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Ratio, in math, is a term that is used to compare two or more numbers. It is used to indicate how big or small a quantity is when compared to another. In a ratio, two quantities are compared using division. Here the dividend is called the 'antecedent' and the divisor is called the 'consequent'. For example, in a group of 30 people, 17 of them prefer to walk in the morning and 13 of them prefer to cycle. To represent this information as a ratio, we write it as 17: 13. Here, the symbol ': ' is read as "is to". So, the ratio of people who prefer walking to the people who prefer cycling is read as '17 is to 13'.
|1.||What is Ratio?|
|2.||Calculation of Ratios|
|3.||How to Simplify Ratios?|
|5.||FAQs on Ratio|
What is Ratio?
The ratio is defined as the comparison of two quantities of the same units that indicates how much of one quantity is present in the other quantity. Ratios can be classified into two types. One is part to part ratio and the other is part to whole ratio. The part-to-part ratio denotes how two distinct entities or groups are related. For example, the ratio of boys to girls in a class is 12: 15, whereas, the part-to-whole ratio denotes the relationship between a specific group to a whole. For example, out of every 10 people, 5 of them like to read books. Therefore, the part to the whole ratio is 5: 10, which means every 5 people from 10 people like to read books.
We use the ratio formula while comparing the relationship between two numbers or quantities. The general form of representing a ratio of between two quantities say 'a' and 'b' is a: b, which is read as 'a is to b'.
The fraction form that represents this ratio is a/b. To further simplify a ratio, we follow the same procedure that we use for simplifying a fraction. a:b = a/b. Let us understand this with an example.
Example: In a class of 50 students, 23 are girls and the remaining are boys. Find the ratio of the number of boys to the number of girls.
Total number of students = 50; Number of girls = 23.
Total number of boys = Total number of students - Total number of girls
= 50 - 23
Therefore, the desired ratio is, (Number of boys: Number of girls), which is 27:23.
Calculation of Ratios
In order to calculate the ratio of two quantities, we can use the following steps. Let us understand this with an example. For example, if 15 cups of flour and 20 cups of sugar are needed to make fluffy pancakes, let us calculate the ratio of flour and sugar used in the recipe.
- Step 1: Find the quantities of both the scenarios for which we are determining the ratio. In this case, it is 15 and 20.
- Step 2: Write it in the fraction form a/b. So, we write it as 15/20.
- Step 3: Simplify the fraction further, if possible. The simplified fraction will give the final ratio. Here, 15/20 can be simplified to 3/4.
- Step 4: Therefore, the ratio of flour to sugar can be expressed as 3: 4.
Use Cuemath's free online ratios calculator to verify your answers while calculating ratios.
How to Simplify Ratios?
A ratio expresses how much of one quantity is required as compared to another quantity. The two terms in the ratio can be simplified and expressed in their lowest form. Ratios when expressed in their lowest terms are easy to understand and can be simplified in the same way as we simplify fractions. To simplify a ratio, we use the following steps. Let us understand this with an example. For example, let us simplify the ratio 18:10.
- Step 1: Write the given ratio a:b in the form of a fraction a/b. On writing the ratio in the fraction form, we get 18/10.
- Step 2: Find the greatest common factor of 'a' and 'b'. In this case, the GCF of 10 and 18 is 2.
- Step 3: Divide the numerator and denominator of the fraction with the GCF to obtain the simplified fraction. Here, by dividing the numerator and denominator by 2, we get, (18÷2)/(10÷2) = 9/5.
- Step 4: Represent this fraction in the ratio form to get the result. Therefore, the simplified ratio is 9:5.
Use Cuemath's free online simplifying ratios calculator to verify your answers.
Tips and Tricks on Ratio:
- In case both the numbers 'a' and 'b' are equal in the ratio a: b, then a: b = 1.
- If a > b in the ratio a : b, then a : b > 1.
- If a < b in the ratio a : b, then a : b < 1.
- It is to be ensured that the units of the two quantities are similar before comparing them.
Equivalent ratios are similar to equivalent fractions. If the antecedent (the first term) and the consequent (the second term) of a given ratio are multiplied or divided by the same number other than zero, it gives an equivalent ratio. For example, when the antecedent and the consequent of the ratio 1:3 are multiplied by 3, we get, (1 × 3) : (3 × 3) or 3: 9. Here, 1:3 and 3:9 are equivalent ratios. Similarly, when both the terms of the ratio 20:10 are divided by 10, it gives 2:1. Here, 20:10 and 2:1 are equivalent ratios. An infinite number of equivalent ratios of any given ratio can be found by multiplying the antecedent and the consequent by a positive integer.
A ratio table is a list containing the equivalent ratios of any given ratio in a structured manner. The following ratio table gives the relation between the ratio 1:4 and four of its equivalent ratios. The equivalent ratios are related to each other by the multiplication of a number. Equivalent ratios are obtained by multiplying or dividing the two terms of a ratio by the same number. In the example shown in the figure, let us take the ratio 1:4 and find four equivalent ratios, by multiplying both the terms of the ratio by 2, 3, 6, and 9. As a result, we get 2:8, 3:12, 6:24, and 9:36.
Use Cuemath's free online equivalent ratios calculator to verify your answers.
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Example 1: There are 49 boys and 28 girls in a school auditorium. Express the ratio of the number of boys to that of girls.
Given, the number of boys = 49; and the number of girls = 28. The GCF of 49 and 28 is 7. Now, to simplify, divide the two terms by their GCF which is 7. This means, (49 ÷ 7)/(28 ÷ 7) = 7/4. Therefore, the ratio of the number of boys to that of girls = 7:4.
Example 2: A music class has 30 students. 10 of them were adults and the rest were children. What is the ratio of the number of children to the total number of students in the music class?
Given, the total number of students in the music class = 30 and the total number of adults = 10. Therefore, the number of children who attended the music class = 30 -10, which is equal to 20. The ratio of the total number of children to the total number of students in the music class = 20: 30, which when simplified gives 2:3.
Example 3: Simplify the given ratio, 87:75.
In order to simplify the given ratio, we will first find the GCF of 87 and 75, which is 3. Then, we will divide each term by 3. This means, (87 ÷ 3)/(75 ÷ 3) = 29/25. Thus, the ratio 87:75 in the simplest form is 29:25.
Practice Questions on Ratio
FAQs on Ratio
What is Ratio in Math?
A ratio can be defined as the relationship or comparison between two numbers of the same unit to check how bigger is one number than the other one. For example, if the number of marks scored in a test is 7 out of 10, then the ratio of marks obtained to the total number of marks is written as 7:10.
What are the Ways of Writing a Ratio?
A ratio can be written by separating the two quantities using a colon (:) or it can be written in the fractional form. For example, if there are 4 apples and 8 melons, then the ratio of apples to melons can be written as 4:8 or 4/8, which can be further simplified as 1:2.
How to Calculate the Ratio Between Two Numbers?
In order to calculate the ratio of two quantities, we can use the following steps. Let us understand this with an example. For example, if 14 cups of butter and 28 cups of sugar are needed to make a frosting cream, what is the ratio of butter and sugar?
- Step 1: Note the quantities of both the ingredients for which we are determining the ratio. In this case, it is 14 and 28.
- Step 2: Write it in the fraction form a/b. So, we write it as 14/28.
- Step 3: Simplify the fraction further, if possible. The simplified fraction will give the final ratio. Here, 14/28 can be simplified to 1/2.
- Step 4: Therefore, the ratio of butter to sugar can be expressed as 1: 2.
How to Find Equivalent Ratios?
Two ratios are said to be equivalent if they represent the same value when simplified. This concept is similar to equivalent fractions. For example, when the ratio 1: 4 is multiplied by 2, it means multiplying both the terms in the ratio by 2. So, we get, (1 × 2)/ (4 × 2) = 2/8 or 2: 8. Here, 1:4 and 2:8 are equivalent ratios. Similarly, the ratio 30: 10, when divided by 10, gives the ratio as 3:1. Here, 30:10 and 3:1 are equivalent ratios. So, equivalent ratios can be found by using the multiplication or division operation depending on the numbers.
What is a Ratio Table?
A ratio table shows a list of equivalent ratios that are obtained either by multiplying or dividing both the quantities by the same value. For example, if the ratio table starts with the ratio 1 : 3, then the successive rows will have 2:6, 3:9, 4:12, and so on. When these ratios are simplified, they represent the same value, that is, 1: 3.
What is the Golden Ratio?
A golden ratio is a distinct number whose value is approximately equal to 1.618. The symbol for this is a Greek letter 'phi' represented as ϕ. It is a special attribute and is used in art, geometry, and architecture because it is believed that the golden ratio makes the most pleasing and beautiful shape. It is also known as a divine proportion that exists between two quantities and the relationship for calculating the golden ratio is represented as ϕ = a/b = (a + b)/a = 1.61803398875... where a and b are the dimensions of two quantities and a is the larger between the two.
Why are Ratios Important?
Ratios are important because they allow us to express quantities in such a way that they are easier to interpret. It is a tool that is used to compare the size of two or more quantities with respect to each other. For example, if there are 30 girls and 20 boys in a class. We can represent the number of girls to the number of boys with the help of the ratio which is 3: 2 in this case.
What is the Ratio Formula?
The ratio formula is used to compare the relationship between two numbers or quantities. The general form of representing a ratio of between two quantities say 'a' and 'b' is a: b, which is read as 'a is to b'.
What is Ratio and Proportion?
The ratio is the relationship or comparison between two quantities of the same unit to check how bigger is one number than the other one. It is written as a/b or a: b where b is not equal to zero. A proportion is an equality of two ratios. Proportions are used to write equivalent ratios which helps to solve the unknown quantities. For example, a proportion is expressed as, a: b = c: d
How to Compare Ratios?
There are various methods to compare ratios. For example, let us compare 1: 2 and 2: 3 using the LCM method.
- Step 1: Write the ratios in the form of a fraction. Here, it means 1/2 and 2/3.
- Step 2: Reduce the fractions separately. Here, both the fractions 1/2 and 2/3 are already in their reduced form.
- Step 3: Now, compare 1/2 and 2/3 by finding the LCM (Least Common Multiple) of the denominators. The LCM of 2 and 3 is 6.
- Step 4: Make the denominators equal by multiplying the numerator and denominator of the first fraction by 3, that is, (1 × 3)/(2 × 3) = 3/6. Then, multiply the numerator and denominator of the second fraction by 2, that is, (2 × 2)/(3 × 2) = 4/6.
- Step 5: Now, 3/6 and 4/6 can be easily compared. This shows that 4/6 is greater than 3/6. Therefore, 2:3 > 1:2.
How to Convert Ratios to Fractions?
Ratios can be written in the form of fractions in a very simple way. The antecedent is written as the numerator and the consequent is written as the denominator. For example, if we take the ratio 3: 5. Here, 3 is the antecedent, and 5 is the consequent. So, we can write it as 3/5.
How to Convert Fractions to Ratios?
Fractions can be written in the form of ratios after simplification. This means, we first reduce the given fraction to its lowest terms and then write the numerator as the antecedent and the denominator as the consequent. For example, the fraction 16/48 will first be reduced to 1/3 and then it can be expressed in the form of a ratio as 1: 3.
How to Convert Ratios to Decimals?
Ratios can be easily converted to decimals by writing the ratio in the form of a fraction, and then the fraction is converted to a decimal by dividing the numerator by the denominator. For example, 3:7 can be written as 3/7. Now, 3/7 = 0.428.
How to Convert Ratios to Percentages?
Ratios can be converted to percentages using the following steps. For example, let us convert 5: 6 in the form of a percentage.
- Step 1: Write the ratio in the form of a fraction. Here, 5: 6 can be written as 5/6.
- Step 2: Multiply this fraction by 100 and add the percentage symbol. In this case, 5/6 × 100 = 83.33%.
Check this article on 'ratio to percent' to learn more.