Ratio, Proportion & Percentage - Cheat Sheet
Your last-minute revision companion for mastering ratio, proportion, and percentage problems in Philippine entrance exams
Ratio Fundamentals
Formulas
a:b = a/b
Meaning
a and b are quantities being compared, ratio can be expressed as fraction
Watch out
Always simplify to lowest terms - don't leave ratios like 6:9 instead of 2:3
When to use
When comparing two quantities in simplest form
a:b:c = xa:xb:xc
Meaning
x is the common factor, a:b:c represents three-part ratio
Watch out
Remember that total parts = a+b+c, not just two parts
When to use
When dealing with three or more quantities in proportion
Important facts
- Ratios must be in same units before comparing
- Order matters in ratios - 3:4 ≠ 4:3
- Cross multiplication works: if a:b = c:d, then ad = bc
- Ratios can be written as fractions, decimals, or with colon notation
Key definitions
Ratio
Example
Boys to girls = 3:4 means 3 boys for every 4 girls
Comparison of two or more quantities of the same kind
Equivalent Ratios
Example
2:4 = 1:2 = 3:6
Ratios that express the same relationship when simplified
Diagrams to know
- Part-to-part vs part-to-whole ratio diagrams
- Ratio distribution charts
Proportion
Formulas
a:b = c:d or a/b = c/d
Meaning
Two ratios are equal, forming a proportion
Watch out
Cross multiply correctly: ad = bc, not ac = bd
When to use
When two ratios are stated to be equal
x = (a×d)/b when a:b = c:d and c = x
Meaning
Solving for unknown term in proportion using cross multiplication
Watch out
Make sure you multiply and divide in correct order
When to use
When one term in a proportion is unknown
Important facts
- Cross multiplication: if a:b = c:d, then ad = bc
- Three terms proportion: a:b = b:c means b² = ac
- Check if ratios are proportional by cross multiplying
- Word problems often hide proportional relationships
Key definitions
Proportion
Example
5:10 = 3:6 (both equal 1:2)
Statement that two ratios are equal
Direct Proportion
Example
More workers, more work done (at same rate)
As one quantity increases, the other increases proportionally
Inverse Proportion
Example
More speed, less time for same distance
As one quantity increases, the other decreases proportionally
Diagrams to know
- Cross multiplication diagrams
- Direct vs inverse proportion graphs
Reactions or equations
If a:b = c:d, then ad = bc
Note
Fundamental property for solving proportion problems
Conditions
All terms must be non-zero
Percentage Basics
Formulas
Percentage = (Part/Whole) × 100%
Meaning
Part is the portion, Whole is the total, result is percentage
Watch out
Don't forget to multiply by 100 - decimal 0.25 = 25%
When to use
Finding what percent one number is of another
Part = (Percentage/100) × Whole
Meaning
Finding the actual amount when percentage and total are known
Watch out
Convert percentage to decimal first: 25% = 0.25
When to use
When calculating percentage of a given amount
Whole = (Part × 100)/Percentage
Meaning
Finding the total when part and percentage are known
Watch out
This is the BASE in percentage problems
When to use
Finding original amount from percentage information
Common values
Value
50%
Symbol
1/2
Quantity
Half
Value
25%
Symbol
1/4
Quantity
Quarter
Value
75%
Symbol
3/4
Quantity
Three quarters
Important facts
- 100% = 1.00 = whole amount
- 50% = 0.5 = half
- 25% = 0.25 = one quarter
- Percent means 'per hundred'
- Converting: multiply by 100 to get %, divide by 100 to get decimal
Key definitions
Percentage
Example
25% = 25/100 = 0.25
A fraction expressed as parts per hundred
Base
Example
In '20% of 500', the base is 500
The original amount or whole quantity (100%)
Rate
Example
In '20% of 500', the rate is 20%
The percentage itself
Percentage (Amount)
Example
In '20% of 500 = 100', the percentage amount is 100
The result of percentage calculation
Diagrams to know
- Percentage circle diagrams
- Part-whole relationship charts
Reactions or equations
Decimal × 100 = Percentage
Note
0.75 × 100 = 75%
Conditions
For conversion between forms
Percentage ÷ 100 = Decimal
Note
45% ÷ 100 = 0.45
Conditions
For calculation purposes
Percentage Applications
Formulas
Discount = Original Price × Discount Rate
Meaning
Amount saved when discount rate is applied to original price
Watch out
Sale price = Original price - Discount, not just the discount amount
When to use
Calculating money saved in sales/discounts
Sale Price = Original Price × (100% - Discount Rate)
Meaning
Final price after discount is applied
Watch out
Convert to decimal: 30% discount means multiply by 0.70, not 0.30
When to use
Finding final price after discount
Tax Amount = Base Price × Tax Rate
Meaning
Additional amount added as tax
Watch out
Final price = Base price + Tax amount
When to use
Calculating VAT, sales tax, or other taxes
Profit % = ((Selling Price - Cost Price)/Cost Price) × 100
Meaning
Percentage profit based on cost price
Watch out
Loss % uses same formula but result is negative
When to use
Calculating profit percentage in business problems
Common values
Value
12%
Symbol
VAT
Quantity
Philippine VAT
Important facts
- Philippine VAT rate is typically 12%
- Discount is subtracted, tax is added
- Profit/Loss % is always calculated on Cost Price
- Successive discounts: apply one after another, not add percentages
Key definitions
Markup
Example
Cost ₱100, 20% markup = Selling price ₱120
Percentage added to cost price to get selling price
Discount
Example
₱100 item, 20% discount = Pay ₱80
Percentage reduction from marked price
VAT (Value Added Tax)
Example
₱100 + 12% VAT = ₱112 total
Tax added to the price of goods and services
Diagrams to know
- Price breakdown charts
- Tax and discount flow diagrams
Reactions or equations
Final Price = Base × (1 + Tax Rate) × (1 - Discount Rate)
Note
Order matters: usually discount first, then tax
Conditions
When both tax and discount apply
Percentage Change
Formulas
Percentage Change = ((New Value - Old Value)/Old Value) × 100
Meaning
Shows increase or decrease as percentage of original
Watch out
Negative result means decrease, positive means increase
When to use
Comparing two values to show change
New Value = Old Value × (1 + Rate of Change)
Meaning
Calculating new value after percentage increase/decrease
Watch out
For decrease, use (1 - Rate), for increase use (1 + Rate)
When to use
Finding final value after percentage change
Important facts
- Always calculate change based on original value, not new value
- Population growth, inflation are common percentage change problems
- 50% increase then 50% decrease ≠ back to original (ends up at 75% of original)
Key definitions
Percentage Increase
Example
From ₱100 to ₱120 is 20% increase
How much a value has grown expressed as percentage of original
Percentage Decrease
Example
From ₱100 to ₱80 is 20% decrease
How much a value has fallen expressed as percentage of original
Diagrams to know
- Before and after comparison charts
- Growth/decline trend diagrams
Must remember
- Base formula: Percentage = (Part/Whole) × 100%
- Cross multiplication: if a:b = c:d, then ad = bc
- Sale Price = Original Price × (100% - Discount%)
- Tax Amount = Base Price × Tax Rate
- Profit% = ((SP - CP)/CP) × 100
- For 30% discount, multiply by 0.70, not 0.30
- Always simplify ratios to lowest terms
- Philippine VAT is 12%
- Percentage change = ((New - Old)/Old) × 100
- In proportions, check your answer by cross multiplication
Last minute tips
- For percentage problems, identify if you're finding Base, Rate, or Percentage first
- In ratio word problems, set up the proportion carefully - make sure units match on both sides
- When solving discount problems, remember: you pay (100% - discount%), not the discount%
- For complex percentage problems, work step by step: find the decimal first, then multiply
- Always check if your percentage answer makes sense (can't be negative in most real situations)
Comparison tables
Ratio vs Proportion vs Percentage
| Aspect | Ratio | Proportion | Percentage |
|---|---|---|---|
| Definition | Comparison of quantities | Equality of two ratios | Parts per hundred |
| Format | a:b or a/b | a:b = c:d | x% or x/100 |
| Example | 3:4 | 3:4 = 6:8 | 75% |
| Use Case | Comparing quantities | Solving for unknowns | Finding parts of whole |
Direct vs Inverse Proportion
| Aspect | Direct Proportion | Inverse Proportion |
|---|---|---|
| Relationship | Both increase/decrease together | One increases, other decreases |
| Formula | y = kx (k constant) | xy = k (k constant) |
| Example | Distance and time at constant speed | Speed and time for fixed distance |
| Graph Shape | Straight line through origin | Hyperbola curve |
Diagrams
Ratio, Proportion & Percentage Concept Map
Complete overview of all major concepts in ratio, proportion, and percentage
Percentage Problem Solving Steps
Step-by-step approach to solve any percentage problem by first identifying what you need to find
Common Percentage Values
Visual representation of the most commonly used percentage values in exams