Ratio Comparisons

Ratio comparisons help us understand how two or more quantities relate to each other. By comparing ratios, we can determine whether one ratio is greater, smaller, or equal to another. This concept is widely used in everyday life, from calculating recipe ingredients to analyzing financial performance.

A ratio is essentially a division between two numbers that represents their relative sizes. For example, if a recipe calls for a ratio of 2:3 for sugar to flour, it means for every 2 parts of sugar, 3 parts of flour are needed. To compare two ratios, such as 2:5 and 3:4, we can use methods like cross-multiplication or finding the least common multiple (LCM). These methods help determine which ratio is larger or if they are equivalent.

In this article, we will learn about Ratio Comparisons in great detail. We will try to understand the concept and solve examples related to ratio comparisons for a better understanding and strengthening of our knowledge of this topic.

What are Ratios?

Ratios are a way to compare two or more quantities that describe the relative size or amount of each.

A ratio shows how many times one quantity is contained in another. It can be expressed in several forms, such as:

• Colon notation: a:b, where a and b represent two quantities.
  • For example, if there are 3 apples for every 5 oranges, the ratio of apples to oranges is 3:5.
• Fractional notation: a/b​, which expresses the ratio as a fraction.
  • For instance, the ratio of apples to oranges can also be written as 3/5.
• Word form: Ratios can be expressed as "a to b".
  • In our example, the ratio of apples to oranges would be read as "3 to 5."

How to Compare Ratios?

Comparing ratios involves two main steps:

Step 1: Make the second numbers (consequents) in both ratios the same.

• First, we find the least common multiple (LCM) of the second numbers in the ratios.
• Divide this LCM by each of the second numbers to get a quotient.
• Multiply both the second and first numbers (antecedents) of each ratio by their respective quotients.

Step 2: Compare the first numbers (antecedents) of the adjusted ratios.

• After adjusting the ratios in Step 1, compare the first numbers of each ratio to determine which is greater.

Let's consider an example for better understanding:

Example: Compare the ratios 2:6 and 5:4.

Solution:

Comparing the ratios: 2:6 and 5:4.

1. Find the LCM of 6 and 4, which is 12.
2. Divide 12 by 6 to get 2, and 12 by 4 to get 3.
3. Adjust the ratios:
   • For 2:6, multiply both numbers by 2 to get 4:12.
   • For 5:4, multiply both numbers by 3 to get 15:12.
4. Compare the first numbers (antecedents): 15 (from 5:4) is greater than 4 (from 2:6).

So, 5:4 is greater than 2:6.

Converting Ratios to Decimal Numbers

We can compare ratios by converting them into decimal numbers. To covert ratio into decimal we can se the following steps:

1. Divide the numerator by the denominator.
2. Find the decimal numbers.
3. Round the decimals if required and compare.

Let's consider an example.

Example: Compare 8 : 5 and 3 : 2.

Solution:

• 8 : 5  = 8/5 =  1.6
• 3 : 2  = 3/2 =  1.5

Since 1.5 < 1.6, we can say that 3 : 2 < 8 : 5. 

Solved Examples on Ratio Comparisons

Example 1: Compare the given ratios and find which of the following is greater: 12:16 or 18:20?

Solution:

Let us find the LCM of the consequents of both the ratios. LCM of 16 and 20 is 80. Divide the LCM with the consequents, 80 ÷ 16 = 5 and 80 ÷ 20 = 4. Multiply the answers with the ratios. 

(12 × 5):(16 × 5) = 60 and 80 

(18 × 4):(20 × 4) = 72 and 80 

Since 72 > 60, the ratio 18:20 is greater than 12:16.

Example 2: Use the cross multiplication method for comparison of ratios and find which ratio is greater? 5:18 or 9:25

Solution:

Given ratios are 5:18 and 9:25. It can be written as 5/18 or 9/25. By using the cross multiplication method, we get 5 × 25 and 9 × 18 = 125 and 162 

Since 162 is greater than 125. Therefore, 9:25 is greater.

Example 3: Compare the ratios 3:4 and 5:6.

Solution:

Using LCM Method:

1. Find the LCM of 4 and 6, which is 12.
2. Divide 12 by 4 to get 3 and 12 by 6 to get 2.
3. Adjust the ratios:
   • For 3:4: Multiply by 3 → 9:12.
   • For 5:6: Multiply by 2 → 10:12.
4. Compare the first numbers: 10 > 9.

So, 5:6 is greater than 3:4.

Example 4: Compare the ratios 7:10 and 8:12.

Solution:

Using Cross Multiplication Method:

1. Multiply 7 by 12 and 8 by 10:
   • 7 × 12 = 84
   • 8 × 10 = 80
2. Compare the results: 84 > 80.

So, 7:10 is greater than 8:12.

Example 5: Compare the ratios 2:3 and 4:5.

Solution:

Using LCM Method:

1. Find the LCM of 3 and 5, which is 15.
2. Divide 15 by 3 to get 5 and 15 by 5 to get 3.
3. Adjust the ratios:
   • For 2:3: Multiply by 5 → 10:15.
   • For 4:5: Multiply by 3 → 12:15.
4. Compare the first numbers: 12 > 10.

So, 4:5 is greater than 2:3.

Practice Problems on Ratio Comparisons

Problem 1: How do the ratios 5:8 and 7:10 compare?

Problem 2: How do the ratios 9:12 and 15:20 compare?

Problem 3: Compare the ratios 6:7 and 4:5.

Problem 4: How does the ratio 11:15 compare to 8:10?

Problem 5: Compare the ratios 3:5 and 7:12.

Problem 6: How do the ratios 14:18 and 16:20 compare?

Problem 7: Compare the ratios 10:12 and 15:18.

Problem 8: How do the ratios 2:9 and 3:13 compare?

Answer Key

1. 5:8 is greater than 7:10.
2. They are equal (both simplify to 3:4).
3. 6:7 is greater than 4:5.
4. 8:10 is greater than 11:15.
5. 3:5 is greater than 7:12.
6. They are equal (both simplify to 7:9).
7. They are equal (both simplify to 5:6)
8. 2:9 is greater than 3:13.

Read More,

• What are Fractions?
• Comparing Fractions
• Comparing and Ordering Decimals Practice Questions
• Equivalent Fractions
• Comparing Numbers
• Simplifying Fractions Worksheet