Understanding Ratios and Proportions: Explained with Examples

Ratio and proportion are fundamental concepts in mathematics that help us understand the relationship between quantities. The concept of ratio dates back to ancient civilizations, where it was used for trade, construction, and measurement. The Egyptians, Babylonians, and Greeks contributed significantly to the development of ratio and proportion.

The understanding of ratios and proportions expanded, with contributions from mathematicians like Euclid and Archimedes. The development of algebraic notation further enhanced their application and problem-solving capabilities.

Today, ratio and proportion have general applications in diverse fields, such as geometry, statistics, finance, and everyday life. They are crucial for scaling, mixing ingredients, calculating percentages, analyzing data, and solving a wide range of mathematical problems.

In this article, we will discuss the basic definition of ratio and proportion, formulas, and their main difference.

Understanding Ratios and Proportions

Ratio and Proportion

The concept of ratio involves comparing two or more quantities, expressed as a fraction. It represents the relative sizes or values of the quantities being compared. Proportion extends the idea of ratio to compare multiple ratios and state their equality.

Ratio Proportion
  • In mathematics, a ratio is a comparison of two quantities or numbers, typically expressed as a fraction or with a colon (:). It represents the relative sizes or values of the quantities being compared.
  • Proportion is an extension of the concept of a ratio and establishes an equivalence between two or more ratios. It states that the fractions on either side of the proportion sign (=) are equal. In proportion, the multiple of the minimum is equal to the product of the maximum.
  • Example: If there are 4 red balls and 6 blue balls, the ratio of red to blue balls is 4:6 or 2:3.
  • Example: In the proportion 1/2 = 3/6, the ratio of 1 to 2 is equivalent to the ratio of 3 to 6.

The formula of ratio and proportion 

Ratio formula:

 The formula for a ratio is:

Ratio = a/b or a: b

In this formula, 

  • “a” and “b” represent the quantities being compared. 
  • The ratio is calculated by dividing the value of “a” by the value of “b”. 
  • The result is typically expressed as a fraction or using a colon (:).

Proportion formula:

The formula for proportion, which compares two or more ratios to establish equivalence is:

Proportion = a/b = c/d or a: b: c: d

In this formula, 

  • a, b, c, and d represent the quantities or values in the ratios being compared. 
  • The proportion is formed by equating the product of the means (a & d) to the product of the extremes (b & c).

Types of proportion 

Several types of proportions can be faced in mathematics. Here are the commonly discussed types:

  • Direct Proportion
  • Inverse Proportion

Direct Proportion:

Direct proportion also known as, direct variation, this type of proportion occurs when two quantities change in the same ratio. If one quantity rises, the other quantity also increases, and as one quantity decreases, the other quantity decreases. It can be represented as “y = kx,” where “y” and “x” are the two quantities and “k” is the constant of proportionality.

Inverse Proportion:

In inverse to direct proportion, inverse proportion occurs when two quantities change in opposite ratios. If one quantity grows, the other quantity decreases, and vice versa. It can be represented as “y = k/x” or “xy = k,” where “y” and “x” are the two quantities, and “k” is the constant of proportionality.

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How to Calculate Ratio and Proportion?

Below are a few examples to understand how to find ratios and proportions. 

Example 1: For Ratio

In a wooden box, there are 23 fruits. Of these 23 fruits, 7 are mango, 5 are apple, 3 are orange, and 8 are banana. Evaluate the ratio of:

  • Mango to apple
  • Orange to total fruits
  • Banana to apple
  • Orange to apple 

Solution:

Step 1:

First, write the number of fruits.

Total fruits = 23

Mango = 7

Apple = 5

Orange = 3

Banana = 8

Step 2:

Find the ratio of mango to apple.

Number of mango = 7

Number of apple = 5

The ratio is

7: 5 

Or

7/5

Step 3:

Ratio of Oranges with total fruits

Number of total fruits = 23

Number of orange = 3

Subtract the total fruits from orange to find the difference = 23 – 3 = 20

Now, the ratio is

3: 20 

or

3/20

Step 4:

Evaluate the ratio of banana to apple.

Number of banana = 8

Number of apple = 5

The ratio is

8: 5

or

⇒ 8/5

Step 5:

Find the ratio of orange to apple.

Number of orange = 3

Number of apple = 5

The ratio is

3: 5

Or

⇒ 3/5

Example 2: For Proportion

If 15 buses are required to bring the workers of a factory in two weeks, how many buses are required to bring an equal number of workers in eight days?

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Solution:

Step 1: 

Suppose that we write the given data of buses and days.

So, buses required to bring the workers of a factory in two weeks = 15

Buses required to bring the workers of a factory in eight days = x

Step 2: 

We write the formula of proportion and write the data of buses and days allowing the formula.

a: b :: c : d

Buses: days:: buses: days

15: 14: : x : 8

Step 3: 

We write the ratios in the form of fractions and put an equality sign between them.

15/14 = x/8

Step 4: 

Calculate the above expression to find the value of x.

15/14 = x/8

(15/14) * 8 = x

(15/7) * 4 = x

15*4/7 = x

x = 15*4/7

x = 60/7 = 8.57 ≈ 9

x = 9

Therefore, 9 more buses are required to bring the same number of workers. Buses required bringing the workers of a factory in eight days = 15 + 9 = 24.

You can also explore online tools to solve proportions to find the missing value “x” to ease up the calculations. 

Conclusion

In this article, we have discussed the basic definition of ratio and proportion, the formulas of ratio and proportion, the type of proportion, and the main difference between ratio and proportion in detail. Also with the help of an illustration topic will be clarified. After completing this article anyone can simply solve this topic.

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Frequently Asked Questions

Question 1:

What is the ratio? 

Solution:

The ratio is a comparison of two quantities or numbers, typically expressed as a fraction or with a colon (:). It represents the relative sizes or values of the quantities being compared.

Question 2:

What is proportion?

Solution:

Proportion is an extension of the concept of a ratio and establishes an equivalence between two or more ratios. It states that the fractions on either side of the proportion sign (=) are equal

Question 3:

The basic concept of ratio and proportion

Solution:

Ratio and proportion are fundamental concepts in mathematics that help us understand the relationship between quantities. In ratio and proportion, you can find an unknown value by setting up equivalent fractions or using cross-multiplication to solve for the missing value.

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