Mathematics is often called the language of logic, and one of the most powerful ideas within it is the concept of ratios. Ratios allow us to compare equivalent ratios examples quantities in an efficient and meaningful way. However, a single ratio can be expressed in multiple different forms that still represent the same relationship. These are called equivalent ratios.
Equivalent ratios are essential for solving problems in fractions, percentages, proportions, scaling, and everyday life situations such as cooking, budgeting, or measuring distances on a map. In this article, we will explore what equivalent ratios are, how to calculate them, and see a wide range of examples that make this concept easier to understand.
What is a Ratio?
A ratio is a way of comparing two or more numbers or quantities. For instance:
If there are 4 apples and 2 oranges in a basket, the ratio of apples to oranges is written as 4:2.
Ratios can be expressed using a colon (4:2), as fractions (4/2), or in words (“4 to 2”).
Ratios are not about exact amounts but about the relationship between the quantities.
What are Equivalent Ratios?
An equivalent ratio is one that represents the same proportional relationship as another ratio, even though the numbers may look different.
For example:
2:3, 4:6, 6:9, and 8:12 are all equivalent ratios
Each one simplifies back to the basic form 2:3.
This is similar to how fractions like 1/2 and 2/4 are equal even though they use different numbers.
How to Find Equivalent Ratios
There are two main methods to generate equivalent ratios:
1. Multiplying Both Terms by the Same Number
If you multiply both sides of a ratio by the same number, you create an equivalent ratio.
Example: Start with 3:5.
Multiply by 2 → 6:10
Multiply by 3 → 9:15
Multiply by 4 → 12:20
So, 3:5 = 6:10 = 9:15 = 12:20.
2. Dividing Both Terms by the Same Number
Dividing both numbers of a ratio by the same factor also gives an equivalent ratio.
Example: Take 20:30.
Divide by 10 → 2:3
Divide by 5 → 4:6
So, 20:30 = 4:6 = 2:3.
Equivalent Ratios vs. Equal Ratios
It’s important not to confuse the two:
Equivalent ratios express the same proportional relationship. Example: 2:3 and 4:6.
Equal ratios means the numbers themselves are identical. Example: 2:3 and 2:3.
Also remember, the order of numbers matters. 2:3 is not equivalent to 3:2.
Everyday Life Examples of Equivalent Ratios
Ratios are everywhere in daily life. Here are some practical examples:
1. Cooking and Recipes
A cake recipe requires 2 cups of sugar for every 3 cups of flour (ratio 2:3).
Doubling the recipe gives 4:6.
Tripling gives 6:9.
All of these are equivalent ratios that maintain the same sweetness balance.
2. Maps and Scale
If 1 cm on a map represents 100 km in real life, the ratio is 1:100.
2 cm represents 200 km (2:200).
5 cm represents 500 km (5:500).
All equivalent ratios show the same scale.
3. Speed and Distance
A car travels 60 km in 1 hour, giving a ratio of 60:1.
In 2 hours: 120:2
In 3 hours: 180:3
Each represents the same speed of 60 km/h.
4. Money and Savings
If you save $5 for every $10 you spend (5:10), it simplifies to 1:2.
Spending $20 while saving $10 → 10:20 = 1:2.
Spending $30 while saving $15 → 15:30 = 1:2.
The ratio always remains equivalent.
Step-by-Step Equivalent Ratio Examples
Let’s solve some examples step by step.
Example 1:
Find two equivalent ratios for 4:7.
Solution:
Multiply both numbers by 2 → 8:14
Multiply both numbers by 3 → 12:21
So, the equivalent ratios are 8:14 and 12:21.
Example 2:
Are 15:20 and 3:4 equivalent?
Solution:
Simplify 15:20 by dividing by 5 → 3:4.
Since they reduce to the same form, they are equivalent.
Example 3:
Complete this ratio: 5:9 = ? : 27
Solution:
9 × 3 = 27.
Multiply the first number (5) by 3 → 15.
So, the ratio is 15:27.
Example 4:
If 6 pencils cost $12, how much do 9 pencils cost?
Solution:
Ratio pencils:cost = 6:12 = 1:2.
For 9 pencils, multiply both by 9 → 9:18.
So, 9 pencils cost $18.
Using Ratio Tables
Tables help visualize equivalent ratios easily.
Example: Write a table for 2:3
MultiplierFirst TermSecond TermRatio1232:32464:63696:948128:125101510:15
This method is particularly useful for solving word problems involving proportions.
Word Problems on Equivalent Ratios
Problem 1:
A fruit basket has 3 apples for every 2 bananas. If there are 12 apples, how many bananas should there be?
Solution:
Original ratio = 3:2.
Apples = 12, which is 3 × 4.
Multiply bananas (2) × 4 = 8.
So, there are 8 bananas.
Problem 2:
A car consumes 5 liters of fuel to travel 40 km. How much fuel is needed for 100 km?
Solution:
Ratio fuel:distance = 5:40 = 1:8.
For 100 km → 100 ÷ 8 = 12.5 liters.
So, 12.5 liters are required.
Problem 3:
A teacher distributes pencils in a ratio of 2:5 (boys:girls). If 20 pencils are given to boys, how many go to girls?
Solution:
Ratio boys:girls = 2:5.
Boys = 20 → 2 × 10.
Multiply girls (5) × 10 = 50.
So, 50 pencils are given to girls.
Why Equivalent Ratios Matter
Equivalent ratios are critical because they:
Form the basis of proportional reasoning in mathematics.
Simplify real-world problem solving in cooking, construction, budgeting, and travel.
Help in scaling recipes, maps, and models.
Provide a foundation for understanding fractions, percentages, and algebra.
Common Mistakes to Avoid
Changing only one number — You must multiply or divide both terms by the same factor.
Example: 2:3 → 4:3 (❌ wrong).
Mixing up order — 2:3 is not equivalent to 3:2.
Always maintain the order.
Forgetting to simplify — Ratios should often be reduced to lowest terms for clarity.
Example: 10:20 simplifies to 1:2.
Conclusion
Equivalent ratios are one of the cornerstones of mathematics. They allow us to represent the same relationship in multiple forms, making it easier to compare, calculate, and apply proportional reasoning. By multiplying or dividing both terms of a ratio by the same number, we can generate new equivalent ratios that are useful in solving word problems and handling real-life situations.
