Ratio, Rate and Proportion
Complete Guide with Exam-Style Examples
Introduction
Ratios, rates, and proportions are fundamental mathematical concepts used to compare quantities and solve real-world problems. This guide covers all key concepts with clear explanations and exam-style examples to help you master these topics.
Key Concepts at a Glance
Ratio
Compares quantities in a specific order (a:b)
Scale
Ratio between model/map and real-life measurements
Rate
Compares different quantities with units (km/h, $/kg)
Proportion
Shows how quantities change in relation to each other
1. Ratio
A ratio compares two or more quantities in a specific order. It is written as a:b and has no units.
Simplifying Ratios
Divide all parts by their highest common factor (HCF).
Example: Simplify the ratio 20:100
Solution: 20:100 = 1:5 (dividing both by 20)
Equivalent Ratios
Multiply or divide all parts by the same number.
Example: Find the missing value: 1:4 = x:20
Solution: 1:4 = 5:20 (multiplying both parts by 5)
Dividing a Quantity in a Given Ratio
Two methods:
- Unitary Method: Add parts, find value of one part, then multiply
- Fraction Method: Express each share as a fraction of total parts
Exam Style Example: $200 is shared between Ana, Ben, and Chris in the ratio 3:4:3. How much does Ben receive?
Solution:
Total parts = 3 + 4 + 3 = 10 parts
Value of 1 part = $200 ÷ 10 = $20
Ben's share (4 parts) = 4 × $20 = $80
Key Point: Ratios must be expressed in the same units. Convert different units before simplifying.
2. Scale
Scale is a ratio between a measurement on a model/map and the corresponding real-life measurement. Usually expressed as 1:n.
Scale Length = Real Length ÷ n
Important: Always ensure units are the same before calculating!
Exam Style Example: A map has a scale of 1:50,000. The distance between two towns on the map is 8 cm. Calculate the actual distance in kilometers.
Solution:
Actual distance = Map distance × Scale
= 8 cm × 50,000 = 400,000 cm
Convert to km (1 km = 100,000 cm): 400,000 ÷ 100,000 = 4 km
Scale Conversion
To express a ratio in the form 1:n, divide both parts by the first number.
Example: Express 4mm:50cm as a ratio scale in the form 1:n
Solution: First convert to same units: 4mm:500mm = 4:500 = 1:125
3. Rate
A rate compares two quantities with different units (e.g., km/h, $/kg). Often expressed as a quantity "per" one unit of another.
Speed
A common rate. Use the formula triangle:
Distance = Speed × Time
Time = Distance ÷ Speed
Exam Style Example: A cyclist travels 45 km in 1 hour and 30 minutes. Calculate her average speed in km/h.
Solution:
Time = 1.5 hours (30 mins is 0.5 hours)
Speed = Distance ÷ Time = 45 km ÷ 1.5 h = 30 km/h
Other Rates
Rates are used in various contexts:
- Population density (people/km²)
- Unit price ($/kg)
- Flow rate (liters/minute)
- Exchange rates ($/£)
Example: 492 people live in an area of 12 km². Express this as a rate in simplest form.
Solution: 492 ÷ 12 = 41 people/km²
4. Direct Proportion
Two quantities are in direct proportion if they increase or decrease at the same rate. Their ratio remains constant.
where k is the constant of proportionality
Exam Style Example: y is directly proportional to x. When x = 5, y = 30.
a) Find the equation connecting y and x.
b) Find y when x = 8.
Solution:
a) y = kx → 30 = k × 5 → k = 6
∴ Equation: y = 6x
b) When x = 8, y = 6 × 8 = 48
Direct Proportion Graphs
Graphs of directly proportional relationships are straight lines that pass through the origin (0,0).
Real-world examples: Exchange rates, earnings based on hours worked, distance traveled at constant speed.
5. Inverse Proportion
Two quantities are in inverse proportion if one increases at the same rate as the other decreases. Their product is constant.
where k is the constant of proportionality
Exam Style Example: It takes 6 workers 10 days to build a wall. How long would it take 15 workers to build the same wall? (Assume constant rate of work).
Solution:
This is inverse proportion (more workers, less time)
Find constant k: workers × days = k → 6 × 10 = 60
For 15 workers: 15 × days = 60 → days = 60 ÷ 15 = 4 days
Inverse Proportion Graphs
Graphs of inversely proportional relationships are hyperbolic curves.
Real-world examples: Speed and time for a fixed distance, number of people and time to complete a task, pressure and volume of a gas.
6. Kinematic Graphs
A. Distance-Time Graphs
- Gradient = Speed
- A steeper gradient means a higher speed
- A horizontal line means the object is stationary (speed = 0)
- A downward slope means motion back toward the start
B. Speed-Time Graphs
- Gradient = Acceleration
- Positive gradient: acceleration
- Negative gradient: deceleration
- Area under the graph = Distance traveled
Exam Style Example: The speed-time graph for a car's journey is shown below (simplified version). Calculate the total distance traveled.
[Imagine a graph showing a triangle with base 10s and height 15m/s]
Solution:
The graph forms a triangle
Area under graph = Distance traveled
Area of triangle = ½ × base × height = ½ × 10 s × 15 m/s = 75 m
Interpreting Graphs
Different graph shapes tell us about motion:
| Graph Shape | Interpretation |
|---|---|
| Straight line with positive gradient | Constant acceleration |
| Horizontal line | Constant speed (zero acceleration) |
| Curved line | Changing acceleration |
| Steep downward slope | Rapid deceleration |
Key Exam Tips
- Show your working clearly. You can often earn method marks even with a final arithmetic error.
- Check your units. Convert them to be consistent (e.g., cm to m, minutes to hours).
- Label your answers. Include units ($, km, m/s, etc.).
- For graphs, read questions carefully to see if you need to find gradient (speed/acceleration) or area (distance).
- For proportion questions, always start by finding the constant k before answering the question.
- Practice with past papers to familiarize yourself with question formats and time management.
Remember: Ratio has no units, while rate always includes units for both quantities.
Common Mistakes to Avoid in Mathematical Calculations
Understanding these common mathematical mistakes will improve your accuracy in calculations and data interpretation. Below we explore three critical areas where errors frequently occur.
The order of terms in a ratio is critical because it defines the relationship between quantities. Reversing the order changes the meaning entirely.
Scenario: In a basket, there are 12 apples and 8 oranges.
Correct: The ratio of apples to oranges is 12:8, which simplifies to 3:2.
Incorrect: The ratio of apples to oranges is 2:3.
Key Point: Always ensure the order of terms in your ratio matches the order of quantities mentioned in the question or statement.
When calculating ratios or percentages, all measurements must be in the same units. Comparing different units leads to incorrect results.
Scenario: What is the ratio of 500 grams to 2 kilograms?
Incorrect: 500:2 (mixing grams and kilograms)
Correct: Convert 2kg to 2000g, then ratio is 500:2000, which simplifies to 1:4.
Key Point: Always convert all measurements to the same unit before calculating ratios or percentages.
Percentage change and percentage points are different concepts that are often confused. Percentage change is relative, while percentage points are absolute.
Scenario: An interest rate increases from 5% to 7%.
The change is 2 percentage points (7 - 5 = 2).
The percentage increase is 40% ((7-5)/5 × 100 = 40%).
Incorrect: "The interest rate increased by 2%."
Key Point: Use "percentage points" when referring to an absolute difference between percentages, and "percentage" when describing a relative change.
Summary
Understanding these common mathematical mistakes will improve your accuracy in calculations and data interpretation:
- Order of Terms: Ratios are order-sensitive - A:B is not the same as B:A
- Units Consistency: Always convert to the same units before calculating
- Percentage vs. Percentage Points: Know the difference between relative change and absolute difference
Mastering these concepts is essential for accurate mathematical reasoning in academics, finance, and everyday life.
Ratio and Proportion - Learner Success Notes
Key Concepts
Understanding Ratios
A ratio compares two or more quantities, showing their relative sizes.
Example: The ratio 3:2 means for every 3 parts of the first quantity, there are 2 parts of the second.
Simplifying Ratios
Divide all terms by their greatest common divisor to simplify ratios.
Example: 12:18 simplifies to 2:3 (dividing both by 6).
Sharing in a Ratio
To divide a quantity in a given ratio, add the ratio parts and divide the quantity by this total.
Example: Share $240 in ratio 7:3 → 7+3=10 parts → $240/10 = $24 per part → $168:$72
Percentage Calculations
Percentage = (Part/Whole) × 100%
Example: 198 tigers as percentage of 106 tigers: (198/106)×100% ≈ 186.79%
Percentage Increase/Decrease
Percentage change = [(New - Original)/Original] × 100%
Example: Increase from 2226 to 2967: [(2967-2226)/2226]×100% ≈ 33.29%
Reverse Percentage
If A is x% greater than B, then B = A ÷ (1 + x/100)
Example: 2226 is 30.48% greater than 2010 value: 2226 ÷ 1.3048 ≈ 1706
Worked Examples
Example 1: Ratio Simplification
Problem: Find the ratio tigers in Bangladesh : Indonesia : India (106:371:2226) in simplest form.
Solution:
Find GCD of 106, 371, and 2226:
106 = 2 × 53
371 = 7 × 53
2226 = 42 × 53 (since 53 × 42 = 2226)
All divisible by 53 → 106:371:2226 = 2:7:42
Example 2: Map Scale Conversion
Problem: Map scale 1:50,000. Map area is 1.2 cm². Find actual area in km².
Solution:
Linear scale: 1 cm = 50,000 cm = 0.5 km
Area scale: 1 cm² = (0.5 km)² = 0.25 km²
Actual area = 1.2 × 0.25 = 0.3 km²
Example 3: Combined Ratios
Problem: Group A:B = 7:10 and B:C = 4:3. Find A:C.
Solution:
Make the B parts equal in both ratios:
A:B = 7:10 = 14:20 (multiplying by 2)
B:C = 4:3 = 20:15 (multiplying by 5)
So A:B:C = 14:20:15
Therefore A:C = 14:15
Key Formulas
| Concept | Formula | Application |
|---|---|---|
| Ratio Division | Part = (Ratio part/Total parts) × Whole | Sharing quantities |
| Percentage | Percentage = (Part/Whole) × 100% | Comparing quantities |
| Percentage Change | % Change = [(New - Original)/Original] × 100% | Increase/decrease calculations |
| Reverse Percentage | Original = New ÷ (1 + Percentage/100) | Finding original values |
| Map Scales | Actual = Measurement × Scale Factor | Converting map measurements |
Problem-Solving Strategies
1. Understand the Problem
Identify what is being compared and what you need to find.
2. Set Up Ratios Correctly
Ensure the order of terms matches the quantities being compared.
3. Work with Common Units
Convert all measurements to the same units before calculating.
4. Check Your Answers
Verify that your results make sense in context.
Real-World Application: Speed Calculations
A train takes 65 minutes to travel 52 km. Calculate average speed in km/h.
Solution:
1. Convert time to hours: 65 minutes = 65/60 hours ≈ 1.0833 hours
2. Speed = Distance/Time = 52 km / 1.0833 h ≈ 48 km/h
Practice Quiz
Question 1
Find the number of tigers in Nepal (198) as a percentage of those in Bangladesh (106).
Question 2
Simplify the ratio 106:371:2226 to its simplest form.
Question 3
The number of tigers in India increased from 2226 to 2967. Calculate the percentage increase.
Question 4
If A:B = 7:10 and B:C = 4:3, find A:C in simplest form.
Question 5
Ahmed and Babar share 240g of sweets in ratio 7:3. Calculate how much Ahmed receives.
Quiz Answers
- (198/106) × 100% ≈ 186.79%
- GCD is 53 → 106:371:2226 = 2:7:42
- [(2967-2226)/2226] × 100% ≈ 33.29%
- A:B = 7:10 = 14:20, B:C = 4:3 = 20:15, so A:C = 14:15
- Total parts = 7+3 = 10, value per part = 240/10 = 24g, Ahmed gets 7×24 = 168g
Common Mistakes to Avoid
Order of Terms
Ensure the order of terms in your ratio matches the order of quantities mentioned.
Units Consistency
Always convert measurements to the same units before calculating ratios or percentages.
Percentage vs. Percentage Points
Understand the difference between percentage increase and percentage points.
Simplifying Completely
Always simplify ratios to their simplest form unless instructed otherwise.
Exam Tips
1. Show your working - Even if the final answer is wrong, you may get credit for correct method.
2. Label your answers - Include units (%, km, g, etc.) in your final answer.
3. Check calculations - If time allows, recheck your work for errors.
4. Estimate first - Make a rough estimate to check if your final answer is reasonable.