Ratio & Proportion - Study notes
Ratio and proportion are fundamental mathematical concepts that appear in nearly every UPCAT math section. They form the foundation for solving problems involving mixtures, scale drawings, speed, percentages, and many word problems. Mastering these concepts will significantly boost your problem-solving ability across multiple math topics. This comprehensive guide will teach you to recognize, set up, and solve ratio and proportion problems using step-by-step methods.
Understanding Ratios - The Foundation
A ratio compares two quantities of the same kind, telling us how many times one quantity contains another. Think of it as a mathematical way to express relationships between numbers.
Three Ways to Write Ratios:
- Using a colon: 3:5 (read as "3 to 5")
- As a fraction: 3/5
- With words: 3 to 5
Simplifying Ratios:
Just like fractions, ratios can be simplified by dividing both terms by their greatest common divisor (GCD).
Step-by-Step Example:
Simplify the ratio 24:36
Step 1: Find the GCD of 24 and 36
Factors of 24: 1, 2, 3, 4, 6, 8, 12, 24
Factors of 36: 1, 2, 3, 4, 6, 9, 12, 18, 36
GCD = 12
Step 2: Divide both terms by the GCD
24 ÷ 12 = 2
36 ÷ 12 = 3
Step 3: Write the simplified ratio
24:36 = 2:3
Continued Ratios:
When comparing three or more quantities, we use continued ratios like 2:3:5. These are simplified by dividing all terms by the same number.
Example: Simplify 6:9:15
Step 1: Find GCD of all three numbers (GCD = 3)
Step 2: Divide each term: 6÷3 : 9÷3 : 15÷3 = 2:3:5
Examples
- Speed ratio: If Car A travels 60 km and Car B travels 80 km in the same time, the speed ratio is 60:80 = 3:4
- Money ratio: Ana has ₱150 and Ben has ₱200. Their money ratio is 150:200 = 3:4
- Ingredient ratio: A recipe calls for 2 cups flour, 3 cups sugar, 1 cup milk. The ratio is 2:3:1
Key points
- Ratios compare quantities of the same kind
- Can be written as fractions, with colons, or in words
- Always simplify ratios to lowest terms
- Continued ratios involve three or more quantities
- GCD method is most reliable for simplification
Understanding Proportions - Equal Ratios
A proportion is a mathematical statement that two ratios are equal. This is where ratios become powerful problem-solving tools.
Proportion Format:
a/b = c/d or a:b = c:d
The Cross-Product Rule (Most Important Tool):
If a/b = c/d, then a × d = b × c
This cross-multiplication method solves most proportion problems.
Step-by-Step Solution Process:
Example 1: Solve for x in 3/4 = x/20
Step 1: Write the proportion clearly
3/4 = x/20
Step 2: Cross-multiply
3 × 20 = 4 × x
60 = 4x
Step 3: Solve for x
x = 60 ÷ 4 = 15
Step 4: Verify (always check!)
3/4 = 15/20
3/4 = 3/4 ✓
Example 2: If 5 notebooks cost ₱125, how much do 8 notebooks cost?
Step 1: Set up the proportion
5 notebooks / ₱125 = 8 notebooks / x pesos
Step 2: Cross-multiply
5x = 125 × 8
5x = 1000
Step 3: Solve
x = 1000 ÷ 5 = ₱200
Step 4: Verify
Cost per notebook = ₱125 ÷ 5 = ₱25
8 notebooks = 8 × ₱25 = ₱200 ✓
Examples
- Map scale: If 2 cm represents 5 km, how many km does 7 cm represent? (2/5 = 7/x, so x = 17.5 km)
- Recipe scaling: If 3 eggs serve 4 people, how many eggs for 10 people? (3/4 = x/10, so x = 7.5 eggs)
- Unit conversion: If $1 = ₱56, how many pesos is $12? (1/56 = 12/x, so x = ₱672)
Key points
- Proportions state that two ratios are equal
- Cross-product rule: if a/b = c/d, then a×d = b×c
- Always verify your answer by substitution
- Set up proportions carefully - units must match
- Cross-multiplication is the fastest solution method
Direct vs Inverse Proportion - Recognizing the Relationship
Understanding whether quantities have a direct or inverse relationship is crucial for setting up correct proportions.
Direct Proportion:
As one quantity increases, the other increases at the same rate.
Formula: y = kx (where k is constant)
Symbol: y ∝ x
Inverse Proportion:
As one quantity increases, the other decreases at the same rate.
Formula: xy = k (where k is constant)
Symbol: y ∝ 1/x
Recognition Strategy:
Ask: "If I double the first quantity, what happens to the second?"
- If it doubles → Direct proportion
- If it halves → Inverse proportion
Direct Proportion Examples with Solutions:
Example 1: Speed and Distance (time constant)
If a car traveling at 60 km/h covers 180 km, how far will it travel at 80 km/h?
Step 1: Recognize direct relationship (faster speed = more distance)
Step 2: Set up proportion: 60/180 = 80/x
Step 3: Cross-multiply: 60x = 180 × 80 = 14,400
Step 4: Solve: x = 14,400 ÷ 60 = 240 km
Inverse Proportion Examples with Solutions:
Example 2: Workers and Time
If 6 workers complete a job in 12 days, how long will 9 workers take?
Step 1: Recognize inverse relationship (more workers = less time)
Step 2: Use inverse formula: workers × time = constant
6 × 12 = 9 × x
Step 3: Solve: 72 = 9x, so x = 8 days
Example 3: Speed and Time (distance constant)
A journey takes 4 hours at 75 km/h. How long at 100 km/h?
Step 1: Recognize inverse relationship (faster speed = less time)
Step 2: Speed × time = constant distance
75 × 4 = 100 × x
Step 3: Solve: 300 = 100x, so x = 3 hours
Examples
- Direct: More hours worked = more pay earned
- Direct: Larger area = more paint needed
- Inverse: More people sharing pizza = smaller slices each
- Inverse: Higher speed = less travel time
- Inverse: More workers = job finished faster
Key points
- Direct: both quantities change in the same direction
- Inverse: quantities change in opposite directions
- Recognition test: double one, see what happens to the other
- Direct formula: y = kx; Inverse formula: xy = k
- Word problems often disguise the relationship type
Partitive Proportion - Dividing by Ratios
Partitive proportion problems ask you to divide a total amount into parts that follow a given ratio. This is extremely common in UPCAT word problems.
The Four-Step Method:
- Add up all parts of the ratio
- Divide the total by the sum to find one 'share'
- Multiply each ratio part by the share value
- Verify that parts add up to the original total
Detailed Examples:
Example 1: Divide ₱840 among three friends A, B, and C in the ratio 2:3:7
Step 1: Add ratio parts
2 + 3 + 7 = 12 parts total
Step 2: Find value of one part
₱840 ÷ 12 = ₱70 per part
Step 3: Calculate each person's share
A gets: 2 × ₱70 = ₱140
B gets: 3 × ₱70 = ₱210
C gets: 7 × ₱70 = ₱490
Step 4: Verify
₱140 + ₱210 + ₱490 = ₱840 ✓
Example 2: A school has 450 students. The ratio of Grade 11 to Grade 12 is 4:5. How many students are in each grade?
Step 1: Add ratio parts
4 + 5 = 9 parts total
Step 2: Find students per part
450 ÷ 9 = 50 students per part
Step 3: Calculate each grade
Grade 11: 4 × 50 = 200 students
Grade 12: 5 × 50 = 250 students
Step 4: Verify
200 + 250 = 450 ✓
Example 3: An inheritance of ₱1,200,000 is divided among three heirs in the ratio 3:5:4. Find each heir's share.
Step 1: Total parts = 3 + 5 + 4 = 12
Step 2: Value per part = ₱1,200,000 ÷ 12 = ₱100,000
Step 3: Individual shares:
First heir: 3 × ₱100,000 = ₱300,000
Second heir: 5 × ₱100,000 = ₱500,000
Third heir: 4 × ₱100,000 = ₱400,000
Step 4: Check: ₱300,000 + ₱500,000 + ₱400,000 = ₱1,200,000 ✓
Examples
- Business profit sharing: Partners invest in ratio 2:3:5, profit ₱500,000
- Recipe scaling: Ingredients in ratio 1:2:3, total mixture 600g
- Land division: Property split 3:4:5, total area 2400 sq meters
Key points
- Always add up ratio parts first
- One share = total ÷ sum of parts
- Multiply each ratio part by the share value
- Always verify your answer adds to the original total
- Works for any number of parts in the ratio
Percentages as Ratios - The Hidden Connection
Every percentage problem is actually a ratio problem in disguise. Understanding this connection makes percentage calculations much easier.
Key Insight:
Percent means 'per hundred', so any percent is a ratio with 100 as the second term.
25% = 25:100 = 25/100 = 1/4
Converting Between Forms:
- Percent to ratio: 30% = 30:100 = 3:10
- Ratio to percent: 3:20 = 15:100 = 15%
- Percent to fraction: 40% = 40/100 = 2/5
Solving Percentage Problems Using Proportions:
Example 1: What is 35% of ₱800?
Method 1 (Direct): 35% × ₱800 = 0.35 × ₱800 = ₱280
Method 2 (Proportion): 35/100 = x/800
Cross-multiply: 35 × 800 = 100x
28,000 = 100x
x = ₱280
Example 2: ₱150 is what percent of ₱500?
Set up proportion: 150/500 = x/100
Cross-multiply: 150 × 100 = 500x
15,000 = 500x
x = 30%
Example 3: If 40% of a number is 120, what is the number?
Set up: 40/100 = 120/x
Cross-multiply: 40x = 120 × 100 = 12,000
x = 12,000 ÷ 40 = 300
Percentage Change Problems:
Example 4: A price increases from ₱200 to ₱250. What is the percentage increase?
Step 1: Find the increase
Increase = ₱250 - ₱200 = ₱50
Step 2: Set up proportion (increase compared to original)
50/200 = x/100
Step 3: Solve
x = (50 × 100) ÷ 200 = 25%
Formula: Percentage change = (New value - Old value)/Old value × 100%
Examples
- Grade calculation: 85% of 120 points = what score?
- Discount problems: 20% off ₱500 = ₱100 discount
- Tax calculations: 12% VAT on ₱1000 = ₱120 tax
- Population growth: City grows from 100,000 to 115,000 = 15% increase
Key points
- Every percent is a ratio with 100 as denominator
- Percentage problems can always be solved using proportions
- Three types: find part, find whole, find percentage
- Percentage change = (difference/original) × 100%
- Converting between percents, ratios, and fractions is crucial
Common UPCAT Traps and How to Avoid Them
UPCAT test makers deliberately include traps in ratio and proportion problems. Recognizing these common mistakes can save you valuable points.
Trap 1: Unit Mismatch
Problem: Different units in the same ratio
Example: If a car travels 2 hours at 60 km/h, how far in 30 minutes at the same speed?
Wrong approach: 2/distance = 30/x (mixing hours and minutes)
Correct approach: Convert first
30 minutes = 0.5 hours
Then: 2 hours/120 km = 0.5 hours/x km
x = 30 km
Trap 2: Ratio vs Total Confusion
Problem: Students think the ratio numbers are the actual quantities
Example: In a class, the ratio of boys to girls is 3:2. If there are 40 students total, how many boys?
Wrong thinking: 3 boys (using ratio number directly)
Correct solution:
Step 1: Total parts = 3 + 2 = 5
Step 2: Students per part = 40 ÷ 5 = 8
Step 3: Boys = 3 × 8 = 24
Trap 3: Direct vs Inverse Misidentification
Problem: Assuming wrong relationship type
Example: If 4 pipes fill a tank in 6 hours, how long for 6 pipes?
Wrong (direct): 4/6 = 6/x → x = 9 hours
Correct (inverse): More pipes = less time
4 × 6 = 6 × x → x = 4 hours
Trap 4: Forgetting to Verify
Always plug your answer back into the original problem
Example: Solve 3/7 = x/21
Solution: x = 9
Verification: 3/7 = 9/21 = 3/7 ✓
Trap 5: Percentage Base Confusion
Problem: Using wrong base for percentage calculations
Example: A shirt costs ₱500. After 20% discount, VAT of 12% is added. Final price?
Wrong: ₱500 × 0.8 × 1.12 = ₱448 (applying VAT to original price)
Correct: Discounted price = ₱500 × 0.8 = ₱400
Final price = ₱400 × 1.12 = ₱448
Problem-Solving Checklist:
- Read the problem completely before writing anything
- Identify if it's direct or inverse proportion
- Check units and convert if necessary
- Set up the proportion carefully
- Cross-multiply and solve
- Verify your answer makes sense
- Check units in your final answer
Examples
- Unit trap: Speed in km/h vs time in minutes
- Ratio trap: Boys:Girls = 2:3 doesn't mean 2 boys and 3 girls
- Inverse trap: More workers usually means less time, not more
- Percentage trap: Discount first, then tax on discounted price
Key points
- Always convert units before setting up ratios
- Ratio parts are not the actual quantities
- Carefully determine direct vs inverse relationships
- Always verify answers by substitution
- Watch out for percentage base confusion
- Read word problems twice before solving
Diagrams
Ratio & Proportion Concept Map
This mind map shows the complete structure of ratio and proportion concepts, illustrating how different topics connect and build upon each other for comprehensive understanding.
Problem-Solving Decision Tree
This flowchart provides a systematic approach to solving any ratio and proportion problem, helping students choose the right method and avoid common mistakes.
UPCAT Ratio & Proportion Question Types Distribution
This pie chart shows the typical distribution of ratio and proportion question types in UPCAT exams, helping students prioritize their study focus.
Step-by-Step Proportion Solving Method
This flowchart breaks down the systematic process of solving proportions using cross-multiplication, emphasizing the importance of verification.
In summary
Ratio and proportion mastery requires understanding four key concepts: (1) Ratios compare quantities and can be simplified like fractions, (2) Proportions are equal ratios solved using cross-multiplication, (3) Direct proportion means quantities change together while inverse means they change oppositely, and (4) Partitive proportion divides totals according to given ratios. Remember that percentages are ratios in disguise (25% = 25:100), and always verify your answers. The cross-product rule (if a/b = c/d, then a×d = b×c) is your most powerful tool. Watch for UPCAT traps: unit mismatches, confusing ratio parts with totals, misidentifying direct vs inverse relationships, and percentage base errors. Success comes from careful problem reading, proper setup, systematic solving, and thorough verification.