Ratio & Proportion - Revision notes
Ratio and proportion are fundamental mathematical concepts that appear in almost every UPCAT Math section. They form the foundation for solving problems involving mixtures, scale drawings, speed, percentages, and many real-world applications. Understanding these concepts thoroughly will help you tackle multiple question types with confidence.
Understanding Ratios
Formulas
a:b = c:d (equivalent ratios)
Example
If 3:4 = x:12, then x = 9 (by cross multiplication: 3×12 = 4×x)
Variables
a, b, c, d are quantities being compared
Application
Used to find missing terms in proportional relationships
Simplified ratio = a/gcd : b/gcd
Example
18:24 = 3:4 (dividing both by gcd = 6)
Variables
gcd = greatest common divisor of a and b
Application
Reducing ratios to simplest form
Exam tips
- Always check if units are the same before creating ratios
- Simplify ratios immediately to avoid calculation errors
- Label your ratios clearly to avoid confusion
- Use cross multiplication for quick solving
Key points
- A ratio compares two quantities of the same kind
- Ratios can be written in three ways: 3:5, 3/5, or '3 to 5'
- Ratios must be simplified to their lowest terms like fractions
- Continued ratios involve three or more parts (e.g., 2:3:5)
- Units must be the same when comparing quantities
Definitions
Ratio
A comparison of two quantities of the same kind, expressing how many times one contains the other
Importance
Foundation for all proportion problems and percentage calculations
Continued Ratio
A ratio involving three or more quantities, written as a:b:c
Importance
Essential for dividing amounts among multiple parties
Common mistakes
- Forgetting to convert units before comparing (e.g., comparing minutes to hours)
- Not simplifying ratios to their lowest terms
- Confusing ratio parts with actual values
- Writing ratios in wrong order (antecedent vs consequent)
Proportions and Cross Multiplication
Formulas
Cross Product Rule: a/b = c/d → ad = bc
Example
Find x: 5/8 = x/24. Cross multiply: 5×24 = 8×x, so 120 = 8x, therefore x = 15
Variables
a, b, c, d are the terms of the proportion
Application
Solving for unknown terms in proportions
Mean Proportional: if a/b = b/c, then b² = ac
Example
If 4/b = b/9, then b² = 36, so b = 6
Variables
b is the mean proportional between a and c
Application
Finding geometric means in sequences
Exam tips
- Double-check your cross multiplication setup
- Verify answers by substituting back into original proportion
- Use estimation to check if your answer makes sense
- Practice cross multiplication until it becomes automatic
Key points
- A proportion states that two ratios are equal
- Cross multiplication: if a/b = c/d, then ad = bc
- Cross multiplication is the fastest way to solve proportions
- Always verify your answer by substituting back
- Proportions appear in every type of ratio word problem
Definitions
Proportion
A mathematical statement that two ratios are equal
Importance
Core tool for solving ratio problems and scaling calculations
Cross Multiplication
Method where you multiply diagonally across the equal sign in a proportion
Importance
Most efficient technique for solving proportion equations
Common mistakes
- Mixing up which numbers to multiply when cross multiplying
- Forgetting to check the reasonableness of the answer
- Not setting up the proportion correctly from word problems
- Arithmetic errors during cross multiplication
Direct and Inverse Proportion
Formulas
Direct Proportion: y₁/x₁ = y₂/x₂ or y = kx
Example
If 3 books cost ₱150, then 7 books cost ₱350 (150/3 = 350/7 = 50)
Variables
x and y are the variables, k is the constant of proportionality
Application
When quantities change in the same direction
Inverse Proportion: x₁y₁ = x₂y₂ or xy = k
Example
If 6 workers finish in 8 days, then 12 workers finish in 4 days (6×8 = 12×4 = 48)
Variables
x and y are inversely related variables, k is constant
Application
When one quantity increases, the other decreases proportionally
Exam tips
- Ask yourself: 'If I double one, does the other double or halve?'
- Look for keywords: 'together' suggests direct, 'shared work' suggests inverse
- Practice identifying proportion types from word problems
- Always state which type you're using before solving
Key points
- Direct proportion: both quantities increase or decrease together
- Inverse proportion: one increases while the other decreases
- Direct formula: y = kx (k is constant)
- Inverse formula: xy = k (k is constant)
- Key is identifying which type from the problem context
Definitions
Direct Proportion
Relationship where two quantities increase or decrease at the same rate
Importance
Common in cost calculations, speed problems, and scaling
Inverse Proportion
Relationship where one quantity increases as the other decreases proportionally
Importance
Essential for work-rate problems, pipe filling, and gear problems
Common mistakes
- Confusing direct and inverse relationships
- Not identifying the type of proportion from context
- Using wrong formula for the relationship type
- Misinterpreting 'more workers, less time' scenarios
Partitive Proportion
Formulas
Share value = Total amount ÷ Sum of ratio parts
Example
Divide ₱840 in ratio 2:3:7. Share value = 840 ÷ (2+3+7) = 840 ÷ 12 = ₱70
Variables
Total amount is what's being divided, ratio parts are the given proportions
Application
First step in all partitive proportion problems
Each part = Ratio part × Share value
Example
First part = 2 × ₱70 = ₱140, Second part = 3 × ₱70 = ₱210, Third part = 7 × ₱70 = ₱490
Variables
Ratio part is each number in the ratio, share value from previous calculation
Application
Finding actual amount for each party
Exam tips
- Always write out the step: 'Sum of parts = ...'
- Label each part clearly (A gets..., B gets..., C gets...)
- Check your work: do all parts add up to the original total?
- Use this method for any 'divide in ratio' problem
Key points
- Used to divide a total into parts following a given ratio
- Add all ratio parts to find total shares
- Divide the total amount by total shares to get one share value
- Multiply each ratio part by the share value
- Always verify that parts add up to the original total
Definitions
Partitive Proportion
Method of dividing a quantity into parts that maintain a specified ratio
Importance
Essential for business partnerships, inheritance problems, and resource allocation
Share Value
The value of one unit in the ratio when dividing a total amount
Importance
Key intermediate step that makes calculations systematic and error-free
Common mistakes
- Forgetting to add all ratio parts before dividing
- Using individual ratio parts as final answers instead of calculating actual amounts
- Not checking that final parts sum to original total
- Mixing up which part corresponds to which person or item
Applications and Word Problems
Formulas
Percentage as ratio: x% = x/100
Example
25% = 25/100 = 1/4, so 25% of 80 = (1/4) × 80 = 20
Variables
x is the percentage value
Application
Converting between percentages and ratios
Scale problems: Map distance/Actual distance = Scale ratio
Example
If 2 cm represents 5 km, then x cm represents 12.5 km. Set up: 2/5 = x/12.5
Variables
Distances must be in same units
Application
Map reading and scale drawing problems
Exam tips
- Read the problem twice before writing anything
- Identify exactly what the question is asking for
- Check units and convert if necessary before setting up ratios
- Use estimation to verify your final answer makes sense
Key points
- Ratio and proportion appear in mixture problems, map scales, and speed calculations
- Percentages are ratios with denominator 100
- Always identify what quantities are being compared
- Set up proportions systematically from word problems
- Practice translating word problems into mathematical expressions
Definitions
Scale Ratio
Ratio comparing a representation (like a map) to actual size
Importance
Critical for geography problems and technical drawing questions
Mixture Ratio
Ratio of different components in a mixture or solution
Importance
Common in chemistry word problems and cooking/recipe contexts
Common mistakes
- Not reading the problem completely before starting calculations
- Mixing up units (cm vs km, minutes vs hours)
- Setting up proportions with terms in wrong positions
- Forgetting to convert final answer to required units
Connections
- Ratios connect to fractions, decimals, and percentages - they're all ways of expressing relationships between quantities
- Proportion skills are essential for algebra, particularly solving linear equations and systems
- Direct and inverse variation concepts prepare you for functions and graphing in advanced mathematics
- Partitive proportion is fundamental for statistics, probability, and data analysis
- Scale and map problems connect mathematics to geography and technical drawing
Quick review questions
Simplify the ratio 24:36 to its lowest terms.
Answer
2:3
Find GCD of 24 and 36, which is 12. Divide both terms: 24÷12 = 2, 36÷12 = 3
If 3/5 = x/25, find x using cross multiplication.
Answer
x = 15
Cross multiply: 3 × 25 = 5 × x, so 75 = 5x, therefore x = 15
If 4 pens cost ₱60, how much do 7 pens cost? (Direct proportion)
Answer
₱105
Set up proportion: 4/60 = 7/x. Cross multiply: 4x = 420, so x = 105
Divide ₱450 among A, B, and C in the ratio 2:3:4.
Answer
A gets ₱100, B gets ₱150, C gets ₱200
Sum of parts = 2+3+4 = 9. Share value = 450÷9 = 50. A: 2×50=100, B: 3×50=150, C: 4×50=200
If 8 workers complete a job in 15 days, how long will 12 workers take? (Inverse proportion)
Answer
10 days
Use xy = k: 8×15 = 12×d, so 120 = 12d, therefore d = 10 days
Diagrams
Ratio & Proportion Complete Overview
This mind map shows the complete structure of ratio and proportion topics, helping you see how all concepts connect together.
Problem-Solving Decision Tree for Ratio & Proportion
This flowchart helps you quickly identify which type of ratio/proportion problem you're facing and which method to use.
Step-by-Step Cross Multiplication Process
This step-by-step flowchart shows exactly how to use cross multiplication to solve proportion problems systematically.
Exam strategy
For UPCAT success in ratio and proportion: (1) Master the cross multiplication technique - it's your fastest tool, (2) Practice identifying direct vs inverse relationships quickly, (3) Always simplify ratios first to avoid large number calculations, (4) Use the three-step method for partitive proportion: find sum, find share value, multiply each part, (5) Verify every answer by substitution or logical checking, (6) Watch for unit conversion traps - UPCAT loves to mix different units in the same problem.