Ratios and Proportional Reasoning
5. Real-World Applications of Ratios
Learning outcomes
- I can use ratios in maps, recipes, and scale drawings.
- I can solve problems involving mixtures and comparisons.
- I can apply ratios to sports and statistics.
- I can interpret ratio information from graphs and tables.
- I can explain how ratios help solve real-world problems.
Why Are Ratios Useful?
Ratios allow us to compare quantities and describe how they are related.
In real life, we rarely use ratios only to simplify numbers such as:
12 : 18 = 2 : 3
Instead, ratios help us answer practical questions such as:
- How much of each ingredient should a recipe use?
- How far apart are two places on a map?
- How large should a scale model be?
- How should substances be mixed?
- How can athletes with different numbers of attempts be compared?
- How can data from different-sized groups be compared?
Ratios are useful because they describe a relationship that can be scaled up or down.
Ratios in Recipes
Recipes are one of the most familiar applications of ratios.
Suppose a pancake recipe uses:
flour : milk = 2 : 3
This means that for every:
2 parts flour
we use:
3 parts milk
The size of each "part" can change.
For example:
2 cups flour : 3 cups milk
4 cups flour : 6 cups milk
6 cups flour : 9 cups milk
All represent the same ratio:
2 : 3
Scaling a Recipe Up
Suppose a recipe for 4 people requires:
rice : vegetables = 3 : 2
If we double the recipe for 8 people, multiply both quantities by 2:
3 × 2 : 2 × 2
= 6 : 4
The relationship remains:
3 : 2
Scaling a recipe works because all ingredients are changed by the same scale factor.
Scaling a Recipe Down
Suppose a recipe uses:
600 g flour
300 g sugar
The ratio is:
600 : 300
Simplify:
2 : 1
If we want to make half as much:
300 g flour
150 g sugar
The ratio remains:
300 : 150 = 2 : 1
The quantities changed, but the proportions did not.
Finding a Missing Ingredient
A drink recipe uses:
juice : water = 2 : 5
If we use 8 cups of juice, how much water is needed?
Start:
2 : 5
The juice increased from:
2 → 8
Scale factor:
×4
Therefore:
5 × 4 = 20
Answer:
20 cups of water
Ratios and Total Amounts
Suppose fruit punch contains:
juice : water = 3 : 7
You want to make:
20 L
of punch.
Total ratio parts:
3 + 7 = 10
One part:
20 ÷ 10 = 2 L
Juice:
3 × 2 = 6 L
Water:
7 × 2 = 14 L
Check:
6 + 14 = 20 L
Therefore:
6 L juice and 14 L water
Ratios in Mixtures
Ratios are widely used when substances need to be combined in particular proportions.
Examples include:
- drinks
- paint
- concrete
- cleaning solutions
- fertilizers
- fuels
- laboratory solutions
- alloys
A mixture ratio tells us how much of each component should be used relative to the others.
Mixing Paint
Suppose a paint color requires:
blue : yellow = 3 : 2
If we use:
12 cups of blue paint
then:
3 → 12
Scale factor:
×4
Yellow paint:
2 × 4 = 8 cups
Therefore:
12 cups blue : 8 cups yellow
The ratio simplifies back to:
3 : 2
Three-Part Mixtures
Ratios can contain more than two quantities.
Suppose a mixture uses:
A : B : C = 2 : 3 : 5
If we need:
50 L
altogether, first find the total number of parts:
2 + 3 + 5 = 10 parts
One part:
50 ÷ 10 = 5 L
Therefore:
A:
2 × 5 = 10 L
B:
3 × 5 = 15 L
C:
5 × 5 = 25 L
Check:
10 + 15 + 25 = 50 L
Concentration and Ratios
Ratios can describe how concentrated a mixture is.
Consider:
Mixture A:
concentrate : water = 1 : 4
Mixture B:
concentrate : water = 1 : 8
Mixture A contains more concentrate relative to the amount of water.
This illustrates an important point:
Ratios help us compare relative amounts, not simply total quantities.
Ratios on Maps
Maps represent very large real-world distances on much smaller surfaces.
A map scale describes the relationship between:
distance on the map : actual distance
For example:
1 cm : 5 km
means:
1 cm on the map represents 5 km in reality.
Calculating Actual Distance from a Map
Suppose a map uses:
1 cm : 8 km
Two towns are:
6 cm
apart on the map.
Calculate:
6 × 8 = 48 km
Therefore:
actual distance = 48 km
Calculating Map Distance
Suppose:
1 cm : 10 km
Two locations are actually:
75 km
apart.
Calculate:
75 ÷ 10 = 7.5 cm
Therefore, their distance on the map should be:
7.5 cm
Ratio Scales
Maps sometimes use scales such as:
1 : 100,000
This means:
1 unit on the map represents 100,000 of the same units in reality.
For example:
1 cm : 100,000 cm
Since:
100,000 cm = 1 km
the scale can also be interpreted as:
1 cm : 1 km
Units are extremely important when working with map scales.
Another Map Scale Example
Suppose a map uses:
1 : 50,000
A road measures:
8 cm
on the map.
Actual distance:
8 × 50,000 = 400,000 cm
Convert:
400,000 cm = 4,000 m = 4 km
Therefore:
actual distance = 4 km
Scale Drawings
A scale drawing represents an object using dimensions that are proportional to the real object.
Scale drawings are used in:
- architecture
- engineering
- construction
- product design
- maps
- floor plans
- technical drawings
For example:
1 cm : 2 m
means every 1 cm on the drawing represents 2 m in reality.
Floor Plan Example
A floor plan uses:
1 cm : 2 m
A room measures:
4 cm × 3 cm
on the plan.
Actual length:
4 × 2 = 8 m
Actual width:
3 × 2 = 6 m
Therefore, the real room measures:
8 m × 6 m
Scale Models
Models also use ratios.
Suppose a model car has a scale of:
1 : 20
This means:
1 unit on the model represents 20 of the same units on the real car.
If the model is:
22 cm long
then the actual car would be:
22 × 20 = 440 cm
Convert:
440 cm = 4.4 m
Finding the Size of a Model
A real aircraft is:
36 m long
A model is built at:
1 : 100
Convert:
36 m = 3600 cm
Model length:
3600 ÷ 100 = 36 cm
Therefore:
model length = 36 cm
Scale Factors
The number used to enlarge or reduce quantities proportionally is called the scale factor.
Suppose:
Small drawing length = 5 cm
Actual length = 20 cm
Scale factor:
20 ÷ 5 = 4
Therefore, every measurement on the drawing must be multiplied by:
4
to find the corresponding actual measurement.
Scale Factors and Similar Shapes
Scale factors are also important in geometry.
Suppose a small rectangle measures:
3 cm × 5 cm
A larger similar rectangle uses a scale factor of:
4
New dimensions:
3 × 4 = 12 cm
5 × 4 = 20 cm
All corresponding lengths are multiplied by the same factor.
Comparing Quantities Using Ratios
Ratios allow us to compare groups even when the groups have different sizes.
Suppose:
Class A has:
12 laptops for 24 students.
Class B has:
15 laptops for 25 students.
Class A:
12 : 24 = 1 : 2
This means:
0.5 laptops per student
Class B:
15 : 25 = 3 : 5
This means:
0.6 laptops per student
Converting to a common form makes the comparison easier.
Ratios in Shopping
Suppose two packages contain:
Package A:
6 bottles for $12
Package B:
10 bottles for $18
Calculate the cost per bottle.
Package A:
$12 ÷ 6 = $2.00 per bottle
Package B:
$18 ÷ 10 = $1.80 per bottle
Rates and ratios help us compare packages of different sizes.
Ratios in Sports
Sports statistics frequently involve ratios and rates.
Examples include:
- successful shots : attempts
- wins : losses
- goals : games
- assists : games
- hits : attempts
- points : minutes played
Ratios allow performance to be compared even when athletes have participated different amounts.
Successful Shots
Player A makes:
18 shots from 30 attempts
Success ratio:
18 : 30
Simplify:
3 : 5
Success fraction:
18/30 = 0.60
Success percentage:
60%
Player B makes:
24 shots from 40 attempts
Ratio:
24 : 40 = 3 : 5
Percentage:
60%
Although the players made different numbers of shots, their success rates are equivalent.
Comparing Athletes Fairly
Suppose:
Player A:
15 goals in 20 games
Player B:
18 goals in 30 games
Looking only at total goals gives incomplete information.
Calculate goals per game.
Player A:
15 ÷ 20 = 0.75 goals/game
Player B:
18 ÷ 30 = 0.60 goals/game
The ratio provides another way of comparing performance relative to opportunities.
Context still matters: opponents, playing time, role, and other factors may also be relevant.
Win-Loss Ratios
Suppose a team has:
18 wins and 12 losses
Win-to-loss ratio:
18 : 12
Simplify:
3 : 2
This means that proportionally, the team recorded:
3 wins for every 2 losses
However, this does not mean the team literally followed a repeating pattern of three wins and two losses.
A ratio describes the overall relationship.
Ratios and Statistics
Ratios are widely used to summarize data.
Suppose a survey contains:
120 people who chose A
80 people who chose B
The ratio:
A : B = 120 : 80
Simplify:
3 : 2
This means that for every 3 responses for A, there were proportionally 2 responses for B.
Converting Statistical Ratios to Fractions
Suppose:
A : B = 3 : 2
Total parts:
3 + 2 = 5
Fraction choosing A:
3/5
Fraction choosing B:
2/5
Convert to percentages:
3/5 = 60%
2/5 = 40%
Ratios, fractions, decimals, and percentages can often describe the same data in different ways.
Ratios in Tables
Tables are useful for displaying proportional data.
Suppose a factory produces 40 components every 2 hours.
| Time |
Components |
|---|---|
| 2 | 40 |
| 4 | 80 |
| 6 | 120 |
| 8 | 160 |
| 10 | 200 |
The ratio remains:
40 components : 2 hours
Simplify to the unit rate:
20 components : 1 hour
or:
20 components/hour
Reading Ratio Information from a Table
Consider:
| Distance (km) | Time |
|---|---|
| 60 | 1 |
| 120 | 2 |
| 180 | 3 |
| 240 | 4 |
Calculate:
distance/time
For each row:
60/1 = 60
120/2 = 60
180/3 = 60
240/4 = 60
The ratio is constant.
Therefore, the table represents a proportional relationship with an average rate of:
60 km/h
Spotting a Non-Proportional Table
Consider:
| Items | Cost |
|---|---|
| 1 | $6 |
| 2 | $10 |
| 3 | $14 |
| 4 | $18 |
Check:
6/1 = 6
10/2 = 5
14/3 ≈ 4.67
18/4 = 4.5
The ratios are not equal.
Therefore, the relationship is not proportional.
The table may follow another pattern, but it does not maintain a constant ratio.
Ratios in Bar Graphs
Graphs can also communicate ratios.
Consider this illustrative survey of preferred after-school activities:
Illustrative activity preferences
From the graph:
Sports : Music
30 : 20
Simplify:
3 : 2
Gaming : Art
25 : 15
Simplify:
5 : 3
Graphs provide the quantities, while ratio reasoning allows us to compare them.
Interpreting Ratios from Graphs
When using a graph:
1. Identify the quantities being compared.
2. Read their values carefully.
3. Write the ratio in the requested order.
4. Simplify if appropriate.
5. Explain what the ratio means in context.
For example:
If a graph shows:
40 students prefer A
and:
24 students prefer B
then:
A : B = 40 : 24
Simplify:
5 : 3
Meaning:
For every 5 students preferring A, there are proportionally 3 preferring B.
Ratios in Pie Charts
A pie chart represents parts of a whole.
Suppose a survey gives:
Walking = 40%
Bus = 30%
Car = 20%
Cycling = 10%
The ratio:
Walking : Bus = 40 : 30
Simplify:
4 : 3
The ratio:
Car : Cycling = 20 : 10
Simplify:
2 : 1
Percentages can therefore be converted into ratios for comparison.
Ratios and Population Data
Suppose a region has:
60,000 adults
and:
20,000 children
The ratio:
adults : children = 60,000 : 20,000
Divide by 20,000:
3 : 1
This communicates the relative sizes of the two groups without needing to repeat the large numbers.
Ratios in Science
Ratios are extremely important in science.
Examples include:
- mass : volume
- distance : time
- force : area
- reactants in chemical equations
- genetic ratios
- scale models
- concentrations
- energy input : useful output
For example:
A material with:
mass = 200 g
and:
volume = 50 cm³
has:
mass : volume = 200 : 50
Simplify:
4 : 1
Its density is:
4 g/cm³
Ratios in Chemistry
Chemical equations contain ratios between reacting particles.
For example:
2H₂ + O₂ → 2H₂O
The particle ratio is:
H₂ : O₂ : H₂O = 2 : 1 : 2
This means the balanced equation represents:
2 hydrogen molecules reacting with 1 oxygen molecule to produce 2 water molecules.
Ratios become especially important later when studying stoichiometry.
Ratios in Biology
Genetics frequently involves ratios.
For example, a simple genetic model might predict a phenotype ratio of:
3 : 1
This means that over many expected outcomes, approximately three parts may show one phenotype for every one part showing another, under the assumptions of that model.
It does not guarantee that every group of four offspring will contain exactly three of one type and one of another.
Ratios in Engineering
Engineers use ratios when designing:
- buildings
- bridges
- machines
- gears
- models
- technical drawings
- structures
A gear system might contain:
Gear A = 20 teeth
Gear B = 60 teeth
Ratio:
20 : 60
Simplify:
1 : 3
Gear ratios help engineers analyze how rotational speed and torque change between connected gears.
Ratios in Photography and Screens
Screen dimensions are often described using aspect ratios.
Common examples include:
16 : 9
and:
4 : 3
A 16 : 9 screen means that for every:
16 units of width
there are:
9 units of height
The actual screen could be small or large. The ratio describes its shape.
Ratios in Construction
Suppose concrete is mixed using:
cement : sand : aggregate = 1 : 2 : 4
If one ratio part represents:
5 kg
then:
Cement:
1 × 5 = 5 kg
Sand:
2 × 5 = 10 kg
Aggregate:
4 × 5 = 20 kg
Total:
35 kg
The ratio ensures that the components maintain the intended proportions.
Ratios in Probability
Suppose a bag contains:
6 red counters
and:
4 blue counters
Ratio:
red : blue = 6 : 4 = 3 : 2
Total counters:
10
Probability of red:
6/10 = 3/5
Probability of blue:
4/10 = 2/5
Remember:
3 : 2
compares red with blue.
But:
3/5
compares red with the total.
Ratio vs Actual Quantity
Suppose two classrooms both have:
boys : girls = 2 : 3
Class A could contain:
10 boys and 15 girls.
Class B could contain:
20 boys and 30 girls.
The ratio is the same, but the actual numbers are different.
Ratios describe relative quantities, not necessarily group size.
Comparing Different-Sized Groups
Suppose:
School A:
300 students, 30 computers.
School B:
500 students, 40 computers.
Simply comparing computers:
30 vs 40
suggests School B has more.
But compare computers to students.
School A:
30 : 300 = 1 : 10
School B:
40 : 500 = 2 : 25
This is equivalent to:
0.08 computers per student
compared with:
0.10 computers per student
Ratio and rate comparisons reveal information that totals alone may hide.
Choosing the Useful Form
Different situations may be easier to understand using different representations.
For example:
3 : 5
might be useful when mixing paint.
3/8
might be useful when describing the fraction of the total represented by the first quantity.
37.5%
might be useful when reporting the first quantity as a percentage of the total.
0.6
might be useful when comparing the first quantity directly with the second through division.
The best representation depends on the question being asked.
Worked Example 1: Recipe
A recipe uses:
flour : sugar = 5 : 2
If 20 cups of flour are used:
Scale factor:
5 → 20 = ×4
Sugar:
2 × 4 = 8 cups
Answer:
8 cups of sugar
Worked Example 2: Mixture
A solution uses:
concentrate : water = 1 : 6
You need:
21 L
altogether.
Total parts:
1 + 6 = 7
One part:
21 ÷ 7 = 3 L
Concentrate:
3 L
Water:
18 L
Check:
3 : 18 = 1 : 6
Worked Example 3: Map
Scale:
1 cm : 12 km
Map distance:
7.5 cm
Actual distance:
7.5 × 12 = 90 km
Answer:
90 km
Worked Example 4: Scale Drawing
A building is:
30 m high
A drawing uses:
1 cm : 5 m
Drawing height:
30 ÷ 5 = 6 cm
Answer:
6 cm
Worked Example 5: Sports
Player A scores:
24 goals in 32 games
Player B scores:
30 goals in 50 games
Player A:
24/32 = 0.75 goals/game
Player B:
30/50 = 0.60 goals/game
Using rates allows their scoring records to be compared relative to games played.
Worked Example 6: Table
A table shows:
| Time | Distance |
|---|---|
| 2 h | 140 km |
| 4 h | 280 km |
| 6 h | 420 km |
Calculate:
140 ÷ 2 = 70 km/h
280 ÷ 4 = 70 km/h
420 ÷ 6 = 70 km/h
The ratio is constant.
Therefore, the data represent a proportional relationship with an average rate of:
70 km/h
Worked Example 7: Three-Part Ratio
A school club has students in the ratio:
Grade 8 : Grade 9 : Grade 10 = 2 : 3 : 4
There are:
45 students
altogether.
Total parts:
2 + 3 + 4 = 9
One part:
45 ÷ 9 = 5
Grade 8:
2 × 5 = 10
Grade 9:
3 × 5 = 15
Grade 10:
4 × 5 = 20
Check:
10 + 15 + 20 = 45
Worked Example 8: Scale Model
A model bridge uses a scale:
1 : 200
A section measures:
35 cm
on the model.
Actual length:
35 × 200 = 7000 cm
Convert:
7000 cm = 70 m
Answer:
70 m
Worked Example 9: Interpreting Data
A survey records:
Music = 48 students
Sports = 72 students
Art = 36 students
Find:
Sports : Music
Start:
72 : 48
Divide by 24:
3 : 2
Interpretation:
For every 3 students choosing sports, there are proportionally 2 students choosing music.
Worked Example 10: Comparison
Drink A uses:
2 parts concentrate : 7 parts water
Drink B uses:
3 parts concentrate : 9 parts water
Compare the amount of concentrate relative to water.
Drink A:
2/7 ≈ 0.286
Drink B:
3/9 = 1/3 ≈ 0.333
Drink B has a greater amount of concentrate relative to its water content.
A Reliable Real-World Ratio Strategy
When solving a practical ratio problem:
Step 1: Identify the quantities being compared.
Step 2: Write them in the correct order.
Step 3: Check the units.
Step 4: Convert units if necessary.
Step 5: Simplify the ratio if useful.
Step 6: Identify the scale factor or unit rate.
Step 7: Calculate the missing quantity.
Step 8: Include appropriate units.
Step 9: Check that the final quantities preserve the original ratio.
Step 10: Explain what the answer means in context.
Checking Your Answer
Suppose:
red paint : white paint = 2 : 5
You calculate that:
8 L red requires 20 L white
Check:
8 : 20
Divide both by 4:
2 : 5
The original ratio has been preserved.
Therefore, the answer is consistent.
Estimating Before Calculating
Suppose a map scale is:
1 cm : 9 km
and a route measures:
5.2 cm
Estimate:
5 × 9 ≈ 45 km
Exact calculation:
5.2 × 9 = 46.8 km
The exact answer is close to our estimate, so it appears reasonable.
When Ratio Reasoning Does Not Apply
Not every real-world relationship is proportional.
Suppose a delivery company charges:
$5 fixed fee + $2 per kilometre
For 1 km:
$7
For 2 km:
$9
For 3 km:
$11
The ratios:
7/1, 9/2, 11/3
are not constant.
Therefore, total cost and distance are not directly proportional.
The fixed $5 charge changes the relationship.
Common Mistakes
Mistake 1: Reversing the ratio
If the question asks for:
juice : water
do not write:
water : juice
Mistake 2: Changing only one quantity
If a recipe doubles, every ingredient must be scaled appropriately.
Mistake 3: Forgetting total parts
For:
2 : 3
the total is:
5 parts
not 3 parts.
Mistake 4: Ignoring units
For:
1 cm : 2 m
the units are different and must be handled carefully during calculations.
Mistake 5: Comparing totals instead of relative values
40 successes may sound better than 30 successes, but the number of attempts also matters.
Mistake 6: Assuming every table is proportional
Check whether the ratio or unit rate remains constant.
Mistake 7: Assuming a ratio tells you the total
A ratio of:
3 : 2
could describe:
3 and 2
30 and 20
300 and 200
or infinitely many other equivalent pairs.
Mistake 8: Confusing part-to-part with part-to-whole
If:
red : blue = 3 : 2
then the fraction that is red is:
3/5
not:
3/2
Why Ratios Are Powerful
Ratios allow us to take a known relationship and apply it to a different-sized situation.
A recipe can be doubled.
A map can represent an entire country on one page.
A model can represent a huge building.
Athletes can be compared despite playing different numbers of games.
Mixtures can be reproduced consistently.
Data from groups of different sizes can be compared.
The numbers change, but the relationship remains consistent.
Did You Know?
Many objects around us depend on ratios.
A widescreen display may use an aspect ratio of:
16 : 9
A map may use a scale such as:
1 : 50,000
A model may use:
1 : 100
A recipe may use:
2 : 3
A sports statistic may compare:
successes : attempts
Ratios provide a common mathematical language for describing all of these relationships.
Key Terms
- Ratio: Comparison between two or more quantities.
- Equivalent ratios: Ratios representing the same proportional relationship.
- Scale factor: Number used to multiply or divide related quantities.
- Map scale: Ratio comparing map distance with actual distance.
- Scale drawing: Drawing whose dimensions are proportional to the real object.
- Scale model: Smaller or larger representation of an object that preserves proportional dimensions.
- Mixture: Combination of two or more substances.
- Part-to-part ratio: Comparison between different parts of a group.
- Part-to-whole ratio: Comparison between one part and the total.
- Rate: Ratio comparing quantities, often with different units.
- Unit rate: Rate expressed per one unit.
- Proportional relationship: Relationship that maintains a constant ratio.
- Statistics: Numerical information collected and analyzed to describe data.
- Aspect ratio: Ratio comparing width with height.
- Concentration: Amount of one component relative to a mixture or solution.
Real-World Ratio Guide
Recipes
Keep ingredient ratios constant when scaling.
Mixtures
Maintain the correct proportion of each component.
Maps
Use the scale to convert between map and actual distance.
Scale Drawings
Multiply or divide dimensions using the scale factor.
Sports
Compare performance relative to attempts, games, time, or opportunities.
Statistics
Use ratios to compare groups of different sizes.
Tables
Look for a constant ratio or unit rate.
Graphs
Read the quantities first, then form and simplify the required ratio.
Science
Use ratios to describe quantities such as mass-to-volume, distance-to-time, chemical proportions, and concentrations.
Key Takeaways
- Ratios are used throughout everyday life to compare related quantities.
- Recipes use ratios to maintain consistent ingredient proportions.
- Mixture ratios allow substances to be combined consistently.
- Map scales compare distances on a map with actual distances.
- Scale drawings and models preserve proportional dimensions.
- Scale factors allow objects and relationships to be enlarged or reduced.
- Sports ratios allow performance to be compared relative to opportunities.
- Statistical ratios help compare groups of different sizes.
- Tables can show whether ratios remain constant.
- Graphs can provide quantities that can be compared using ratios.
- Ratios can often be converted into fractions, decimals, percentages, or unit rates.
- Units must be considered carefully when using maps and scale drawings.
- Part-to-part ratios should not be confused with part-to-whole fractions.
- Ratios describe relative quantities rather than necessarily describing the actual size of a group.
- Not every real-world relationship is proportional.
- A fixed fee or changing rate can prevent a relationship from maintaining a constant ratio.
- Answers should be checked by simplifying the final quantities back to the original ratio.
- Ratios help us solve real-world problems because they allow a known relationship to be scaled, compared, interpreted, and applied to new situations.