Complete Guide to Ratios
1. Introduction to Ratios
A ratio is a way to compare two or more quantities. It shows the relationship between numbers or quantities by dividing one by the other.
Basic Definition:
A ratio compares how much of one thing there is to another. For example, if a class has 20 boys and 15 girls, the ratio of boys to girls is 20:15 (or 20 to 15, or 20/15).
Ways to Write Ratios:
There are three common ways to write ratios:
| Format | Example | Pronounced |
|---|---|---|
| With a colon | 3:2 | "three to two" |
| With the word "to" | 3 to 2 | "three to two" |
| As a fraction | 3/2 | "three over two" or "three divided by two" |
Ratios vs. Fractions:
Although ratios can be written in a form that looks like a fraction, they express a different relationship:
- Fractions represent parts of a whole. The denominator represents the total number of equal parts, and the numerator represents how many of those parts we have.
- Ratios compare quantities that may or may not be part of the same whole. They show how quantities relate to each other in terms of size or number.
For example, if we have 3 red apples and 2 green apples:
- The ratio of red to green apples is 3:2
- The fraction of apples that are red is 3/5 (because there are 5 apples total)
2. Types of Ratios
Part-to-Part Ratios:
These ratios compare one part of a whole to another part of the same whole.
In a bag containing 8 red marbles and 12 blue marbles:
The ratio of red marbles to blue marbles is 8:12 or 2:3 (simplified)
Part-to-Whole Ratios:
These ratios compare one part of a whole to the entire whole.
In a bag containing 8 red marbles and 12 blue marbles (20 total):
The ratio of red marbles to all marbles is 8:20 or 2:5 (simplified)
The ratio of blue marbles to all marbles is 12:20 or 3:5 (simplified)
Whole-to-Part Ratios:
These ratios compare the whole to one of its parts.
In a bag containing 8 red marbles and 12 blue marbles (20 total):
The ratio of all marbles to red marbles is 20:8 or 5:2 (simplified)
Note: This is the reciprocal of the part-to-whole ratio.
Multiple-Part Ratios:
These ratios compare more than two quantities.
A fruit basket contains 6 apples, 9 oranges, and 3 bananas:
The ratio of apples to oranges to bananas is 6:9:3 or 2:3:1 (simplified)
3. Simplifying Ratios
Finding the Greatest Common Divisor (GCD):
To simplify a ratio, we divide all terms by their greatest common divisor (GCD). There are several ways to find the GCD:
Method 1: List Factors
To find the GCD of 18 and 24:
Factors of 18: 1, 2, 3, 6, 9, 18
Factors of 24: 1, 2, 3, 4, 6, 8, 12, 24
Common factors: 1, 2, 3, 6
GCD = 6
Method 2: Prime Factorization
To find the GCD of 18 and 24:
18 = 2 × 3²
24 = 2³ × 3
GCD = 2¹ × 3¹ = 6
(We take the common prime factors with the smallest exponents)
Method 3: Euclidean Algorithm
To find the GCD of 18 and 24:
24 = 18 × 1 + 6 (Divide 24 by 18, remainder 6)
18 = 6 × 3 + 0 (Divide 18 by 6, remainder 0)
Since the remainder is 0, the GCD is 6
Steps to Simplify a Ratio:
- Find the greatest common divisor (GCD) of all numbers in the ratio.
- Divide each term in the ratio by the GCD.
Simplify the ratio 15:25
GCD of 15 and 25 = 5
15 ÷ 5 = 3
25 ÷ 5 = 5
Simplified ratio: 3:5
Simplify the ratio 36:60:48
GCD of 36, 60, and 48 = 12
36 ÷ 12 = 3
60 ÷ 12 = 5
48 ÷ 12 = 4
Simplified ratio: 3:5:4
Ratios with Different Units:
When a ratio involves different units, convert all terms to the same unit first.
Find the ratio of 2 hours to 30 minutes
Convert: 2 hours = 120 minutes
Ratio: 120 minutes : 30 minutes = 120:30 = 4:1
Find the ratio of 1.5 kg to 500 g
Convert: 1.5 kg = 1500 g
Ratio: 1500 g : 500 g = 1500:500 = 3:1
4. Equivalent Ratios
Definition of Equivalent Ratios:
Equivalent ratios are ratios that express the same relationship between numbers. They can be obtained by multiplying or dividing all terms of a ratio by the same non-zero number.
The following ratios are equivalent:
1:2, 2:4, 3:6, 4:8, 5:10
All these ratios represent the same relationship: the second number is twice the first number.
Creating Equivalent Ratios:
To create an equivalent ratio, multiply or divide all terms by the same non-zero number.
Create equivalent ratios for 3:5
Multiply by 2: 3×2 : 5×2 = 6:10
Multiply by 3: 3×3 : 5×3 = 9:15
Multiply by 4: 3×4 : 5×4 = 12:20
Create equivalent ratios for 6:9:12
Multiply by 2: 6×2 : 9×2 : 12×2 = 12:18:24
Divide by 3: 6÷3 : 9÷3 : 12÷3 = 2:3:4
Testing if Ratios are Equivalent:
To test if two ratios are equivalent, convert them to their simplest form or use cross multiplication.
Are 6:10 and 9:15 equivalent ratios?
Method 1: Convert to simplest form
6:10 simplifies to 3:5
9:15 simplifies to 3:5
Since both simplify to 3:5, they are equivalent.
Method 2: Cross multiply (for 2-term ratios)
For a:b and c:d, check if a×d = b×c
6×15 = 10×9
90 = 90
Since 6×15 = 10×9, the ratios are equivalent.
Finding a Missing Term in an Equivalent Ratio:
To find a missing term in a pair of equivalent ratios, set up a proportion and solve for the unknown.
Find the missing value x in the equivalent ratios 3:4 and x:20
Set up a proportion: 3/4 = x/20
Cross multiply: 3×20 = 4×x
60 = 4x
x = 15
So the equivalent ratio is 15:20 (which simplifies to 3:4)
5. Ratio Calculations
Finding a Quantity When the Ratio and Total are Known:
When you know the ratio of parts and the total of all parts, you can find the individual quantities.
A sum of $800 is divided in the ratio 3:5. How much does each person get?
Sum of ratio parts = 3 + 5 = 8
First part = (3 ÷ 8) × $800 = $300
Second part = (5 ÷ 8) × $800 = $500
Check: $300 + $500 = $800 ✓
A mixture of cement, sand, and gravel is made in the ratio 1:2:3. If 30 kg of the mixture is needed, how much of each component should be used?
Sum of ratio parts = 1 + 2 + 3 = 6
Cement = (1 ÷ 6) × 30 kg = 5 kg
Sand = (2 ÷ 6) × 30 kg = 10 kg
Gravel = (3 ÷ 6) × 30 kg = 15 kg
Check: 5 kg + 10 kg + 15 kg = 30 kg ✓
Finding a Quantity When the Ratio and One Part are Known:
When you know the ratio of parts and one of the parts, you can find the other parts.
Two numbers are in the ratio 2:7. If the smaller number is 10, what is the larger number?
Larger number = (7 ÷ 2) × 10 = 35
Check: 10:35 simplifies to 2:7 ✓
A recipe uses ingredients in the ratio 3:2:1. If it requires 150g of the first ingredient, how much of the other ingredients are needed?
Second ingredient = (2 ÷ 3) × 150g = 100g
Third ingredient = (1 ÷ 3) × 150g = 50g
Check: 150g:100g:50g simplifies to 3:2:1 ✓
Finding the Total When the Ratio and One Part are Known:
When you know the ratio of parts and one of the parts, you can find the total of all parts.
Two numbers are in the ratio 3:5. If the smaller number is 21, what is the sum of the two numbers?
Sum of ratio parts = 3 + 5 = 8
Total = (8 ÷ 3) × 21 = 56
Check: Larger number = (5 ÷ 3) × 21 = 35, and 21 + 35 = 56 ✓
Three partners share profits in the ratio 2:3:5. If the first partner receives $12,000, what is the total profit?
Sum of ratio parts = 2 + 3 + 5 = 10
Total profit = (10 ÷ 2) × $12,000 = $60,000
Check: Second partner = (3 ÷ 2) × $12,000 = $18,000
Third partner = (5 ÷ 2) × $12,000 = $30,000
$12,000 + $18,000 + $30,000 = $60,000 ✓
6. Proportions and Ratios
Definition of a Proportion:
A proportion is a statement that two ratios are equal. It can be written as a:b = c:d or a/b = c/d.
The proportion 2:3 = 10:15 states that the ratios 2:3 and 10:15 are equal.
We can verify this by simplifying 10:15 to 2:3, or by cross multiplication:
2×15 = 3×10
30 = 30 ✓
Solving Proportions:
To solve for a missing value in a proportion, you can use cross multiplication.
Solve for x in the proportion 3:8 = x:40
Cross multiply: 3×40 = 8×x
120 = 8x
x = 15
Direct Proportion:
In a direct proportion, as one quantity increases, the other increases at the same rate. The ratio of corresponding values remains constant.
The cost of apples is directly proportional to their weight. If 3 kg of apples cost $12, how much would 5 kg cost?
Set up a proportion: 3 kg : $12 = 5 kg : $x
Cross multiply: 3×x = 12×5
3x = 60
x = 20
So 5 kg of apples would cost $20.
Inverse Proportion:
In an inverse proportion, as one quantity increases, the other decreases at such a rate that their product remains constant.
The time taken to complete a job is inversely proportional to the number of workers. If 4 workers can complete a job in 15 hours, how long would it take 6 workers to complete the same job?
For inverse proportion, use: Workers × Time = constant
4 × 15 = 6 × x
60 = 6x
x = 10
So 6 workers would take 10 hours to complete the job.
7. Applications of Ratios
Scale Drawings and Maps
Understanding Scale Ratios:
A scale ratio compares the dimensions on a drawing or map to the actual dimensions in the real world.
A scale of 1:100 means that 1 unit on the drawing represents 100 units in reality.
For instance, 1 cm on the drawing represents 100 cm (or 1 m) in reality.
Finding Actual Dimensions:
To find the actual dimension from a scale drawing:
On a map with a scale of 1:50,000, two cities are 8 cm apart. What is the actual distance between the cities?
Actual distance = 8 cm × 50,000 = 400,000 cm = 4 km
Finding Drawing Dimensions:
To find the drawing dimension from actual measurements:
An architect wants to draw a building that is 15 m long on a blueprint with a scale of 1:150. How long should the building be on the blueprint?
Drawing length = 15 m ÷ 150 = 0.1 m = 10 cm
Cooking and Recipes
Scaling Recipes:
Ratios are used to adjust recipe quantities for a different number of servings.
A recipe for 4 servings calls for 2 cups of flour and 3 eggs. How much flour and how many eggs are needed for 6 servings?
Flour: 2 cups × (6 ÷ 4) = 2 cups × 1.5 = 3 cups
Eggs: 3 eggs × (6 ÷ 4) = 3 eggs × 1.5 = 4.5 eggs (use 4 or 5 eggs since you can't have partial eggs)
Recipe Ratios:
Many recipes rely on specific ratios of ingredients for consistent results.
A basic bread recipe might use a flour to water ratio of 5:3 by weight. If you have 500g of flour, how much water should you use?
Water = (3 ÷ 5) × 500g = 300g
Financial Calculations
Sharing Profits:
Ratios are used to distribute profits or costs among partners or investors.
Three investors contribute to a business in the ratio 2:3:5. If the annual profit is $100,000, how much does each investor receive?
Sum of ratio parts = 2 + 3 + 5 = 10
First investor: (2 ÷ 10) × $100,000 = $20,000
Second investor: (3 ÷ 10) × $100,000 = $30,000
Third investor: (5 ÷ 10) × $100,000 = $50,000
Income Tax Calculations:
Ratios help analyze how income is divided between taxes and take-home pay.
If a person pays $15,000 in taxes on an income of $60,000, what is the ratio of taxes to take-home pay?
Take-home pay = $60,000 - $15,000 = $45,000
Ratio of taxes to take-home pay = $15,000 : $45,000 = 15:45 = 1:3
Mixtures and Solutions
Creating Mixtures with Specific Ratios:
Ratios are used to create mixtures with specific proportions of ingredients.
A concrete mix requires cement, sand, and gravel in the ratio 1:2:3. How much of each is needed to make 180 kg of concrete?
Sum of ratio parts = 1 + 2 + 3 = 6
Cement = (1 ÷ 6) × 180 kg = 30 kg
Sand = (2 ÷ 6) × 180 kg = 60 kg
Gravel = (3 ÷ 6) × 180 kg = 90 kg
Determining Concentration Ratios:
Ratios are used to express the concentration of solutions.
If 15 ml of alcohol is mixed with 85 ml of water, what is the ratio of alcohol to water, and what is the concentration of alcohol in the solution?
Ratio of alcohol to water = 15:85 = 3:17 (simplified)
Concentration of alcohol = (15 ml ÷ 100 ml) × 100% = 15%
Mixing Solutions of Different Concentrations:
Ratios help determine the resulting concentration when solutions of different concentrations are mixed.
If 200 ml of a 30% salt solution is mixed with 300 ml of a 10% salt solution, what is the concentration of the resulting mixture?
Salt in first solution = 30% of 200 ml = 0.3 × 200 ml = 60 ml
Salt in second solution = 10% of 300 ml = 0.1 × 300 ml = 30 ml
Total salt = 60 ml + 30 ml = 90 ml
Total volume = 200 ml + 300 ml = 500 ml
Concentration = (90 ml ÷ 500 ml) × 100% = 18%
Speeds, Rates, and Ratios
Speed as a Ratio:
Speed is a ratio of distance to time.
If a car travels 240 kilometers in 3 hours, what is its speed?
Speed = 240 km ÷ 3 h = 80 km/h
Comparing Rates:
Ratios are used to compare rates and determine the better deal.
Store A sells 5 apples for $2, while Store B sells 8 apples for $3. Which store offers the better value?
Store A: $2 ÷ 5 apples = $0.40 per apple
Store B: $3 ÷ 8 apples = $0.375 per apple
Store B offers better value as the cost per apple is lower.
Fuel Efficiency:
Fuel efficiency is often expressed as a ratio of distance to fuel volume.
A car travels 350 kilometers on 25 liters of fuel. What is its fuel efficiency?
Fuel efficiency = 350 km ÷ 25 L = 14 km/L
8. Common Mistakes with Ratios
Confusing Part-to-Part and Part-to-Whole Ratios:
A common mistake is confusing part-to-part ratios with part-to-whole ratios.
If a class has 15 boys and 10 girls:
The part-to-part ratio of boys to girls is 15:10 or 3:2.
The part-to-whole ratio of boys to total students is 15:25 or 3:5.
INCORRECT: "The ratio of boys in the class is 3:2."
CORRECT: "The ratio of boys to girls is 3:2." OR "The ratio of boys to total students is 3:5."
Not Simplifying Ratios:
Ratios should generally be expressed in their simplest form.
INCORRECT: The ratio of 8:12 (not simplified)
CORRECT: The ratio of 8:12 = 2:3 (simplified)
Not Using Consistent Units:
When creating ratios, all quantities must be expressed in the same units.
INCORRECT: The ratio of 2 meters to 50 centimeters = 2:50
CORRECT: The ratio of 2 meters to 50 centimeters = 200 cm : 50 cm = 4:1
Mixing Up Terms in Proportions:
When setting up proportions, it's important to maintain consistent order of terms.
If 3 pens cost $4.50, how much would 8 pens cost?
INCORRECT: 3 pens : 8 pens = $4.50 : $x (mixing quantities)
CORRECT: 3 pens : $4.50 = 8 pens : $x (or 3:$4.50 = 8:$x)
Assuming Linearity for All Relationships:
Not all relationships between quantities are linear or directly proportional.
INCORRECT: "If it takes 1 person 10 hours to complete a task, it will take 10 people 1 hour."
This assumes perfect linearity, which isn't always true due to factors like coordination overhead or resource constraints.
9. Interactive Ratio Quiz
Test Your Ratio Skills
Try these problems and check your answers:
