Comparison of Ratios

A ratio represents the proportion or relative amount of two or more quantities. It can be expressed in a colon or fractional notation, a/b or a:b. For example, 5/6 or 5:6. In word form, a ratio can be expressed as a to b, that is, 5 to 6.

Now, let’s learn how to compare ratios. Comparing ratios involves two main steps. Here we can learn the process in detail. 

Step 1: Make the second term of both ratios the same

To compare two ratios, first we need to make the second terms of both ratios the same. So, we find the LCM of the second terms of both ratios and then divide the LCM by each of the second terms. The respective quotient is used to multiply both terms of each ratio. 

Step 2: Compare the first terms of the ratio

Now that both the second terms of the ratios are the same, we simply compare the first terms. 

For example, let’s now compare 7:9 and 5:6

Step 1: Make the second terms the same

The second terms of the ratios are 9 and 6

The LCM of 9 and 6 is 18

Dividing 18 by 9 is 2, and dividing 18 by 6 is 3

Multiplying the first ratio, 7:9, by 2, we get 14:18

Multiplying the second ratio, 5:6, by 3, we get 15:18

 

Step 2: Compare the first terms.
 
The second terms in both ratios are the same (18), so we compare the first terms. As 15 is greater than 14, we conclude that 5:6 is greater than 7:9

To compare ratios, we have different methods. Commonly used methods are:

• Comparing Ratios Using the LCM Method
   
• Comparing Ratios by Cross-Multiplication 
   
• Comparing Ratios to Decimal Numbers
   
• Comparing Ratios to Percentages

Comparing ratios using the LCM method was discussed earlier. In this method, if the second terms of the ratios are different, we find the LCM of the second terms. Then we divide the LCM by the second term, and then multiply the ratio by the quotient. At last, we compare the first term to compare the ratios.

For example, comparing 5:8 and 4:6

The LCM of 8 and 6 is 24.

Dividing 24 by 8 is 3, and dividing 24 by 6 is 4

Multiplying the ratio 5:8 by 3, we get 15:24

Multiplying the ratio 4:6 by 4, we get 16:24

As the second numbers are the same, we compare the first numbers of the ratios. As 16 is greater than 15, we conclude that 4:6 is greater than 5:8

In the cross-multiplication method, always cross-multiply the ratios to compare them. In other words, we first multiply the antecedent of the first ratio by the consequent of the second ratio, and vice versa. Then we compare the product to compare the ratios.

• If the products of the cross-multiplication are the same, then the two ratios are the same. This means if ad = bc then a:b = c:d

• If ad < bc, then a:b is less than c:d

• If ad > bc, then a:b is greater than c:d

In this method, we convert each ratio to a decimal by dividing the first number by the second number. Then we compare the decimals.
 
For example, comparing 12:7 and 9:5

First, we convert the ratios to decimals, 

For 12:7, we divide 12 by 7; 12/7 = 1.714

For 9:5, we divide 9 by 5; 9/5 = 1.8

Here, 1.8 is greater than 1.714.

Therefore, 9:5 is greater than 12:7
 

In this method, we convert each ratio to a decimal by dividing the first number by the second number. Then we compare the decimals. 

For example, comparing 12:7 and 9:5

First, we convert the ratios to decimals, 

For 12:7, we divide 12 by 7; 12/7 = 1.714

For 9:5, we divide 9 by 5; 9/5 = 1.8

Here, 1.8 is greater than 1.714.

Therefore, 9:5 is greater than 12:7

When comparing the ratios to percentages, we first convert the ratios to percentages. To convert, we divide the first number by the second and multiply by 100. That is, a:b can be converted to a percentage as a/b ×100%.

For example, comparing 6:8 and 5:6

Converting 6:8 to percentage = 6/8 × 100 = 75%

Converting 5:6 to percentage = 5/6 × 100 = 83.333%

As 83.33% is greater than 75%, we can conclude that 5:6 is greater than 6:8.

In real life, we compare ratios in the fields of cooking, art, science, and finance. Let’s learn how we use comparing ratios in real life. 

• In cooking, we use comparing ratios to measure ingredients and adjust them according to the number of servings. 

• We use comparative ratios in the arts for creating visually appealing art forms.

• To form new colors by mixing colors, we use comparing ratios to achieve the required tone.

• Ratios are used to compare prices and determine the best deal.