Ratios, in general, are used to compare any two quantities. It is used to show how larger or smaller a quantity is relative to another quantity. These ratios in mathematics are represented using notations like; p:q, a:b, x:y, etc.

For example, if a statement says in an institution out of every 14 individuals, 7 of them like to play any type of sports. Thus, the ratio of individuals who like to play any type of sports to the total number of individuals is 7: 14. This further implies that 7 individuals from every 14 like to play a sport in that particular institute.

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In this article, you will learn how to do the comparison of two ratios with the steps and methods for comparing ratios using LCM and Cross Multiplication which is similar to Equivalent Ratios. We will also practice some questions to understand the topic in a much better way.

Comparison of Ratios

Terms like ratio, proportion, percentage and fraction are frequently used in mathematics. Whenever there is a situation when we have to compare two numbers or elements, to see how larger or smaller a quantity is with respect to another we use ratios. The ratio of two quantities is obtained by dividing one quantity by the other quantity; \(\frac{A}{B}\) and is represented as A:B.

In the comparison of ratios, say A:B and P:Q we compare the two fractions\(\frac{A}{B}\text{and}\ \frac{P}{Q}\). In comparing the two given ratios we may find out the faster rate of variation or higher magnitude of a quantity per unit other quantity. We often need to compare two ratios like the rate of change of position with time, i.e speed of two bodies. Here the comparison is done between the speed of two bodies to estimate the faster-moving body or the slower-moving body. Similarly, a packet of chocolates when distributed between two groups of people. The comparison of the two ratios gives the ratio of the distribution of the chocolates per person.

There are many such mathematical as well as real-life situations when we come across a comparison of two to more ratios to reach a certain conclusion. To compare such two ratios or fractions we use the LCM or cross multiplication approach. We will learn these methods in the coming headers themselves.

Check out this article on Ratio to Percentage conversion.

Learn about Ratios and Proportions in the video below!

Ratios as known to us are employed to compare two numbers or quantities. The digits used in the building of the ratio are named as the terms of the ratio. To compare the two given ratios follow the below steps.

Step 1: Take the two ratios to be compared.

Step 2: Simplify each of the ratios in the simplest form.

Step 3: Now either apply the least common multiple approaches and make the denominators of the ratios equal and compare the numerator to find the bigger ratio. Or use the cross multiplication method for the same.

Check out this article on the different Operations on functions here.

Methods Used to Compare Ratios

In the previous headers, we read about the steps to compare two ratios and introduce the methods to find the same. Let us understand those methods in detail now.

•  LCM Method
• Cross Multiplication Method

LCM Method of Comparing Ratios

The steps to compare two ratios using the LCM method are as follows:

Step 1: Fetch the given ratios in the simplest form possible. For example, consider we are asked to compare the ratios 4: 5 and 1: 7.

Step 2: Start with taking the LCM of the denominators i.e., 5 and 7 which is equal to 35.

Step 3: Next, divide LCM by the denominator of each term.

For the ratio, 4: 5, the denominator is 5, thus, \(\frac{35}{5}=7\) and for the ratio 1: 7, 7 is the denominator thus, \(\frac{35}{7}=5\).

Step 4: Express the ratios in the form of fractions and multiply both the numerator and denominator of the ratio 4:5 by 7 we get; \(\frac{4\times7}{5\times7}=\frac{28}{35}\). In a similar way multiply both the numerator and denominator of the ratio 1: 7 by 5 we get; \(\frac{1\times5}{7\times5}=\frac{5}{35}\).

Step 5: In the modified fraction of the ratios we can see that both the terms in the denominators are the same. Thus compare the numerators and for the fraction with a larger numerator will be bigger than the other one.

Step 6: On comparing 28 and 5 we can say that 28 >5, thus \(\frac{28}{35}>\frac{5}{35}\), that is \(\frac{4}{5}>\frac{1}{7}\).

Learn more about the Addition and Subtraction of Polynomials here.

Comparing Ratios By Cross Multiplication Method

The approach of cross multiply is also used for comparing ratios and finding the bigger one. The steps involved in the method are:

Step 1: Bring the given ratios in the simplest form possible.

Step 2: Next multiply the numerator of the first ratio by the denominator of the second ratio. Similarly, multiply the denominator of the first ratio by the numerator of the second ratio.

Step 3: The result obtained in the above step can be compared using these three simple points.

If we are given two ratios \(\frac{a}{b\ }\text{ and }\frac{c}{d}\), then after cross multiplication;

If ad>bc then \(\frac{a}{b\ }>\frac{c}{d}\)

If ad<bc then \(\frac{a}{b\ }<\frac{c}{d}\)

If ad=bc then \(\frac{a}{b\ }=\frac{c}{d}\)

Read more about the different Operations on Real Numbers here.

Converting Ratios to Decimal Numbers 

Ratios are a way to compare two quantities by division. To analyze or compare different ratios more precisely, one common method is to convert them into decimal form. This method makes it easier to compare numerically, particularly where the ratios contain dissimilar terms.

Steps:

Divide the first number by the second.

Express the result in decimal form.

Compare the decimal forms of each ratio and decide which is larger.

Example: Compare 8:5 and 3:2

Step 1: Express both ratios in decimal form

8÷5=1.6

3÷2=1.5

Step 2: Compare the decimals

1.6>1.5

Conclusion:

Since 1.6 is greater than 1.5, we can say that:
8:5 is greater than 3:2

Applications of Comparison of Ratios

Comparison of ratios is used in many real-life situations where we need to understand the relationship between two quantities. This method helps us decide which value is larger, smaller, or if they are equal in proportion.

1. Business and Finance

• Budgeting: Companies compare income and expenses using ratios to manage profits.

• Investment Analysis: Investors compare the return ratios of different assets before investing.

• Pricing: Cost and selling price ratios help determine profit margins.

2. Education and Exams

• Marks Comparison: Students and teachers compare test scores using ratios to evaluate performance.

• Admission Criteria: Some exams use ratio-based scoring systems for seat allotment or ranking.

3. Daily Life

• Cooking Recipes: Ingredients are often mixed in specific ratios, and comparing them helps maintain taste and quantity.
   

• Fuel Efficiency: Comparing distance-to-fuel ratios helps evaluate which vehicle is more efficient.

4. Construction and Architecture

• Mixing Materials: The correct ratio of cement, sand, and gravel is crucial for strong construction.

• Scaling Designs: Architects use ratio comparisons to create scale models of buildings.

5. Science and Engineering

• Chemical Mixtures: Ratios are compared in chemical formulas to ensure correct reactions.

• Speed and Distance: In physics, comparing time-to-distance or speed-to-time ratios helps in calculations.

Solved Examples 

Throughout the article, we learnt how to do the comparison of two objects in ratio, along with the methods and steps. Let us go through some solved examples for more practice on the topic.

Solved Example 1: Compare the ratios 2: 5 and 3: 4, and obtain the bigger one using the LCM method.

Solution: The given ratios are 2: 5 and 3: 4.

First express the ratios in their respective fraction form;\(2:5=\frac{2}{5}\text{ and }3:4=\frac{3}{4}\).

Check if the denominator of both the factors is the same, if not then take the LCM of the denominators.

LCM of 5 and 4=20

Next divide the LCM by each fraction denominator, and we get\(\frac{20}{5}=4\), and \(\frac{20}{4}\)=5.

Multiplying the term 4 to both numerator and denominator of \(\frac{2}{5}\) and 5 to both numerator and denominator of \(\frac{3}{4}\) we get the fractions; \(\frac{2\times4}{5\times4}=\frac{8}{20}\) and \(\frac{3\times5}{4\times5}=\frac{15}{20}\).

As 15 is greater than 8, \(\frac{15}{20}>\frac{8}{20}\) and so is \(\frac{3}{4}>\frac{2}{5}\).

Check out this article on Comparison of Quantity.

Solved Example 2: Compare the given ratios 2: 9 and 4: 7 and obtain the bigger one using the cross multiplication method.

Solution: Given ratios are 2: 9 and 4: 7.

Express the ratios as fractions we get; \(2:9=\frac{2}{9}\text{ and }4:7=\frac{4}{7}\)

Next multiply the numerator element of the first ratio by the denominator of the element of the second ratio and vice versa for the second ratio.

\(\frac{2}{9}\times\frac{4}{7}\)

Comparing with the formula if ad<bc then \(\frac{a}{b\ }<\frac{c}{d}\) \(\frac{a}{b\ }<\frac{c}{d}\).

Here as 14<36, \(\frac{2}{9}<\frac{4}{7}\).

Solved Example 3: Compare the ratios 4: 9 and 2: 9 and check which one is greater.

Solution: Given ratios are 4: 9 and 2: 9.

Expressing the ratios in fractions we get; \(4:9=\frac{4}{9}\text{ and }2:9=\frac{2}{9}\)

In both the factions we can check that the denominators are the same. Thus no need to make the necessary edits. We can directly compare the numerators to get the bigger one from the two factions.

As 4>2, \(\frac{4}{9}>\frac{2}{9}\).

We hope that the above article is helpful for your understanding and exam preparations. Stay tuned to the Testbook App for more updates on related topics from Mathematics, and various such subjects. Also, reach out to the test series available to examine your knowledge regarding several exams. For better practice, solve the below provided previous year papers and mock tests for each of the given entrance exam: