Ratio & Proportion - Cheat sheet
Your last-minute revision companion for mastering ratio and proportion problems in UPCAT Math
Basic Ratio Concepts
Formulas
a : b = c : d ⟺ a/b = c/d
Meaning
a, b, c, d are terms in proportion; a and d are extremes, b and c are means
Watch out
Don't confuse ratio order - first term goes with first term
When to use
When two ratios are equal or when solving for unknown terms
Cross Product Rule: a × d = b × c
Meaning
Product of extremes equals product of means
Watch out
Always cross-multiply correctly - diagonal terms only
When to use
To solve for any missing term in a proportion
Ratio simplification: a : b = (a÷k) : (b÷k)
Meaning
k is the common factor of both terms
Watch out
Find the GCD first - don't divide by wrong numbers
When to use
To reduce ratios to lowest terms
Important facts
- Ratios can be written as fractions, with colons, or with 'to'
- Always simplify ratios like fractions (divide by GCD)
- Ratios have no units - they're pure numbers
- Order matters in ratios: 3:5 ≠ 5:3
Key definitions
Ratio
Example
3:5 or 3/5 or 3 to 5
Comparison of two quantities of the same kind
Proportion
Example
3:5 = 6:10
Statement that two ratios are equal
Continued Ratio
Example
2:3:5 means three quantities in this ratio
Ratio with three or more parts
Diagrams to know
- Ratio representation as parts of a whole
- Cross-multiplication diagram for proportions
Direct and Inverse Proportion
Formulas
Direct: y = kx or y₁/x₁ = y₂/x₂
Meaning
k is constant of proportionality; as x increases, y increases
Watch out
Make sure relationship is actually direct - test with doubling
When to use
When quantities increase or decrease together
Inverse: xy = k or x₁y₁ = x₂y₂
Meaning
k is constant; as x increases, y decreases proportionally
Watch out
Workers-time, speed-time problems are usually inverse
When to use
When one quantity increases while the other decreases
Important facts
- Direct: doubling one doubles the other
- Inverse: doubling one halves the other
- Always identify type before setting up equation
- Graph of direct proportion passes through origin
Key definitions
Direct Proportion
Example
More pens cost more money
Two quantities increase or decrease at the same rate
Inverse Proportion
Example
More workers finish job in less time
One quantity increases while the other decreases at same rate
Constant of Proportionality
Example
In y = 3x, k = 3
The fixed value k in proportion equations
Diagrams to know
- Direct proportion graph (straight line through origin)
- Inverse proportion graph (hyperbola)
Partitive Proportion
Formulas
Total ÷ Sum of Ratio Parts = Value of One Share
Meaning
Find value of each part by dividing total by sum of all ratio parts
Watch out
Always add up ALL ratio parts, including parts with larger numbers
When to use
When dividing a total amount according to a given ratio
Each Part = (Ratio Part) × (Value of One Share)
Meaning
Multiply each ratio part by the share value to get actual amounts
Watch out
Check that all parts add up to the original total
When to use
After finding the share value, to get individual amounts
Important facts
- Sum all ratio parts first before dividing
- Each person gets (their ratio part) × (share value)
- All distributed amounts must sum to original total
- Share value = Total ÷ (sum of ratio parts)
Key definitions
Partitive Proportion
Example
Divide ₱600 in ratio 2:3:4
Dividing a total into parts following a given ratio
Share Value
Example
If total is 90 and ratio is 2:3:4, one share = 10
Value of one unit in the ratio
Diagrams to know
- Visual representation of parts as segments
- Step-by-step partitive proportion solution flow
Percentage as Ratios
Formulas
Percentage = (Part/Whole) × 100%
Meaning
Convert fraction to percentage by multiplying by 100
Watch out
Don't forget to multiply by 100, not just convert decimal
When to use
When converting fractions or decimals to percentages
x% = x/100 = x:100
Meaning
Any percentage can be written as fraction or ratio with denominator 100
Watch out
25% means 25 out of 100, not 25 out of 1
When to use
When converting percentage problems to proportion problems
Common values
Value
50% = 1/2, 25% = 1/4, 75% = 3/4
Symbol
%
Quantity
Common Percentages
Value
10% = 0.1, 1% = 0.01
Quantity
Decimal Equivalents
Important facts
- Every percent problem is a proportion problem in disguise
- Percent means 'per hundred' or 'out of 100'
- Convert percentages to fractions for easier calculation
- 100% = whole quantity, 50% = half, 25% = quarter
Key definitions
Percentage
Example
25% = 25:100 = 1:4
Ratio with second term as 100
Percent Change
Example
From 80 to 100 is 25% increase
Percentage increase or decrease from original value
Diagrams to know
- Percentage circle or bar representation
- Before and after comparison for percent change
Must remember
- Cross-multiply rule: if a/b = c/d, then a×d = b×c
- Direct proportion: y = kx (same direction change)
- Inverse proportion: xy = k (opposite direction change)
- Partitive proportion: Total ÷ Sum of parts = One share value
- Always simplify ratios to lowest terms using GCD
- Percentages are ratios with 100 as denominator
- Check answers by substituting back into original problem
- Order matters in ratios: 3:5 ≠ 5:3
- For continued ratios, add ALL parts before dividing
- Convert units to be the same before forming ratios
Last minute tips
- Read the problem twice - identify if it's direct or inverse by asking 'if I double this, what happens to that?'
- Always check if your answer makes logical sense - more workers should mean less time
- Look for key words: 'varies directly' = direct proportion, 'inversely proportional' = inverse
- In partitive proportion problems, verify all parts add up to the given total
- Convert all percentages to fractions immediately - it makes calculations easier
Comparison tables
Direct vs Inverse Proportion
| Aspect | Direct Proportion | Inverse Proportion |
|---|---|---|
| Relationship | Both increase/decrease together | One increases, other decreases |
| Formula | y = kx | xy = k |
| Graph Shape | Straight line through origin | Hyperbola |
| Example | Cost and quantity | Workers and time |
| Test Method | Double x, y also doubles | Double x, y becomes half |
Ratio Writing Methods
| Method | Example | When to Use |
|---|---|---|
| Colon notation | 3:5 | Most common in word problems |
| Fraction form | 3/5 | When doing calculations |
| Word form | 3 to 5 | In written explanations |
Diagrams
Ratio & Proportion Concept Map
Complete overview of all ratio and proportion concepts showing relationships between different types and applications
Problem-Solving Decision Tree
Step-by-step decision tree to identify which type of ratio/proportion problem you're dealing with and how to solve it
Cross-Multiplication Method
Visual guide for the cross-multiplication process used to solve proportion problems