Complete Guide to Ratios
1. Introduction to Ratios
A ratio is a way to compare two or more quantities. A two-term ratio can be represented by a quotient; a multi-part ratio records the relative amounts of all its terms. This guide uses nonnegative quantities, with at least one nonzero term. Quotients and reciprocal ratios require a nonzero denominator.
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:
A two-term ratio a:b has the fractional value a/b when b is nonzero. The interpretation depends on which quantities are compared:
- Fractions can represent a quotient, a number, or a part of a whole. In a part-of-a-whole model, the denominator counts equal parts in one whole and the numerator counts the parts used; fractions can also exceed one.
- 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):
For whole-number terms, simplify a ratio by dividing all terms by their greatest common divisor (GCD). For decimal terms, first multiply every term by the same power of 10 to clear decimal places: 1.5:2.5 becomes 15:25, then 3:5. For fractional terms, multiply all terms by a common denominator first. 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 comparing quantities of the same kind, convert them to a common unit first. Rates compare different kinds of quantities, such as kilometers per hour; keep both units rather than trying to convert distance into time.
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 with b and d nonzero, 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 ✓
In a mathematical model, dry cement, sand, and gravel are mixed in a 1:2:3 mass ratio. 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, with b and d nonzero.
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, y = kx, so y/x is constant for x nonzero. In the positive-quantity examples here, k is positive: multiplying x by a factor multiplies y by that same factor. A straight-line relationship with a nonzero intercept is not directly proportional.
Assume a constant price per kilogram with no fixed charge or bulk discount. The cost of apples is then 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.
Assume equally productive workers, a fixed amount of divisible work and no coordination losses. Under this model, the time 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 (beat an egg and use half of it, or weigh beaten egg; rounding to 4 or 5 changes the ratio)
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 agree to share profits in the same 2:3:5 ratio as their contributions. 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.
In a simplified example with no other deductions, 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.
In a mathematical model, dry cement, sand, and gravel have a 1:2:3 mass ratio. How much of each makes 180 kg of this dry mixture? This is a ratio exercise, not a construction specification; water is excluded.
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.
In an idealized volume model, mix 15 mL of alcohol with 85 mL of water and assume volumes add to 100 mL. What is the initial volume ratio and modeled alcohol percentage? Real alcohol-water volumes can contract on mixing.
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 g of a 30% salt solution by mass is mixed with 300 g of a 10% salt solution by mass, what is the mass percentage of salt, assuming no material is lost?
Salt in first solution = 30% of 200 g = 0.3 × 200 g = 60 g
Salt in second solution = 10% of 300 g = 0.1 × 300 g = 30 g
Total salt = 60 g + 30 g = 90 g
Total mass = 200 g + 300 g = 500 g
Concentration = (90 g ÷ 500 g) × 100% = 18% by mass
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 average 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
For comparable apples, Store B has the lower unit price; actual spending also depends on the quantities purchased.
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:
Simplifying gives a convenient equivalent ratio, but an unsimplified ratio is still mathematically valid. Use simplest form when the question requests it.
VALID, NOT SIMPLEST: The ratio 8:12
SIMPLEST FORM: 8:12 = 2:3
Not Using Consistent Units:
For quantities of the same kind, use a common unit. Rates such as dollars per apple keep different units; corresponding terms in a proportion must use consistent 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?
ALSO CORRECT: 3 pens : 8 pens = $4.50 : $x. Both sides compare the first purchase with the second, so 3/8 = 4.50/x gives x = 12.
CORRECT: 3 pens : $4.50 = 8 pens : $x also gives x = 12. A genuinely incorrect setup is 3/8 = x/4.50, which reverses only the cost comparison. Assume the same price per pen and no fixed fee.
Assuming Linearity for All Relationships:
Not all relationships between quantities are linear or directly proportional.
NEEDS ASSUMPTIONS: "If it takes 1 person 10 hours to complete a task, it will take 10 people 1 hour."
This assumes constant worker productivity and perfectly divisible work with no coordination losses. Under those assumptions it is correct; otherwise the worker-time product need not remain constant.
9. Interactive Ratio Quiz
Test Your Ratio Skills
Try these problems and check your answers:
