Ratio, Proportion & Percentage - Study Notes
Ratio, Proportion, and Percentage are fundamental mathematical concepts that appear frequently in entrance exams like UPCAT, CSE, LET, and other Philippine examinations. These concepts help us compare quantities, solve real-world problems, and understand relationships between numbers. Mastering these topics is essential for success in numerical ability sections of various competitive exams.
Understanding Ratios
A ratio is a comparison of two or more quantities that shows their relative sizes. It expresses how many times one quantity contains another. Ratios can be written in three ways: using a colon (3:4), as a fraction (3/4), or using the word 'to' (3 to 4). When we say the ratio of boys to girls in a class is 3:2, it means for every 3 boys, there are 2 girls. The total parts would be 3 + 2 = 5 parts. To find actual quantities, we multiply each part by a common factor. For example, if there are 50 students total, each part represents 50 ÷ 5 = 10 students, so there would be 30 boys and 20 girls.
Examples
- If the ratio of apples to oranges is 4:3 and there are 28 fruits total, find the number of each fruit. Solution: Total parts = 4 + 3 = 7, Each part = 28 ÷ 7 = 4, Apples = 4 × 4 = 16, Oranges = 3 × 4 = 12
- Simplify the ratio 18:24:30. Solution: Find GCD of 18, 24, 30 which is 6. Divide each term: 18÷6 : 24÷6 : 30÷6 = 3:4:5
Key points
- Ratio compares two or more quantities
- Can be written as a:b, a/b, or a to b
- The order matters - 3:2 is different from 2:3
- Total parts = sum of all ratio parts
- Ratios can be simplified by dividing by common factors
Understanding Proportions
A proportion is an equation that states two ratios are equal. If a:b = c:d, then we have a proportion. The key property of proportions is cross-multiplication: if a/b = c/d, then a×d = b×c. This property is extremely useful for solving problems with unknown values. Proportions are particularly useful in problems involving similar quantities, rates, and scaling. For example, if 3 notebooks cost ₱45, how much do 7 notebooks cost? We set up the proportion: 3:45 = 7:x, then cross-multiply: 3x = 45 × 7, so x = 315 ÷ 3 = ₱105.
Examples
- Solve for x: 5:x = 3:9. Solution: Cross-multiply: 3x = 5 × 9 = 45, so x = 15
- If 12 workers can complete a job in 8 days, how many workers are needed to complete it in 6 days? Solution: 12:8 = x:6, Cross-multiply: 8x = 12 × 6 = 72, so x = 9 workers
Key points
- Proportion shows that two ratios are equal
- Cross-multiplication: if a/b = c/d, then a×d = b×c
- Useful for finding unknown quantities
- Common in rate problems and scaling situations
- Always check if your answer makes logical sense
Understanding Percentages
Percentage means 'per hundred' and is a way of expressing a fraction or ratio as parts out of 100. The symbol % represents percentage. To convert a fraction to percentage, multiply by 100. To convert percentage to decimal, divide by 100. Percentages are widely used in everyday life: exam scores, discounts, taxes, interest rates, and statistical data. Common percentage calculations include finding what percent one number is of another, finding a percentage of a number, and finding the original number when given a percentage.
Examples
- What percent is 25 out of 80? Solution: (25/80) × 100 = 31.25%
- Find 15% of 240. Solution: 15% = 0.15, so 0.15 × 240 = 36
Key points
- Percentage means 'out of 100'
- To convert fraction to %: multiply by 100
- To convert % to decimal: divide by 100
- Formula: Percentage = (Part/Whole) × 100
- Used in discounts, taxes, interest, and statistics
Percentage Applications: Profit, Loss, and Discount
Profit and loss calculations are common in business mathematics and entrance exams. Profit% = (Profit/Cost Price) × 100, Loss% = (Loss/Cost Price) × 100. Discount is the reduction in marked price. Discount% = (Discount/Marked Price) × 100. Sales tax is added to the selling price. When solving problems involving successive discounts or taxes, calculate each step separately. For original price problems when given the final price after discount, use: Original Price = Final Price ÷ (1 - Discount%). For problems involving both discount and tax, first apply discount then add tax to the discounted price.
Examples
- A item bought for ₱800 is sold for ₱1000. Find profit%. Solution: Profit = 1000 - 800 = ₱200, Profit% = (200/800) × 100 = 25%
- A dress marked at ₱2000 is sold at 25% discount. Find selling price. Solution: Discount = 25% of 2000 = ₱500, Selling Price = 2000 - 500 = ₱1500
Key points
- Profit% = (Selling Price - Cost Price)/Cost Price × 100
- Loss% = (Cost Price - Selling Price)/Cost Price × 100
- Discount% = Discount Amount/Marked Price × 100
- Tax is calculated on the final selling price
- For original price: Original = Final ÷ (100% - Discount%)
Advanced Applications and Problem-Solving
Advanced percentage problems often involve multiple steps, rates, depreciation, and compound scenarios. Depreciation problems involve calculating the decrease in value over time. Rate problems combine percentages with ratios to find unknown quantities. When solving complex problems, identify what is given, what is asked, set up equations carefully, and verify your answer logically. Time-based percentage problems might involve population growth, compound interest, or depreciation where the percentage change applies repeatedly over multiple periods.
Examples
- In a school of 800 students, 60% are boys. If 20% of boys and 25% of girls participate in sports, how many students participate? Solution: Boys = 60% of 800 = 480, Girls = 40% of 800 = 320, Sports participants = 20% of 480 + 25% of 320 = 96 + 80 = 176 students
- A car depreciates 20% each year. If it costs ₱500,000 initially, what is its value after 2 years? Solution: After 1 year = 500,000 × 80% = ₱400,000, After 2 years = 400,000 × 80% = ₱320,000
Key points
- Break complex problems into smaller steps
- Identify given information and what needs to be found
- Use appropriate formulas for different scenarios
- Check if your answer makes practical sense
- Practice with various problem types for exam preparation
Problem-Solving Strategies and Tips
Success in ratio, proportion, and percentage problems requires systematic approaches and practice. Always read problems carefully and identify the type of problem. For ratio problems, find the total parts and use them to calculate actual values. For proportion problems, set up the equation correctly and cross-multiply. For percentage problems, identify whether you need to find the percentage, the part, or the whole. Draw diagrams when helpful, especially for complex ratio problems. Practice mental math for common percentages (10%, 25%, 50%, etc.). Time management is crucial in exams - solve easier problems first and return to difficult ones later.
Examples
- Quick mental calculation: 25% of any number = divide by 4, 10% = move decimal one place left, 1% = move decimal two places left
- For complex ratio problems, assign variables to each part and use the total constraint to solve
Key points
- Read problems carefully and identify the problem type
- Use systematic approaches for each type
- Draw diagrams for complex problems
- Practice mental calculations for common percentages
- Manage time effectively during exams
Diagrams
Ratio, Proportion & Percentage Concept Map
This mind map shows the relationship between ratio, proportion, and percentage concepts, highlighting their definitions, properties, and common applications in problem-solving.
Percentage Problem-Solving Flowchart
This flowchart provides a step-by-step approach to solving percentage problems, helping students identify the problem type and apply the correct formula.
Ratio and Proportion Solution Process
This flowchart guides students through the systematic approach to solving ratio and proportion problems, from problem identification to solution verification.
Common Exam Question Types Distribution
This pie chart shows the typical distribution of different types of ratio, proportion, and percentage problems in Philippine entrance examinations, helping students prioritize their study focus.
In summary
Ratio, Proportion, and Percentage are interconnected concepts essential for numerical ability sections in Philippine entrance examinations. Ratios compare quantities, proportions show equal ratios and help solve for unknowns, and percentages express parts out of 100. These concepts appear in various applications including profit/loss calculations, discount problems, tax computations, and real-world scenarios. Master the fundamental formulas, practice systematic problem-solving approaches, and develop mental calculation skills for common percentages. Regular practice with exam-style questions will build confidence and speed, crucial for success in competitive examinations like UPCAT, CSE, and other standardized tests.