rule: Multiply numerator × numerator and denominator × denominator
2×3=6
3412=1/2
Dividing Fractions
Rule: Keep, Change, Flip (KCF)
Keep the first fraction, change ÷ to ×, flip the second fraction.
2/3 ÷ 1/4 = 2/3 × 4/1 = 8/3 = 2⅔
Percentage
Percent means "per hundred" (%). It shows a part of 100.
Conversions
Fraction to %
Divide, then × 100
3/4 = 0.75 = 75%
Decimal to %
Move decimal 2 places right
0.45 = 45%
% to Decimal
Move decimal 2 places left
25% = 0.25
Common Percentage Equivalents
10%
1/10
20%
1/5
25%
1/4
33⅓%
1/3
50%
1/2
66⅔%
2/3
75%
3/4
100%
1
Finding Percentage of a Number
What is 25% of 80?
Method 1: 25/100 × 80 = 2000/100 = 20
Method 2: 0.25 × 80 = 20
Ratio
A ratio compares two or more quantities. It can be written as a:b, a to b, or a/b.
Example: A class has 12 boys and 18 girls.
Ratio of boys to girls: 12:18 or 2:3 (simplified)
Ratio of girls to boys: 18:12 or 3:2
Ratio of boys to total: 12:30 or 2:5
Simplifying Ratios
Divide both parts by their Greatest Common Factor (GCF).
12:18
÷6 ÷6
2:3
15:25
÷5 ÷5
3:5
24:36
÷12 ÷12
2:3
Proportion
A proportion is an equation showing that two ratios are equal.
a=c
bd
Cross multiply: a × d = b × c
Solving Proportions
Example: Find x if 3/4 = x/20
Cross multiply: 4 × x = 3 × 20
4x = 60
x = 60 ÷ 4
x = 15
Word Problem
If 3 pencils cost ₱15, how much do 7 pencils cost?
3/15 = 7/x
3x = 15 × 7
3x = 105
x = 35
Answer: ₱35
Area of Complex Shapes
Triangle
A = ½ × base × height
Example: base = 6cm, height = 4cm
A = ½ × 6 × 4 = 12 cm²
Parallelogram
A = base × height
Example: base = 8cm, height = 5cm
A = 8 × 5 = 40 cm²
Trapezoid
A = ½ × (b₁ + b₂) × h
Example: bases = 5cm, 7cm; height = 4cm
A = ½ × (5+7) × 4 = 24 cm²
Circle
A = π × r²
Example: radius = 5cm (π ≈ 3.14)
A = 3.14 × 5 × 5 = 78.5 cm²
For complex shapes: Break them into simpler shapes, calculate each area, then add or subtract.
Volume
Volume is the amount of space inside a 3D shape. Measured in cubic units (cm³, m³).
Rectangular Prism (Box)
V = length × width × height
Example: 5cm × 4cm × 3cm
V = 5 × 4 × 3 = 60 cm³
Cube
V = side × side × side = s³
Example: side = 4cm
V = 4³ = 4 × 4 × 4 = 64 cm³
Relationship: Area vs Volume
Area (2D)
Measures surface in square units (cm², m²)
Example: floor of a room
Volume (3D)
Measures space in cubic units (cm³, m³)
Example: water in a tank
Data & Graphs
Types of Graphs
Bar Graph
Compares different categories using bars.
Example: Number of students per subject
Line Graph
Shows changes over time using connected points.
Example: Temperature changes over a week
Pie Chart
Shows parts of a whole as slices of a circle.
Example: Expenses breakdown
Pictograph
Uses pictures/symbols to represent data.
Example: Books read (1 symbol = 5 books)
Measures of Central Tendency
Mean (Average)
Sum of all values ÷ number of values
Data: 5, 8, 10, 12, 15 → Mean = 50 ÷ 5 = 10
Median
Middle value when data is arranged in order
Data: 5, 8, 10, 12, 15 → Median = 10
Mode
Value that appears most often
Data: 3, 5, 5, 7, 5, 9 → Mode = 5 (appears 3 times)
Range
Highest value - Lowest value
Data: 5, 8, 10, 12, 15 → Range = 15 - 5 = 10