English:Ratios, Rates, and Proportions
Ratios, Rates, and Proportions
Introduction
Ratios, Rates, and Proportions help you compare quantities, describe how quickly or cheaply something happens, and predict values when quantities change together. In Grades 7–8, these ideas connect arithmetic, fractions, decimals, percentages, linear relationships, geometry, science, maps, recipes, and everyday decision-making.
By the end of this aiMOOC, you should be able to:
- Ratio: Write, simplify, and interpret ratios in several forms.
- Rate: Compare quantities with different units and calculate unit rates.
- Proportion: Recognize and solve equations made from equivalent ratios.
- Proportional relationship: Identify proportional relationships in tables, equations, and graphs.
- Scale factor: Use multiplicative reasoning in scale drawings, recipes, and similar figures.
- Problem solving: Explain your reasoning, check whether an answer is reasonable, and use units correctly.
Ratios: Comparing Quantities
A ratio compares two quantities by division. If a class has 12 students wearing blue and 18 students wearing another color, the ratio of blue to other colors is 12:18. Dividing both parts by 6 gives the equivalent ratio 2:3.
Ratios can be written in several ways: 2 to 3, 2:3, or 2/3. The order matters. A ratio of cats to dogs is not the same as a ratio of dogs to cats unless the two quantities are equal.
A ratio may compare:
- Part-to-part ratio: One part of a group with another part, such as 8 red tiles to 12 blue tiles.
- Part-to-whole ratio: One part with the total, such as 8 red tiles to 20 tiles altogether.
- Equivalent ratio: Ratios that represent the same multiplicative comparison, such as 2:3 and 8:12.
The image above shows how ratios can describe the shape of a rectangle. An aspect ratio such as 4:3 or 16:9 compares width with height. This is the same ratio idea used in screens, photographs, and design.
Simplifying and Building Equivalent Ratios
To simplify a ratio, divide both terms by the same nonzero number. For example, 24:36 simplifies to 2:3 because both numbers are divided by 12.
To build an equivalent ratio, multiply both terms by the same nonzero number. From 2:3, multiplying by 5 gives 10:15. This is useful in ratio tables, where each row keeps the same multiplicative relationship.
A ratio table for 3 notebooks costing €6 could include 1 notebook for €2, 5 notebooks for €10, and 8 notebooks for €16. Each pair has the same cost per notebook, so the relationship is proportional.
Rates and Unit Rates
A rate is a ratio that compares quantities measured in different units. Examples include kilometers per hour, euros per kilogram, words per minute, and liters per second.
A unit rate tells the amount of one quantity for exactly one unit of another quantity. To find a unit rate, divide.
Example: A cyclist travels 150 km in 3 hours.
150 km ÷ 3 h = 50 km/h.
The unit rate is 50 kilometers per hour.
A speedometer reports a rate: distance traveled per unit of time. Rates always need units because the units explain what is being compared.
Comparing Unit Prices
Unit rates help you compare offers fairly.
Store A sells 6 notebooks for €9. The unit price is €9 ÷ 6 = €1.50 per notebook.
Store B sells 10 notebooks for €14. The unit price is €14 ÷ 10 = €1.40 per notebook.
If the notebooks are otherwise comparable, Store B has the lower unit price. A good comparison also considers quality, size, waste, and whether you actually need the larger package.
Proportions: Equal Ratios
A proportion is an equation stating that two ratios are equal. For example:
3/5 = 12/20
Both ratios have the same value, so the equation is true.
You can solve a proportion by using scaling, equivalent fractions, or cross multiplication. Choose a method that makes the structure easy to see.
Example:
3/5 = x/20
Because 5 is multiplied by 4 to get 20, multiply 3 by 4 as well. Therefore x = 12.
You can also use cross products:
3 × 20 = 5 × x
60 = 5x
x = 12
The cross-product method works because multiplying both sides of an equation such as a/b = c/d by bd gives ad = bc, as long as the denominators are not zero.
Checking a Proportion
A proportion should make sense both numerically and in context. If 4 movie tickets cost €36, then one ticket costs €9. So 7 tickets should cost 7 × €9 = €63. If an algebraic method gave €6.30 or €630, the unit rate helps you notice that the result is unreasonable.
Always check:
- Are corresponding quantities written in the same order?
- Are the units compatible?
- Are the ratios equivalent?
- Is the answer reasonable in the situation?
Proportional Relationships
Two variables have a proportional relationship when one variable is always a constant multiple of the other. The equation is:
y = kx
The number k is the constant of proportionality. It is also a unit rate and the slope of the graph.
Example: If each kilogram of apples costs €2.50, then the total cost y for x kilograms is:
y = 2.5x
The constant of proportionality is 2.5 euros per kilogram.
A graph of a proportional relationship is a straight line through the origin. The point (0,0) matters because buying zero kilograms costs zero euros. A straight line that does not pass through the origin may be linear, but it is not proportional.
Tables, Equations, and Graphs
You can test a table for proportionality by dividing y by x for each pair where x is not zero. If the value y/x stays constant, the relationship is proportional.
For example, the pairs (2, 6), (4, 12), and (7, 21) all have y/x = 3. Their equation is y = 3x.
If the ratios change, the relationship is not proportional. For example, a taxi fare with a fixed starting fee plus a cost per kilometer is linear but not proportional because the cost is not zero when the distance is zero.
Scale Factors and Scale Drawings
A scale factor tells how much a figure, map, model, or drawing is enlarged or reduced. If every length in a drawing is twice the matching length in the original, the scale factor is 2.
In a scale of 1:100, one unit on the drawing represents 100 of the same units in reality. For example, 1 cm on the drawing represents 100 cm, or 1 m, in reality.
Map scales are ratios between distances on a map and real-world distances. The units must be handled carefully before comparing or converting.
Example: On a map with scale 1:100,000, 1 cm represents 100,000 cm in reality. Since 100,000 cm = 1 km, a map distance of 6.4 cm represents 6.4 km.
Ratios in Recipes and Measurement
Recipes are a useful example of proportional reasoning. Suppose a soup recipe for 4 servings uses 600 mL of broth. To make 10 servings, the scale factor is 10/4 = 2.5. Multiply the broth amount by 2.5:
600 mL × 2.5 = 1500 mL
So you need 1500 mL, or 1.5 L, of broth.
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When scaling a recipe, multiply every ingredient by the same factor. Changing only one ingredient would change the recipe's ratios.
Ratios in Paper, Screens, and Design
Aspect ratios keep shapes similar when they are resized. If a rectangle has width-to-height ratio 3:2, a 15 cm width corresponds to a 10 cm height because 15:10 simplifies to 3:2.
Proportional resizing appears in printing, photographs, video, digital design, architecture, and technical drawings.
Percent as a Rate per Hundred
A percent is a special ratio with a denominator of 100. For example, 35% means 35 per 100, or 35/100 = 0.35.
This connects percent problems to proportions. If 35% of a number is 84, you can write:
35/100 = 84/x
You can also use the decimal equation 0.35x = 84. Both methods express the same proportional relationship.
Choosing a Strategy
Different problems may be solved efficiently in different ways.
- Equivalent fractions: Useful when one denominator or numerator is an obvious multiple of the other.
- Unit rate: Useful for speed, price, density, productivity, and fair comparisons.
- Cross multiplication: Useful for a proportion with one unknown when simple scaling is not obvious.
- Equation: Useful when a proportional relationship is described by y = kx.
- Graph: Useful for seeing whether the relationship is proportional and comparing rates visually.
Strong mathematical reasoning includes more than getting a number. Explain what each number represents, include units, show why the relationship is proportional, and check the result.
Common Mistakes and How to Avoid Them
Mistake 1: Reversing the order of a ratio. If a problem asks for boys to girls, keep that order in every equivalent ratio.
Mistake 2: Adding instead of multiplying. Proportional reasoning is multiplicative. Going from 3 to 9 uses a factor of 3, so the matching quantity must also be multiplied by 3.
Mistake 3: Ignoring units. A rate such as 60 is incomplete if you do not know whether it means 60 km/h, 60 words/minute, or another unit.
Mistake 4: Assuming every straight-line pattern is proportional. A proportional graph must pass through the origin.
Mistake 5: Cross multiplying without understanding the quantities. First make sure corresponding quantities are placed in matching positions.
Interactive Tasks
Quiz: Test Your Knowledge
What is the simplified form of the ratio 8:12? (2:3) (!3:2) (!4:5) (!8:3)
A car travels 180 km in 3 hours. What is its unit rate? (60 km per hour) (!90 km per hour) (!177 km per hour) (!540 km per hour)
Which equation is a true proportion? (3/4 = 9/12) (!3/4 = 8/12) (!2/5 = 6/10) (!5/8 = 15/16)
In the equation y = 5x, what is the constant of proportionality? (5) (!0) (!x) (!y)
Six notebooks cost €9. What is the unit price? (€1.50 per notebook) (!€3 per notebook) (!€9 per notebook) (!€54 per notebook)
Which graph feature is required for a proportional relationship? (It passes through the origin) (!It is always horizontal) (!It has a negative slope) (!It crosses the y-axis above zero)
A length changes from 6 cm to 15 cm. What scale factor was used? (2.5) (!1.5) (!9) (!21)
Solve the proportion 5/8 = x/24. (15) (!10) (!19) (!40)
What does 30 percent mean as a ratio? (30 per 100) (!30 per 10) (!3 per 1000) (!100 per 30 only)
A recipe uses 2 cups of rice for 6 servings. How many cups are needed for 15 servings? (5 cups) (!3 cups) (!7 cups) (!12 cups)
Memory Game
| Ratio | A comparison of two quantities by division |
| Rate | A ratio comparing quantities with different units |
| Unit rate | A rate for one unit of a quantity |
| Proportion | An equation stating that two ratios are equal |
| Scale factor | A multiplier used to enlarge or reduce corresponding lengths |
| Constant of proportionality | The fixed value k in a relationship of the form y equals kx |
Drag and Drop
| Match the correct terms. | Topic |
|---|---|
| Ratio | Comparison by division |
| Unit rate | Amount for one unit |
| Proportion | Equality of two ratios |
| Constant of proportionality | Fixed multiplier in a proportional relationship |
| Scale factor | Multiplier connecting corresponding lengths |
...
Crossword Puzzle
| Ratio | What compares two quantities by division? |
| Rate | What compares quantities measured in different units? |
| Proportion | What equation says that two ratios are equal? |
| Equivalent | What word describes ratios that have the same value? |
| Scaling | What process multiplies related quantities by the same factor? |
| Constant | What fixed value appears as k in the equation y equals kx? |
LearningApps
Cloze Text
Open-Ended Tasks
Easy
- Ratio Hunt: Find five ratios in your classroom, home, or school environment, photograph or sketch each example, and write one sentence explaining what the two quantities compare.
- Unit Rate Poster: Create a one-page poster that explains one real-life unit rate such as speed, price per item, or words per minute, including a worked example with units.
- Recipe Scaling: Choose a simple recipe and calculate the ingredient amounts for half as many servings and twice as many servings; show the scale factors you used.
- Ratio Interview: Interview a classmate or family member about a situation in which they compare quantities, then rewrite the situation as a ratio or rate problem.
Standard
- Best Buy Investigation: Compare at least four package sizes of the same product using unit prices, record your calculations, and explain which option is the best value under different assumptions.
- Proportional Graph Project: Collect or invent data for a proportional relationship, make a table and graph, write its equation, and explain how the slope relates to the unit rate.
- Map Scale Challenge: Use a printed or digital map with a scale to estimate three real distances, then explain any sources of measurement error.
- Video Explanation of Proportions: Produce a two- to three-minute video teaching how to solve one proportion in two different ways and how to check the answer.
Advanced
- Nonproportional Comparison: Find one proportional and one nonproportional real-world relationship, model both with tables, equations, and graphs, and explain the difference.
- Scale Model Design: Design a scale model of a room, object, or structure, choose a scale, calculate at least six corresponding measurements, and justify why the model stays proportional.
- Rate Experiment: Conduct a safe experiment such as walking a measured distance, filling a container, or typing a passage, calculate rates from repeated trials, and analyze variation in the results.
- Community Data Investigation: Visit or research a relevant local place such as a store, sports facility, transit stop, or museum, collect ratio or rate data, and present a reasoned conclusion supported by calculations.
Learning Assessment
- Model Selection: Given a table, graph, equation, and verbal description, decide which representations show proportional relationships and justify each decision using a constant ratio or the origin test.
- Best Value Reasoning: Compare two pricing plans that include different package sizes or fees, calculate useful unit rates, and explain which plan is better for two different users.
- Error Analysis: Analyze an incorrect proportion solution, identify the exact reasoning error, correct it, and explain how a unit-rate or estimation check reveals the mistake.
- Scale Transfer: Use a scale drawing to calculate an unknown real length, then create a new drawing at a different scale and explain how all corresponding lengths change.
- Multi-Step Rate Problem: Solve a problem that combines a rate with a unit conversion, show each step with units, and explain why the final unit is appropriate.
- Proportionality Argument: Build a mathematical argument from a real or simulated data set showing whether two variables are proportional, using at least two representations as evidence.
Evidence of Learning
Important evidence that you understand the topic includes:
Knowledge: You can define and distinguish ratios, rates, unit rates, proportions, scale factors, and constants of proportionality.
Skills: You can simplify ratios, calculate unit rates, solve proportions, convert units when needed, and recognize proportional relationships in tables, equations, and graphs.
Reasoning: You can explain why two ratios are equivalent, justify whether a relationship is proportional, keep corresponding quantities in the correct order, and check whether an answer is reasonable.
Products: Your ratio tables, graphs, scale drawings, posters, videos, experiments, or comparison reports show accurate calculations and clearly labeled units.
Transfer: You can apply proportional reasoning to unfamiliar situations such as shopping, travel, recipes, maps, science measurements, design, sports statistics, and percentages.
OERs on the Topic
The English Wikipedia article on Ratio provides a broader mathematical reference for ratios and their relationship to proportions.
Linked Learning Areas
Ratios, rates, and proportions connect multiplicative thinking with algebra, geometry, measurement, data, and real-world problem solving. They prepare you for slope, similarity, percent change, probability, dimensional analysis, and many science applications.