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', for example, '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. We find the LCM of the second terms and divide the LCM by each second term. The respective quotient is used to multiply both terms of each ratio. 

Step 2: Compare the first terms of the ratio

Now that both 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. To compare ratios using the LCM method, follow the steps given below:

1. If the second terms of the ratios are different, we find the LCM of the second terms.
    
2. Then we divide the LCM by the second term, and then multiply the ratio by the quotient.
    
3. Finally, we compare the first terms to determine which ratio is greater.

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, we first multiply the antecedent of the first ratio by the consequent of the second ratio, and vice versa. Then we compare the products to determine which ratio is greater.

• 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
 

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.

Comparison of ratios can be confusing for younger students. Here are some tips and tricks to easily remember the concept of ratios and to master it:

• To convert a fraction into decimals that has a power of 10 (10, 100, 1000,…) in the denominator. Count the number of zeroes. Place the decimal before the digits having place value equal to the number of zeroes.
   
• Always reduce the ratio to its simplex form for easy calculation.
   
• For comparing ratios, relate the fraction to items. For example, 1/2 is half of the pizza, and 1/4 is a quarter of a pizza. Now, checking the amount, you will 1/2 is greater.
   
• Use prime factorization methods for calculating LCM.
   
• To convert ratio to percentage, multiply the fraction by 100.

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 ratios to measure ingredients and adjust them according to the number of servings. For example, a recipe requires half a cup of milk, which is 1/2.

• We use ratios in art to create visually appealing compositions. For example, The Golden ratio (1.618:1) is known to be used in famous arts like Mona Lisa and The Last Supper. It is also found in nature. 

• To form new colors, we use ratios to achieve the desired tone. For example, mixing yellow and blue paint in 1:1, it will give a balanced green color.

• Ratios are used to compare prices and determine the best deal. For example, if a store has 20% which is 1/5 off the price of shoes and another have a 25% discount (1/4) on the same shoes, then it is profitable to buy from the second store. 

• We use ratios to convert measurements from one unit to another. Example: Converting 1cm into m = 1cm = 1/100 m