Ratio, Proportion & Percentage - Revision Notes
Ratio, Proportion, and Percentage are fundamental mathematical concepts frequently tested in Philippine entrance exams (UPCAT, CSE, LET, NLE, NMAT, ACET, USTET). These topics involve relationships between quantities, equal ratios, and parts of a whole expressed as fractions of 100. Mastering these concepts is essential for solving word problems in business, statistics, and everyday applications.
Ratio Fundamentals
Formulas
a : b = c : d if and only if ad = bc
Example
5 : x = 3 : 9, so 5 × 9 = 3 × x, therefore x = 15
Variables
a, b, c, d are real numbers where b ≠ 0 and d ≠ 0
Application
Cross-multiplication to verify equal ratios or solve for unknown terms
Total parts = sum of all ratio terms
Example
In ratio 2:3:5 with total 100, parts are 20, 30, 50
Variables
For ratio a:b:c, total parts = a + b + c
Application
Finding actual quantities when given total and ratio
Exam tips
- Always check if ratios need to be simplified
- Draw diagrams for complex ratio problems
- Verify answers using cross-multiplication
- Pay attention to the order of quantities mentioned
Key points
- A ratio compares two or more quantities using division
- Ratios can be written as a:b, a/b, or a to b
- Simplify ratios by dividing by the greatest common factor (GCF)
- Cross-multiplication is used to check if ratios are equal
- Ratios show relative size, not absolute quantities
Definitions
Ratio
A comparison between two or more quantities expressed as a fraction or using colon notation
Importance
Foundation for proportion problems and percentage calculations
Terms of a Ratio
The individual numbers being compared in a ratio
Importance
Understanding which quantities are being related
Common mistakes
- Confusing the order of terms in a ratio
- Not simplifying ratios to lowest terms
- Mixing up cross-multiplication setup
- Adding ratios incorrectly instead of finding equivalent ratios
Proportion Applications
Formulas
If a/b = c/d, then ad = bc
Example
If 5 liters serve 12 guests, then x liters serve 120 guests: 5/12 = x/120
Variables
a, b, c, d are proportional quantities
Application
Solving for unknown values in proportion problems
Direct Proportion: y = kx
Example
Distance = speed × time (constant speed)
Variables
y varies directly with x, k is the constant of proportionality
Application
Problems where quantities increase or decrease together
Exam tips
- Identify what quantities are being compared
- Set up proportion carefully with correct units
- Always check if your answer makes logical sense
- Practice with different types of word problems
Key points
- Proportion states that two ratios are equal
- Used in direct and inverse variation problems
- Essential for scaling, unit conversion, and rate problems
- Cross-multiplication is the primary solving method
- Can involve three or more quantities in continued proportion
Definitions
Proportion
A statement that two ratios are equal, written as a:b = c:d
Importance
Fundamental for solving ratio-based word problems
Mean Proportional
The middle term in a proportion a:b = b:c, where b is the mean proportional
Importance
Used in geometric sequences and scaling problems
Common mistakes
- Setting up incorrect proportions from word problems
- Forgetting to check units when setting up ratios
- Incorrectly identifying which quantities are related
- Not simplifying the final answer
Percentage Calculations
Formulas
Percentage = (Part/Base) × 100%
Example
150 females out of 250 total: (150/250) × 100% = 60%
Variables
Part = portion being considered, Base = total amount
Application
Finding what percent one quantity is of another
Part = (Rate/100) × Base
Example
40% of 500: (40/100) × 500 = 200
Variables
Rate = percentage as a number, Base = total amount
Application
Finding the actual amount when percentage and total are known
Base = Part ÷ (Rate/100)
Example
200 is 40% of what number: 200 ÷ 0.40 = 500
Variables
Part = actual amount, Rate = percentage
Application
Finding the total when part and percentage are known
Exam tips
- Identify whether you're finding rate, base, or percentage
- Convert percentages to decimals before calculating
- Double-check by using a different method
- Pay attention to what the question is asking for
Key points
- Percentage means 'per hundred' or parts out of 100
- Three main types: finding percentage, finding base, finding rate
- Formula: Percentage = (Part/Whole) × 100%
- Converting between fractions, decimals, and percentages
- Applications in business, statistics, and everyday situations
Definitions
Percentage
A fraction expressed as parts per hundred, denoted with % symbol
Importance
Universal way to express proportions and changes
Base
The reference quantity or total amount in percentage problems
Importance
The denominator in percentage calculations
Rate
The percentage number without the % symbol
Importance
Used in calculations after converting from percentage form
Common mistakes
- Confusing which quantity is the base (total)
- Forgetting to convert percentage to decimal
- Using wrong formula for the problem type
- Not expressing final answer in requested format
Business Applications
Formulas
Sale Price = Original Price × (1 - Discount Rate)
Example
₱2,050 with 30% discount: ₱2,050 × 0.70 = ₱1,435
Variables
Discount Rate as decimal (e.g., 20% = 0.20)
Application
Finding final price after discount
Profit Rate = (Selling Price - Cost Price)/Cost Price × 100%
Example
Bought at ₱400, sold at ₱500: (500-400)/400 × 100% = 25%
Variables
Selling Price = final sale amount, Cost Price = original cost
Application
Calculating profit or loss percentage
Original Price = Sale Price ÷ (1 - Discount Rate)
Example
Paid ₱13,600 at 15% discount: ₱13,600 ÷ 0.85 = ₱16,000
Variables
Sale Price = discounted price, Discount Rate as decimal
Application
Finding original price when sale price and discount are known
Exam tips
- Read carefully whether tax is added before or after discount
- Always verify if the calculation makes economic sense
- Practice multi-step problems combining discounts and taxes
- Remember Philippine VAT rate is typically 12%
Key points
- Discount = Original Price × Discount Rate
- Sale Price = Original Price - Discount
- Tax Amount = Price × Tax Rate
- Total Cost = Price + Tax
- Profit/Loss calculations using percentages
Definitions
Discount
Reduction in price, usually expressed as percentage of original price
Importance
Common in retail and business calculations
Value Added Tax (VAT)
Tax added to the price of goods and services, typically 12% in Philippines
Importance
Essential for business and consumer calculations
Common mistakes
- Adding discount instead of subtracting
- Forgetting to add tax after calculating discount
- Using wrong base for profit/loss calculations
- Confusing marked price with selling price
Advanced Applications
Formulas
Final Value = Initial Value × (1 + Rate)^n for growth
Example
Population growing 5% yearly for 3 years
Variables
Rate as decimal, n = number of periods
Application
Compound growth problems
Percentage Change = (New Value - Old Value)/Old Value × 100%
Example
Price changed from ₱80 to ₱100: (100-80)/80 × 100% = 25%
Variables
Positive for increase, negative for decrease
Application
Measuring relative change between two values
Exam tips
- Break complex problems into smaller steps
- Keep track of what percentage refers to what base
- Use estimation to check if answers are reasonable
- Practice problems with multiple percentage operations
Key points
- Compound percentage changes
- Mixture and alligation problems
- Population growth and depreciation
- Multiple ratio comparisons
- Percentage increase and decrease calculations
Definitions
Depreciation
Decrease in value over time, often expressed as percentage
Importance
Important for asset valuation and business accounting
Compound Growth
Growth calculated on both original amount and previously accumulated growth
Importance
Models real-world population and investment growth
Common mistakes
- Treating compound changes as simple addition
- Using wrong base for percentage change calculations
- Confusing appreciation with depreciation
- Not accounting for time periods in growth problems
Connections
- Ratios form the foundation for understanding proportions and percentages
- Percentage problems often require setting up proportions to solve
- Business applications combine multiple concepts: discounts, taxes, and profit calculations
- Proportion concepts are essential for unit conversion and scaling problems
- These topics connect to algebra through cross-multiplication and equation solving
- Statistical analysis frequently uses percentage and ratio comparisons
Quick review questions
If 150 out of 250 female students live in rural areas, what percentage live in rural areas?
Answer
60%
150 ÷ 250 = 0.6 = 60%. Divide the part (150) by the whole (250) and multiply by 100%.
Find x in the proportion 5:x = 3:9
Answer
x = 15
Cross multiply: 5 × 9 = 3 × x, so 45 = 3x, therefore x = 15.
A refrigerator costs ₱15,000. If there's a 30% discount, how much is saved?
Answer
₱4,500
Discount = 30% of ₱15,000 = 0.30 × ₱15,000 = ₱4,500.
800 is 25% of what number?
Answer
3,200
Base = Part ÷ Rate = 800 ÷ 0.25 = 3,200.
If books are in ratio 2:3 and there are 18 red books, how many violet books are there?
Answer
27
Set up proportion: 2:3 = 18:x. Cross multiply: 2x = 54, so x = 27.
Diagrams
Ratio, Proportion & Percentage Concept Overview
Comprehensive mind map showing the relationships between ratios, proportions, and percentages, including their main applications and problem types.
Percentage Problem Solving Strategy
Step-by-step flowchart for systematically approaching percentage problems by first identifying what type of problem it is, then applying the appropriate formula.
Business Problem Solving Process
Decision-making flowchart for solving business-related problems involving discounts, taxes, and profit/loss calculations in the correct sequence.
Common Percentage Problem Types in Exams
Distribution of different types of percentage problems typically found in Philippine entrance examinations, helping students prioritize their study focus.
Exam strategy
Focus on identifying problem types quickly. Practice the three main percentage formulas until they become automatic. For ratio problems, always simplify and check using cross-multiplication. In business problems, read carefully to determine the sequence of operations (discount first, then tax, etc.). Time management is crucial - don't spend too long on complex multi-step problems. Always verify that your answers make logical sense in context.