Ratio & Proportion - Slides
Ratio and proportion are fundamental mathematical concepts that appear in nearly every UPCAT Math problem involving mixtures, scale drawings, speed, percentages, and more. This chapter will teach you to recognize, set up, and solve ratio and proportion problems with confidence. Master these concepts once, and many other mathematical topics become much easier to understand and solve.
Slides
Introduction to Ratio & Proportion
Notes
Start by helping students understand that ratios and proportions are tools for comparing and relating quantities. Emphasize their practical importance in everyday life and academic success.
Examples
- If 3 pens cost ₱45, how much do 7 pens cost?
- A map scale shows 2 cm represents 50 km. What distance does 5 cm represent?
- Mixing paint: 3 parts red to 2 parts blue to make purple
Ratio and proportion form the foundation for many mathematical applications. They help us compare quantities, solve real-world problems, and understand relationships between different values. In UPCAT, these concepts are tested both directly and indirectly through various problem types.
Main points
- Ratio and proportion are among the most tested topics in UPCAT Math
- They appear in word problems involving mixtures, maps, scale drawings, speed, and percentages
- Understanding these concepts makes several other math chapters easier
- Key skills: recognizing ratios, setting up proportions, and solving for unknown values
Diagram
A comprehensive overview of ratio and proportion concepts, showing their applications, types, and required skills
What is a Ratio?
Notes
Emphasize that ratios must compare like quantities and that simplification makes calculations easier. Practice with various examples to build confidence.
Examples
- 12:18 = 2:3 (divide both by 6)
- ₱30:₱45 = 2:3 (divide both by 15)
- 15 minutes:1 hour = 15:60 = 1:4
Think of a ratio as a way to compare two things. Just like fractions, ratios can be simplified to their lowest terms. The key is that both quantities must be of the same type (both lengths, both weights, both amounts, etc.).
Main points
- A ratio compares two quantities of the same kind
- It tells you how many times one quantity contains another
- Can be written three ways: 3:5, 3/5, or 3 to 5
- Ratios can be simplified like fractions by dividing both terms by their GCD
Diagram
Flowchart showing how to simplify ratios and the three different ways to express them
Continued Ratios
Notes
Help students see that continued ratios are like recipes - each number tells us how much of each ingredient or share to use relative to the others.
Examples
- 6:9:12 = 2:3:4 (divide all by 3)
- Mix concrete: 1 part cement : 2 parts sand : 4 parts gravel
- Divide ₱700 among A, B, C in ratio 2:3:2
Continued ratios extend the concept of simple ratios to multiple quantities. They're particularly useful when dividing something among multiple people or comparing multiple components of a mixture.
Main points
- A continued ratio compares three or more quantities
- Written as a:b:c (for three quantities)
- Simplified by dividing all terms by their common factor
- Each term represents a proportional share of the total
Diagram
Process of simplifying continued ratios and understanding how they represent proportional shares
What is a Proportion?
Notes
The cross-product rule is the foundation of proportion solving. Practice this method extensively as it appears in many UPCAT problems. Always verify answers by substituting back.
Examples
- 3/4 = x/20, so 3×20 = 4×x, therefore x = 15
- If 2 kg costs ₱80, then 5 kg costs ₱200
- 5:8 = 15:24 (both ratios equal 5/8)
A proportion is like an equation but with ratios. The cross-product rule is your most powerful tool - it allows you to solve for any missing term when you know three of the four values.
Main points
- A proportion states that two ratios are equal
- Written as a/b = c/d or a:b = c:d
- The cross-product rule: if a/b = c/d, then a×d = b×c
- Used to solve for unknown values in ratio relationships
Diagram
Step-by-step process of solving proportions using cross-multiplication and verification
Direct Proportion - Worked Example
Notes
Emphasize the 'same direction' concept. When solving, always check if your answer makes logical sense - more items should cost more money.
Examples
- Cost and quantity: If 3 notebooks cost ₱60, then 7 notebooks cost ₱140
- Recipe scaling: If 2 cups flour serves 4 people, then 5 cups flour serves 10 people
- Distance at constant speed: 100 km in 2 hours means 250 km in 5 hours
Direct proportion means 'same direction' - if you double one value, the other doubles too. If you triple one, the other triples. The ratio between them stays constant.
Main points
- In direct proportion, as one quantity increases, the other increases proportionally
- Formula: y = kx (where k is the constant of proportionality)
- Both quantities change in the same direction
- Common in: cost vs quantity, distance vs time at constant speed
Diagram
Complete solution process for a direct proportion problem with verification
Inverse Proportion - Worked Example
Notes
Help students identify inverse relationships by asking: 'If I double the first, does the second halve?' Common UPCAT topics include workers/time and speed/time problems.
Examples
- Workers and time: 6 workers finish in 8 days, so 12 workers finish in 4 days
- Speed and time: 60 km/h takes 2 hours, so 120 km/h takes 1 hour
- Pipe filling: 3 pipes fill tank in 4 hours, so 6 pipes fill in 2 hours
Inverse proportion means 'opposite direction' - if you double one value, the other becomes half. If you triple one, the other becomes one-third. Their product stays constant.
Main points
- In inverse proportion, as one quantity increases, the other decreases proportionally
- Formula: xy = k (where k is constant)
- Quantities change in opposite directions
- Common in: workers vs time, speed vs time for fixed distance
Diagram
Complete solution process for an inverse proportion problem showing the constant relationship
Identifying Direct vs Inverse Proportion
Notes
This identification skill is often what separates high scorers from average scorers in UPCAT. Practice with many examples until recognition becomes automatic.
Examples
- Direct: More items → Higher cost, Faster speed → More distance covered
- Inverse: More workers → Less time needed, Faster speed → Less time for same distance
- Tricky: 'Pipes filling tank' is inverse (more pipes = less time)
The ability to quickly identify whether a problem involves direct or inverse proportion is crucial for UPCAT success. This skill determines which formula and approach to use.
Main points
- Direct: Same direction changes (↑↑ or ↓↓)
- Inverse: Opposite direction changes (↑↓ or ↓↑)
- Key question: 'If I double the first, what happens to the second?'
- Look for keywords: 'together', 'shared work', 'combined' often mean inverse
Diagram
Decision tree for identifying whether a proportion problem is direct or inverse
Partitive Proportion - Step by Step
Notes
Partitive proportion appears frequently in UPCAT word problems about sharing money, dividing inheritance, or mixing ingredients. Always verify that parts add up to the original total.
Examples
- Divide ₱840 among A, B, C in ratio 2:3:7
- Mix paint using red:blue:yellow = 3:2:1
- Inheritance divided in ratio 5:3:2
Partitive proportion is like dividing a pizza according to how much each person should get. The ratio tells you the relative sizes of each portion.
Main points
- Used to divide a total into parts following a given ratio
- Step 1: Add all parts of the ratio
- Step 2: Divide total by sum of parts to get 'one share'
- Step 3: Multiply each ratio part by one share
Diagram
Complete step-by-step solution for partitive proportion with verification
Percentages as Ratios in Disguise
Notes
This connection helps students see that they already know how to solve percentage problems if they understand proportions. It reduces the mental burden of learning separate methods.
Examples
- 25% of 80 = 25/100 × 80 = 20
- If 60 is 75% of a number, find the number: 60/x = 75/100
- Increase ₱200 by 15%: ₱200 + (15% of ₱200) = ₱200 + ₱30 = ₱230
Don't treat percentage problems as completely separate from ratios. They're the same concept! This connection makes both topics easier to understand and solve.
Main points
- Every percentage is a ratio with 100 as the denominator
- 25% = 25:100 = 1:4 = 0.25
- 'Percent of' problems become proportion problems
- Convert percentages to ratios to solve systematically
Diagram
Two methods for solving percentage problems using ratio/proportion concepts
Common UPCAT Traps and How to Avoid Them
Notes
Teach students to be extra careful with units and to always do a sanity check on their answers. Many UPCAT questions are designed to catch these specific mistakes.
Examples
- Wrong: 3 hours:45 minutes = 3:45. Right: 180 minutes:45 minutes = 4:1
- Wrong: Boys:Girls = 3:2 means 3 boys, 2 girls. Right: means 3k boys, 2k girls for some value k
- Wrong: More workers, more time. Right: More workers, less time (inverse)
UPCAT test makers deliberately include these common mistakes as answer choices. Recognizing and avoiding these traps can significantly improve your score.
Main points
- Unit mismatch: Convert all units before setting up proportion
- Ratio vs total confusion: 3:2 with 40 students means 24 boys, 16 girls
- Misidentifying direct vs inverse relationships
- Arithmetic errors in cross-multiplication
Diagram
Problem-solving flowchart highlighting common traps and verification steps
Mixed Practice Problem 1: Recipe Scaling
Notes
Show students both the scale factor method and individual proportions. Both give the same answer, but scale factor is often faster for multiple ingredients.
Examples
- Flour: 2/6 = x/10, so x = 20/6 = 3⅓ cups
- Eggs: 3/6 = x/10, so x = 30/6 = 5 eggs
- Milk: 1/6 = x/10, so x = 10/6 = 1⅔ cups
Recipe scaling is a classic direct proportion problem. Each ingredient increases proportionally with the number of servings.
Main points
- Problem: A recipe for 6 servings uses 2 cups flour, 3 eggs, and 1 cup milk. How much of each ingredient for 10 servings?
- This is direct proportion: more servings need more ingredients
- Set up separate proportions for each ingredient
- Scale factor: 10/6 = 5/3
Diagram
Two solution methods for recipe scaling: scale factor multiplication and individual proportions
Mixed Practice Problem 2: Work and Time
Notes
Work problems are very common in UPCAT. Help students remember that the total work (worker-days, machine-hours, etc.) remains constant in these problems.
Examples
- Total work = 8 workers × 15 days = 120 worker-days
- For 10 days: workers × 10 days = 120 worker-days
- Therefore: workers = 120 ÷ 10 = 12 workers
Work problems are typically inverse proportion. The total amount of work (measured in worker-days) stays the same regardless of how it's distributed.
Main points
- Problem: 8 workers can complete a project in 15 days. How many workers needed to complete it in 10 days?
- This is inverse proportion: more workers mean less time
- Use formula: workers₁ × days₁ = workers₂ × days₂
- The total work-days remains constant
Diagram
Complete solution for inverse proportion work problem with verification
Mixed Practice Problem 3: Map Scale
Notes
Map problems often appear in UPCAT. Remind students to check that their answer makes sense - larger map distances should correspond to larger actual distances.
Examples
- Given ratio: 3 cm : 75 km = 1 cm : 25 km
- For 7 cm: 7 × 25 km = 175 km
- Check: 3/75 = 1/25 and 7/175 = 1/25 ✓
Map scale problems are direct proportion. The ratio between map distance and actual distance remains constant throughout the map.
Main points
- Problem: On a map, 3 cm represents 75 km. What actual distance does 7 cm represent?
- This is direct proportion: more map distance means more actual distance
- Set up proportion: map distance / actual distance
- Keep units consistent throughout
Diagram
Map scale problem solved using both proportion and scale factor methods
Quick Problem-Solving Strategy
Notes
This systematic approach should become automatic with practice. Emphasize that the few seconds spent on proper setup saves minutes on calculation and checking.
Examples
- Ask: 'If I double the first quantity, what happens to the second?'
- Look for keywords: 'workers together', 'shared task' → usually inverse
- Always plug your answer back into the original problem
Having a systematic approach saves time and reduces errors in UPCAT. This strategy works for virtually all ratio and proportion problems you'll encounter.
Main points
- Step 1: Read the problem completely before starting
- Step 2: Identify direct vs inverse proportion
- Step 3: Check units and convert if necessary
- Step 4: Set up proportion and solve
- Step 5: Verify answer makes logical sense
Diagram
Comprehensive problem-solving strategy flowchart for all ratio and proportion problems
Summary and Key Takeaways
Notes
This summary ties everything together. Encourage students to practice problems from different contexts to build flexibility and confidence for the UPCAT.
Examples
- Practice identifying problem types quickly
- Master the cross-multiplication technique
- Remember: more practice leads to automatic recognition and faster solving
Ratio and proportion mastery opens doors to solving mixture problems, scale drawings, speed-time-distance, percentage problems, and much more. These concepts appear throughout the UPCAT Math section.
Main points
- Ratios compare quantities; proportions state that two ratios are equal
- Cross-multiplication is your primary solving tool: a/b = c/d → ad = bc
- Direct proportion: same direction changes (y = kx)
- Inverse proportion: opposite direction changes (xy = k)
- Always check units and verify your final answer
Diagram
Comprehensive mind map summarizing all key concepts, formulas, problem types, and success strategies
In summary
Ratio and proportion are fundamental mathematical skills that will serve you well beyond the UPCAT. They provide a logical framework for solving many real-world problems and appear in various forms throughout mathematics. By mastering the techniques of identifying problem types, setting up proportions correctly, and using cross-multiplication systematically, you'll be well-prepared for the wide variety of ratio and proportion questions that appear on the UPCAT. Remember: consistent practice with different problem contexts builds the recognition skills that make these problems quick and manageable on test day.