Ratios and Proportions: What They Are, How to Solve Them, and Exercises

Español| English

Ratios and proportions are fundamental mathematical concepts because they allow us to compare quantities and establish relationships between them. They appear in a wide variety of situations, such as comparing prices, interpreting scales, preparing mixtures, calculating quantities, and solving problems involving different measurements.

A ratio expresses the relationship between two quantities through a comparison. A proportion, on the other hand, establishes an equality between two ratios. Although these concepts are closely related, they do not mean exactly the same thing.

In this article, you will learn what a ratio is, what a proportion is, how to represent and simplify ratios, how to solve proportions, what direct and inverse proportionality are, and how to use the rule of three. You will also find examples and exercises solved step by step.

What Is a Mathematical Ratio?

A ratio is a comparison between two quantities expressed through division. If we have two quantities a and b, where b is not zero, we can express their ratio as:

a/b

We can also represent a ratio using a colon:

a:b

For example, if a group has 8 students studying Mathematics and 4 students studying English, the ratio between the two quantities can be expressed as:

8:4

or:

8/4

When simplified:

8/4 = 2

The ratio can be interpreted as meaning that for every 2 students in the first group, there is 1 student in the second group.

Key idea: A ratio compares two quantities. It does not necessarily represent a part of a whole; it expresses a relationship between two quantities.

Antecedent and Consequent

In a ratio such as:

8:4

the first term, 8, is called the antecedent, and the second term, 4, is called the consequent.

Therefore:

8 → antecedent

4 → consequent

How Do You Simplify a Ratio?

A ratio can be simplified by dividing both of its terms by the same nonzero number.

For example:

12:18

Both terms can be divided by 6:

12 ÷ 6 = 2

18 ÷ 6 = 3

Therefore:

12:18 = 2:3

Both ratios represent the same relationship.

Example of Simplifying a Ratio

Simplify the ratio:

20:30

The greatest common divisor of 20 and 30 is 10. Divide both terms by 10:

20 ÷ 10 = 2

30 ÷ 10 = 3

Therefore:

20:30 = 2:3

What Is a Proportion?

A proportion is an equality between two ratios.

For example:

2/3 = 4/6

The ratio 2/3 and the ratio 4/6 represent the same relationship, so they form a proportion.

We can also write it as:

2:3 = 4:6

Therefore, the fundamental difference can be summarized as follows:

Ratio: compares two quantities.

Proportion: establishes an equality between two ratios.

Terms of a Proportion

In a proportion such as:

2/3 = 4/6

the terms at the ends are called the extremes, while the two middle terms are called the means.

In this proportion:

2/3 = 4/6

the extremes are 2 and 6, while the means are 3 and 4.

The Fundamental Property of Proportions

A fundamental property of proportions states that if:

a/b = c/d

then the product of the extremes is equal to the product of the means:

a × d = b × c

This property allows us to find unknown values in a proportion.

Example

Consider:

2/3 = 4/6

Multiply the extremes:

2 × 6 = 12

Then multiply the means:

3 × 4 = 12

Since both products are equal, the proportion is correct.

12 = 12

How Do You Solve a Proportion?

When a proportion contains an unknown value, we can use the fundamental property of proportions to find it.

For example:

3/5 = x/20

Cross-multiply:

3 × 20 = 5 × x

Then:

60 = 5x

Divide both sides by 5:

x = 12

Therefore:

3/5 = 12/20

The proportion is correct because both ratios represent the same value.

Another Example

Let us solve:

4/7 = 12/x

Cross-multiply:

4x = 7 × 12

4x = 84

Divide by 4:

x = 21

Therefore:

4/7 = 12/21

Equivalent Ratios

Two ratios are equivalent when they represent the same relationship between quantities.

For example:

2:3 = 4:6 = 6:9

We can obtain equivalent ratios by multiplying both terms by the same nonzero number.

Starting with:

2:3

Multiply both terms by 2:

2 × 2 : 3 × 2 = 4:6

We can then multiply by 3:

2 × 3 : 3 × 3 = 6:9

In every case, the same relationship is maintained.

Direct Proportionality

Two quantities have a directly proportional relationship when, under the relevant conditions, increasing one quantity causes the other to increase proportionally, and decreasing one causes the other to decrease proportionally.

For example, if each notebook costs $20:

Number of Notebooks	Price
1	$20
2	$40
3	$60
4	$80

As the number of notebooks increases, the total price increases proportionally.

The constant of proportionality in this case is:

because each notebook costs $20.

Example of Direct Proportionality

If 3 notebooks cost $60, how much do 5 notebooks cost at the same unit price?

Set up the proportion:

3/5 = 60/x

Cross-multiply:

3x = 5 × 60

3x = 300

x = 100

Therefore, 5 notebooks cost:

$100

Inverse Proportionality

Two quantities have an inversely proportional relationship when, under the corresponding conditions, increasing one quantity causes the other to decrease in such a way that their product remains constant.

For example, imagine a task that requires a fixed total amount of work. If the number of people performing the task increases, and everyone works at the same efficiency, the amount of time required may decrease.

Suppose that 4 people complete a task in 12 days. If everyone works at the same rate and the conditions remain unchanged, we can calculate the time required for 6 people using:

4 × 12 = 6 × x

Then:

48 = 6x

x = 8

Therefore, under these conditions, 6 people would complete the task in 8 days.

Important: We should not decide that a situation is directly or inversely proportional simply because one quantity increases or decreases. We must first analyze the relationship between the quantities.

What Is the Rule of Three?

The rule of three is a procedure used to find an unknown value when three related values are known and the quantities are proportional.

In a direct proportionality situation, we can set up a proportion and use the fundamental property.

For example:

If 4 pencils cost $28, how much do 7 pencils cost?

4/7 = 28/x

Cross-multiply:

4x = 7 × 28

4x = 196

x = 49

Therefore:

7 pencils cost $49.

The important point is not to memorize a mechanical formula, but to correctly identify which quantities are related and what type of proportionality exists.

Ratio, Fraction, and Proportion: Are They the Same?

These concepts are related, but they are not synonyms.

Concept	What It Represents	Example
Ratio	Compares two quantities.	2:3
Fraction	May represent a part of a whole, a quotient, or a rational number.	2/3
Proportion	Establishes an equality between two ratios.	2/3 = 4/6

This distinction is important because a ratio can be written using a fraction, but its meaning depends on the context in which it is used.

For example, 2/3 can represent a ratio between two quantities or a part of a whole. The representation is the same, but the mathematical interpretation may be different.

Applications of Ratios and Proportions

Ratios and proportions are not limited to school exercises. They are used to model and solve situations in everyday life and in different fields of knowledge.

Recipes

If a recipe uses a specific relationship between ingredients, we can use proportions to increase or decrease the quantities while maintaining the same relationship.

Scales

Maps, plans, and models use proportional relationships to represent objects or distances at different sizes.

Prices and Quantities

When a product has a constant unit price, we can establish a proportional relationship between the quantity purchased and the total cost.

Percentages

Percentages can also be interpreted through proportional relationships. For example, 25% means 25 out of every 100:

25/100

This allows us to connect the study of ratios and proportions with percentage calculations.

Common Mistakes When Solving Ratios and Proportions

Confusing a Ratio with a Proportion

A ratio compares two quantities, while a proportion establishes an equality between two ratios.

Reversing the Terms Without Considering the Context

The ratio 2:3 does not mean the same thing as 3:2. Although they are related, they represent different comparisons.

Failing to Maintain the Same Order

If a ratio compares Mathematics students with English students, we must maintain the same order when constructing a proportion.

Applying the Rule of Three Without Analyzing the Problem

Not every problem should be solved using the same procedure. We must first identify which quantities are related and whether the relationship is direct or inverse proportionality.

Forgetting the Units

In applied problems, it is important to check the units. Correctly interpreting kilograms, liters, kilometers, hours, or quantities helps us understand the mathematical relationship correctly.

Ratios and Proportions Exercises

Exercise 1

Simplify the ratio 18:24.

The greatest common divisor of 18 and 24 is 6.

18 ÷ 6 = 3

24 ÷ 6 = 4

Answer:

18:24 = 3:4

Exercise 2

Determine whether 3/5 and 12/20 form a proportion.

Compare the cross-products:

3 × 20 = 60

5 × 12 = 60

Since both products are equal:

Yes, they form a proportion.

Exercise 3

Find the value of x:

5/8 = x/24

Cross-multiply:

5 × 24 = 8x

120 = 8x

x = 15

Answer: x = 15.

Exercise 4

If 5 books cost $150, how much do 8 books cost at the same price per book?

5/8 = 150/x

5x = 8 × 150

5x = 1200

x = 240

Answer: 8 books cost $240.

Exercise 5

If 3 people complete a task in 20 days, how many days would 5 people need under the same working conditions?

This is an inversely proportional relationship:

3 × 20 = 5 × x

60 = 5x

x = 12

Answer: 12 days.

Frequently Asked Questions About Ratios and Proportions

What Is a Ratio in Mathematics?

A ratio is a comparison between two quantities expressed through a quotient. It can be written, for example, as 2:3 or 2/3.

What Is a Proportion?

A proportion is an equality between two ratios. For example, 2/3 = 4/6.

What Is the Difference Between a Ratio and a Proportion?

A ratio compares two quantities, while a proportion establishes an equality between two ratios.

How Do You Simplify a Ratio?

Divide both terms by the same common divisor until you obtain an equivalent, simpler expression.

How Do You Solve a Proportion?

You can use the fundamental property of proportions: multiply the terms crosswise and then solve for the unknown value.

What Is Direct Proportionality?

It is a relationship in which two quantities change while maintaining a constant ratio. If one increases, the other also increases proportionally.

What Is Inverse Proportionality?

It is a relationship in which two quantities change in such a way that, under the corresponding conditions, their product remains constant.

Are Ratios and Fractions the Same?

Not necessarily. A ratio can be written as a fraction, but a fraction can have other meanings, such as representing a part of a whole or a rational number.

What Are Ratios and Proportions Used For?

They are used to compare quantities, solve proportionality problems, work with scales, prices, recipes, percentages, and many other situations.

Conclusion

Ratios and proportions allow us to establish mathematical relationships between quantities. Understanding a ratio means knowing how to compare two quantities and recognizing what each term represents. Understanding a proportion means identifying an equality between two ratios and using its properties to find unknown values.

From these concepts, we can study direct and inverse proportionality and solve problems using procedures such as the rule of three. However, solving a problem correctly does not simply mean cross-multiplying. We must first understand which quantities are being compared and what relationship exists between them.

Ratios and proportions also connect different areas of Arithmetic. After studying rational and irrational numbers, fractions, and decimal numbers, these concepts provide a foundation for understanding percentages, scales, proportionality, and other mathematical tools.

In summary:

A ratio compares two quantities.

A proportion establishes an equality between two ratios.

Ratios can be simplified and used to generate equivalent ratios.

Proportions allow us to find unknown values.

Proportionality can be direct or inverse, depending on the relationship between the quantities.

Mastering ratios and proportions means developing a mathematical tool that allows us to interpret relationships between quantities and solve problems logically, accurately, and with sound mathematical reasoning.

Continue Learning with Frecuencia Educativa

This topic is part of the episode:

T2E2 | Fundamentals of Arithmetic: Rational and Irrational Numbers, Fractions, Decimals, Ratios, and Proportions | Frecuencia Educativa

In this episode, you will find an integrated explanation of four fundamental topics in Arithmetic and the relationships between them.

This article explores the topic of Ratios and Proportions in greater depth.

🎙️ Listen to the full episode here:

▶️ T2E2 EPISODE LINK

Module Infographic

Also check out the infographic with a visual summary of the main concepts covered in this topic.

You will be able to review what a ratio is, what a proportion is, how to find unknown values, and how to identify direct and inverse proportional relationships.

🖼️ VIEW INFOGRAPHIC

Short Video

Would you prefer a quick review?

Watch this Short, where we explain what ratios and proportions are and how to solve a basic proportion.

▶️ WATCH SHORT

Academic Resource Information

Collection

Mathematics

Knowledge Center

Arithmetic

Knowledge Area

Ratios and Proportions

Topic

Ratios and Proportions

Level

Basic

Estimated Reading Time

12–15 minutes

Resource Type

Supplementary Article

Skills Developed

Understand the concept of a mathematical ratio.
Identify the antecedent and consequent of a ratio.
Simplify ratios and recognize equivalent ratios.
Understand the concept of a proportion.
Identify the means and extremes of a proportion.
Apply the fundamental property of proportions.
Solve proportions with unknown values.
Distinguish between direct and inverse proportionality.
Solve problems using proportional relationships.
Interpret applications of ratios and proportions in everyday situations.

Library Path

Digital Library → Mathematics Collection → Knowledge Center: Arithmetic → Knowledge Area: Ratios and Proportions