Ratio, Proportion & Percentage - Slides
This chapter covers fundamental concepts of ratio, proportion, and percentage that are essential for numerical ability and problem-solving in various competitive exams in the Philippines. Students will learn to understand relationships between quantities, solve proportion problems, work with percentages, and apply these concepts to real-world scenarios including discount, tax, profit, and depreciation problems.
Slides
Chapter Overview: Ratio, Proportion & Percentage
Notes
Emphasize the interconnected nature of these three concepts and their practical importance in Philippine exam contexts.
Examples
- Student-teacher ratio in schools
- Discount calculations in shopping
- Population growth percentages
This chapter forms the foundation for numerical reasoning skills required in UPCAT, CSE, LET, NLE, NMAT, ACET, and USTET examinations. We'll explore how ratios express relationships, proportions show equality of ratios, and percentages represent parts of a whole.
Main points
- Understanding the relationship between quantities
- Solving proportion and ratio problems
- Mastering percentage calculations
- Real-world applications in business and daily life
- Preparation for Philippine competitive exams
Diagram
Mind map showing the main components and relationships within ratio, proportion, and percentage concepts
Understanding Ratio
Notes
Stress the importance of simplifying ratios and understanding that ratios maintain proportional relationships.
Examples
- In a class of 30 students, 18 are girls and 12 are boys. Ratio = 18:12 = 3:2
- A recipe uses 2 cups flour to 1 cup sugar. Ratio = 2:1
- Jeepney fare to distance: ₱12 for 4 km = 12:4 = 3:1
A ratio is a comparison of two quantities by division. It tells us how much of one thing there is compared to another. For example, if there are 3 boys and 2 girls in a group, the ratio of boys to girls is 3:2.
Main points
- Ratio expresses the relationship between two or more quantities
- Written as a:b or a/b where a and b are numbers
- Shows how many times one quantity contains another
- Can be simplified to lowest terms
- Order matters in ratios
Diagram
Flowchart showing the process of creating and simplifying ratios
Types of Ratios
Notes
Focus primarily on simple ratios as they are most relevant for grade 12 students and Philippine exams.
Examples
- Simple: Boys to girls = 4:3
- Compound: (2:3) × (4:5) = 8:15
- Duplicate: If ratio is 3:4, duplicate ratio is 9:16
Different types of ratios serve different purposes in mathematical problem-solving. Simple ratios are most common in everyday applications, while compound and other types appear in more complex mathematical situations.
Main points
- Simple Ratio: Comparison of two quantities (a:b)
- Compound Ratio: Product of two or more simple ratios
- Duplicate Ratio: Square of a simple ratio (a²:b²)
- Triplicate Ratio: Cube of a simple ratio (a³:b³)
- Sub-duplicate Ratio: Square root of a simple ratio (√a:√b)
Diagram
Classification of different types of ratios and their characteristics
Introduction to Proportion
Notes
Emphasize cross multiplication as the key technique for solving proportion problems.
Examples
- 5:x = 3:9, then 5×9 = 3×x, so x = 15
- Distance:Time = 120km:2hrs = 180km:3hrs
- Recipe scaling: 2 cups:4 servings = 3 cups:6 servings
A proportion is an equation that shows two ratios are equivalent. When we have a proportion a:b = c:d, we can cross multiply to get ad = bc. This property helps us solve for unknown values.
Main points
- Proportion states that two ratios are equal
- Written as a:b = c:d or a/b = c/d
- Cross multiplication: a × d = b × c
- Used to find unknown values in equivalent ratios
- Foundation for solving many real-world problems
Diagram
Process flow for working with proportions and cross multiplication
Solving Proportion Problems
Notes
Practice with Philippine peso amounts and familiar contexts to make problems more relatable.
Examples
- If 5 notebooks cost ₱150, how much do 8 notebooks cost?
- Setup: 5:150 = 8:x
- Cross multiply: 5x = 150×8 = 1200
- Solution: x = ₱240
Proportion problems follow a systematic approach. The key is correctly identifying which quantities are related and setting up the proportion properly before applying cross multiplication.
Main points
- Step 1: Set up the proportion equation
- Step 2: Cross multiply the terms
- Step 3: Solve for the unknown variable
- Step 4: Check your answer by substitution
- Common applications: scaling, rates, similar triangles
Diagram
Sequential steps for solving proportion problems systematically
Understanding Percentages
Notes
Connect percentages to familiar Filipino contexts like grades, discounts, and statistics.
Examples
- Exam score: 85 out of 100 = 85%
- Sale discount: 20% off means pay 80% of original price
- Population: 60% of Filipinos live in urban areas
Percentage is a way of expressing a fraction or decimal as parts per hundred. It's one of the most practical mathematical concepts used daily in shopping, grading, statistics, and financial calculations.
Main points
- Percentage means 'per hundred' or out of 100
- Symbol: % represents division by 100
- 25% = 25/100 = 0.25
- Percentages express parts of a whole
- Essential for statistics, finance, and everyday calculations
Diagram
Pie chart showing how percentages represent parts of a whole (100%)
Converting Between Fractions, Decimals, and Percentages
Notes
Practice with common fractions like 1/2, 1/4, 3/4, 1/5, etc., that appear frequently in exams.
Examples
- 3/4 = 0.75 = 75%
- 0.6 = 60% = 3/5
- 25% = 0.25 = 1/4
- 40% = 0.40 = 2/5
Converting between these forms is crucial for percentage calculations. Understanding the relationships helps in choosing the most efficient method for solving problems.
Main points
- Fraction to Percentage: Multiply by 100
- Decimal to Percentage: Move decimal point 2 places right
- Percentage to Decimal: Divide by 100 or move decimal 2 places left
- Percentage to Fraction: Write over 100 and simplify
- Memorize common conversions for speed
Diagram
Conversion relationships between fractions, decimals, and percentages
Finding the Rate (Percentage)
Notes
Use examples from Philippine education and demographics to make problems more relevant.
Examples
- 250 female students out of 500 total students
- Rate = (250 ÷ 500) × 100% = 50%
- 150 rural females out of 250 female students
- Rate = (150 ÷ 250) × 100% = 60%
When finding what percentage one number is of another, divide the part by the whole and multiply by 100. This is frequently used in analyzing data, survey results, and determining success rates.
Main points
- Formula: Rate = (Part ÷ Whole) × 100%
- Used when you know the part and the whole
- Answer is always expressed as a percentage
- Common in survey results and statistical analysis
- Check if answer makes logical sense
Diagram
Step-by-step process for finding the rate or percentage
Finding the Base (Original Amount)
Notes
Emphasize converting percentages to decimals and checking answers for reasonableness.
Examples
- 200 equals 40% of what original price?
- Base = 200 ÷ 0.40 = 500
- 800 is 25% of what original price?
- Base = 800 ÷ 0.25 = 3,200
Finding the base means determining the original whole amount when you know what percentage a part represents. This is essential for problems involving original prices, total populations, or initial quantities.
Main points
- Formula: Base = Part ÷ Rate (as decimal)
- Used when you know the part and the percentage
- Convert percentage to decimal before dividing
- Common in reverse engineering problems
- Often involves finding original prices or quantities
Diagram
Process for finding the base or original amount from part and rate
Finding the Part (Percentage Amount)
Notes
Use familiar Philippine contexts like restaurant tips, utility bills, and shopping scenarios.
Examples
- What is 25% of 800?
- Part = 800 × 0.25 = 200
- What is 15% tip on a ₱1,200 restaurant bill?
- Part = 1,200 × 0.15 = ₱180
Finding the part involves calculating what amount represents a certain percentage of a base number. This is the most common type of percentage calculation used in shopping, taxes, and tip calculations.
Main points
- Formula: Part = Base × Rate (as decimal)
- Used when you know the whole and the percentage
- Convert percentage to decimal first
- Most straightforward percentage calculation
- Common in discount and tax calculations
Diagram
Straightforward process for finding the part when base and rate are known
Discount and Tax Applications
Notes
Emphasize the 12% VAT rate in the Philippines and common discount percentages in local stores.
Examples
- ₱2,050 lipstick with 30% discount: ₱2,050 × 0.30 = ₱615 off
- Sale price: ₱2,050 - ₱615 = ₱1,435
- With 12% VAT: ₱1,435 × 1.12 = ₱1,607.20 final price
Discount and tax calculations are essential life skills. Discounts reduce the price you pay, while taxes increase it. Understanding both helps in budgeting and making informed purchasing decisions.
Main points
- Discount: Amount reduced from original price
- Sale Price = Original Price - Discount
- Tax: Additional amount added to base price
- Total Price = Base Price + Tax
- VAT (Value Added Tax) = 12% in Philippines
Diagram
Process flow showing how discounts and taxes are applied to original prices
Profit and Loss Calculations
Notes
Use examples with Filipino products and realistic prices to make calculations more meaningful.
Examples
- Bought Gameboy for ₱10,000, sold for ₱12,500
- Profit = ₱12,500 - ₱10,000 = ₱2,500
- Profit % = (₱2,500 ÷ ₱10,000) × 100% = 25%
Profit and loss calculations are fundamental in business and investment analysis. The percentage is always calculated based on the original cost price, not the selling price.
Main points
- Profit = Selling Price - Cost Price
- Loss = Cost Price - Selling Price
- Profit % = (Profit ÷ Cost Price) × 100%
- Loss % = (Loss ÷ Cost Price) × 100%
- Always calculate percentage based on cost price
Diagram
Decision tree for calculating profit or loss and their respective percentages
Depreciation Concepts
Notes
Use examples of common Filipino purchases like cars, gadgets, and appliances that students can relate to.
Examples
- Car bought for ₱1,000,000, sold after 3 years for ₱800,000
- Depreciated Cost = ₱1,000,000 - ₱800,000 = ₱200,000
- Depreciation Rate = (₱200,000 ÷ ₱1,000,000) × 100% = 20%
Depreciation represents how assets lose value over time due to wear, age, or obsolescence. Understanding depreciation helps in making informed decisions about purchases and investments.
Main points
- Depreciation: Decrease in value over time
- Depreciated Cost = Original Value - Current Value
- Depreciation Rate = (Depreciated Cost ÷ Original Value) × 100%
- Common with vehicles, electronics, and machinery
- Opposite of appreciation (increase in value)
Diagram
Timeline showing how asset value decreases over time due to depreciation
Word Problem Solving Strategies
Notes
Emphasize the importance of identifying the problem type before attempting to solve.
Examples
- Survey problem: Finding what percentage of a group has a certain characteristic
- Shopping problem: Calculating final price after discount and tax
- Business problem: Determining profit percentage on investments
Word problems require careful analysis to identify the type of calculation needed. Practice helps in recognizing patterns and selecting the correct approach quickly during exams.
Main points
- Read the problem carefully and identify given information
- Determine what is being asked (rate, base, or part)
- Choose the appropriate formula
- Set up the equation correctly
- Solve step by step and check the answer
Diagram
Sequential approach to solving word problems systematically
Common Exam Question Types
Notes
Focus practice on the most common question types to maximize exam performance.
Examples
- Civil Service: Population statistics and government data
- UPCAT: Academic performance and institutional ratios
- Professional exams: Industry-specific calculations
Philippine competitive exams frequently test these concepts through practical scenarios. Recognizing question patterns helps in applying the correct method quickly and accurately.
Main points
- Finding percentages in surveys and demographics
- Calculating original prices from discounted amounts
- Determining profit/loss in business scenarios
- Solving proportion problems with three terms
- Mixed problems combining multiple concepts
Diagram
Distribution of question types commonly found in Philippine competitive examinations
Practice Problems and Solutions
Notes
Encourage students to practice with actual exam questions from past Philippine competitive exams.
Examples
- If 480 students attended orientation (24% of total), find total population: 480 ÷ 0.24 = 2,000 students
- Guitar with 25% discount saves ₱1,875: Original price = ₱1,875 ÷ 0.25 = ₱7,500
- Profit of ₱100 on ₱400 investment: Profit % = (100 ÷ 400) × 100% = 25%
Regular practice with varied problem types builds confidence and speed. Focus on understanding the logic behind each solution rather than just memorizing procedures.
Main points
- Work through problems systematically
- Show all calculation steps clearly
- Double-check answers using different methods
- Practice mental math for common percentages
- Time management during actual exams
Diagram
Type
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Key Formulas Summary
Notes
Create a formula sheet for quick reference during practice and review sessions.
Examples
- For finding what percent: Use Rate formula
- For finding original amount: Use Base formula
- For finding portion: Use Part formula
- For unknown in equal ratios: Use proportion cross-multiplication
These fundamental formulas form the foundation for solving all ratio, proportion, and percentage problems. Memorize them and understand when to apply each one.
Main points
- Rate = (Part ÷ Base) × 100%
- Base = Part ÷ Rate (as decimal)
- Part = Base × Rate (as decimal)
- Proportion: a:b = c:d, so a×d = b×c
- Profit% = (Profit ÷ Cost Price) × 100%
Diagram
Mind map organizing all key formulas by their applications and relationships
Chapter Review and Key Takeaways
Notes
Encourage continued practice and application of these concepts in daily life situations.
Examples
- Successfully calculate discounts while shopping
- Analyze survey results and statistical data
- Solve business profit and loss problems
- Excel in competitive exam numerical sections
This chapter provides essential numerical skills for both academic success and practical life applications. Master these concepts through consistent practice with varied problem types, especially those common in Philippine competitive examinations.
Main points
- Ratios compare quantities and can be simplified
- Proportions show equal ratios and use cross multiplication
- Percentages represent parts of 100 and have three main calculations
- Real-world applications include business, shopping, and statistics
- Practice with Filipino contexts improves exam performance
Diagram
Type
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In summary
Ratio, Proportion, and Percentage form the cornerstone of numerical ability required for success in Philippine competitive examinations and practical life applications. Students who master these concepts will find themselves well-prepared for UPCAT, CSE, LET, NLE, NMAT, ACET, USTET, and other important assessments. The key to success lies in understanding the relationships between these concepts, practicing with diverse problem types, and applying systematic problem-solving approaches. Regular practice with Filipino contexts and realistic scenarios will build both confidence and competency in handling numerical challenges.