Ratio

A ratio compares two quantities of the same kind by division, showing how many times one value relates to or contains another. It can be written in three forms: as a fraction (10/5), using a colon (1:5), or in words (10 to 5). In the ratio formula a:b, ‘a’ is the antecedent and ‘b’ is the consequent. Ratios help us understand relationships between numbers and are often used when calculating ratio values or filling a ratio table to compare quantities easily.

Ratios can be classified into two types: part-to-part and part-to-whole. A part-to-part ratio compares two separate groups, such as the ratio of the number of boys to girls in a class (12:15). A part-to-whole ratio compares one part to the total, such as 5 out of 10 people liking books (5:10). These ideas are useful in real-life situations, especially when working with the ratio of two quantities, using a ratio calculator, or simply learning how to calculate ratios accurately.

Ratio and proportion are mathematical concepts used to compare quantities. Ratios show relationships, while proportions show equality between two ratios.

 

To calculate a ratio, follow these simple steps:

Step 1: Identify the two quantities

Example: 20 apples and 5 oranges.

Step 2: Write them in ratio form

20 : 5

Step 3: Simplify the ratio (divide both numbers by their GCD)
 

• GCD of 20 and 5 = 5
   
• Divide both:
   
  • 20 ÷ 5 = 4
     
  • 5 ÷ 5 = 1
     

Final Ratio: 4 : 1
 

Another Example:

How to calculate the ratio of 18 to 24:
 

1. Write: 18 : 24
    
2. Find GCD = 6
    
3. Divide:
    
   • 18 ÷ 6 = 3
      
   • 24 ÷ 6 = 4
      
   • Ratio = 3 : 4

Step 1: Write the ratio in numbers.

Example: 12 : 18

Step 2: Find the GCD (Greatest Common Divisor) of the two numbers.

GCD of 12 and 18 = 6

Step 3: Divide both numbers by the GCD.
 

• 12 ÷ 6 = 2
   
• 18 ÷ 6 = 3
   
• Simplified Ratio = 2 : 3
   

Another Example:
 

1. 45 : 60
    
2. GCD = 15
    
3. 45 ÷ 15 = 3
    
4. 60 ÷ 15 = 4
    
5. Simplified Ratio = 3 : 4

Ratios are classified into different types according to their function and purpose. They are used to comparing two or more values.

The main types of ratios are as follows:
 

Simple Ratio:
 

A simple ratio shows the comparison between two numbers. In this type of ratio, the values are expressed in their simplest form. It explains how many times one value contains another value. A simple ratio can be denoted as fractions (/) or with a colon (:)

Take a close look at this example,

Suppose, in a garden, there are 10 roses and 15 sunflowers. The ratio of roses and sunflowers are at 10:15.
 

To get the simplified ratio, we have to find the GCF of the given numbers.
 

For that, we need to list the factors of 10 and 15.
 

• The factors of 10 are 1, 2, 5, and 10.
   
• Factors of 15 are 1, 3, 5, and 15.
   

Here, 1 and 5 are the common factors of the given numbers. So, the greatest common factor of 10 and 15 is 5.
 

Next, we can divide the values by the GCF:
 

• 10 ÷ 5 = 2
   
• 15 ÷ 5 = 3
   

The ratio of roses and sunflowers is 2:3. It means that for every 2 roses, there are 3 sunflowers in the garden.
 

Compound Ratio:

It is a ratio formed by multiplying two or more ratios together. Here, the numerator is the product of the numerators of the original ratios and the denominator is the product of the denominators of the original ratios. For instance, a compound ratio is represented as: a:b and c:d.

When we multiply these ratios, we get a compound ratio. To get a better idea of compound ratio, look at this example:

Milan drinks 2 liters of water every 3 days and eats 4 apples every 5 months. She wants to calculate the ratio of her drinking and eating habits. How can she find the compound ratio?

To find the compound ratio, we have to multiply these two ratios.
 

• (2 : 3) × (4 : 5)
   
• (2 × 4) : (3 × 5) = 8 : 15
   

Hence, the compound ratio of (2 : 3) and (4 × 5) is 8:15.
 

Inverse Ratio:
 

An inverse ratio is also known as an indirect or reciprocal ratio. This ratio expresses the relationship of two quantities, in which one value increases and the other decreases equally.
 

For example:

Building a house takes 100 days, 10 days if 10 workers are employed, and 5 days if 20 workers are engaged.
 

Now we can measure the workers' ratio and days ratio.
 

Workers Ratio = 10:20
 

To simplify this, we have to divide the numbers by the GCF. 10 is the GCF of 10 and 20.
 

• 10 ÷ 10 = 1
   
• 20 ÷ 10 = 2
   

The simplified ratio is 1:2.
 

To find the day's ratio, we have to divide both numbers by their GCF.
 

10:5
 

Here, 5 is the GCF of 10 and 5.
 

• 10 ÷ 5 = 2
   
• 5 ÷ 5 = 1
   

The simplified ratio of days ratio is 2:1
 

Equivalent Ratio:

Equivalent ratios are two or more ratios that express the same relationship between quantities, even though the numbers may be different. They are obtained by multiplying or dividing both terms of a ratio by the same non-zero number, which keeps the comparison unchanged. Equivalent ratios, like equivalent fractions, represent a proportional relationship between two values.

Formulas:
 

1. To form an equivalent ratio (multiplication):
   
   \(a:b=(a × n):(b × n)\)
   
   Where 𝑛 is any non-zero number.
    
2. To form an equivalent ratio (division):
   
   \(a:b=(\frac{a}{n}):(\frac{b}{n}) \)
   
   Where 𝑛 is any common divisor of both numbers.

    

Examples:
 

1. Starting ratio: 2 : 3
   
   Multiply both terms by 2 → 4 : 6
    
2. Starting ratio: 5 : 10
   
   Divide both terms by 5 → 1 : 2

1. Use Real-Life Objects: Teach ratios using fruits, toys, crayons, or blocks. When students see and touch items, the concept becomes concrete.
    
2. Connect Ratios to Daily Life: Cooking, sharing snacks, and comparing heights/ages can all be used to demonstrate how ratios work in the real world.
    
3. Encourage Visual Learning: Draw bar models, pie charts, or ratio tables. Visuals help students understand part-to-part and part-to-whole relationships easily.
    
4. Introduce Equivalent Ratios With Simple Scaling: Show that multiplying or dividing both terms by the same number keeps the ratio constant. Use common examples such as recipes or classroom groups to practice.
    
5. Use Ratio Games & Challenges: Create fun tasks like “Make a 2:3 pattern using blocks” or “Find objects in the room with a 1:2 ratio.” Gamification increases interest.
    
6. Teach Students How to Simplify Ratios Like Fractions: Remind them that simplifying a ratio is the same as reducing a fraction, divide both numbers by their greatest common factor.

Ratios help us in various real-life situations; from cooking and purchasing to solving complex mathematical problems. In the areas of business, construction, physics, and architecture, this essential concept is applicable. 

• In the field of culinary industry, ratios are used to calculate the correct proportion of recipes and food items. 

• To analyze the balance between colors, artists and painters use this concept for their work.

• Companies use ratios to divide financial assets, to check the profit of a company, and to determine discounts. 

• Golden Ratio (1.618:1) is a specific ratio used by artists or sculptors while sculpting or drawing.