Which Ratio In The Form $A: B$ Is Equivalent To The Ratios In The Table?${ \begin{tabular}{|c|c|c|c|c|} \hline A & 2 & 3 & 4 & 5 \ \hline B & 8 & 12 & 16 & 20 \ \hline \end{tabular} }$A. $1: 4$ B. $6:

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Which Ratio in the Form A: B is Equivalent to the Ratios in the Table?

In mathematics, ratios are used to compare the size of two quantities. A ratio is a way of expressing the relationship between two numbers, often in the form of A:B. In this article, we will explore the concept of equivalent ratios and how to determine which ratio in the form A:B is equivalent to the ratios in the given table.

Understanding Equivalent Ratios

Equivalent ratios are ratios that have the same value, even if they are expressed differently. For example, the ratios 2:4 and 3:6 are equivalent because they both represent the same relationship between two quantities. To determine if two ratios are equivalent, we can use the following method:

  • Multiply or divide both numbers in the first ratio by the same number.
  • Check if the resulting ratio is the same as the second ratio.

The Table of Ratios

The table below shows four ratios in the form A:B.

A B
2 8
3 12
4 16
5 20

Determining Equivalent Ratios

To determine which ratio in the form A:B is equivalent to the ratios in the table, we need to find a common factor that can be multiplied or divided by both numbers in each ratio. Let's examine each ratio in the table and find its equivalent ratio.

Ratio 1: 2:8

To find the equivalent ratio of 2:8, we can multiply both numbers by 1, 2, 3, 4, or 5. However, we are looking for a ratio in the form A:B, where A and B are integers. Therefore, we can multiply both numbers by 2 to get 4:16, which is equivalent to the ratio 4:16 in the table.

Ratio 2: 3:12

To find the equivalent ratio of 3:12, we can multiply both numbers by 1, 2, 3, 4, or 5. However, we are looking for a ratio in the form A:B, where A and B are integers. Therefore, we can multiply both numbers by 4 to get 12:48, which is equivalent to the ratio 12:48 in the table.

Ratio 3: 4:16

To find the equivalent ratio of 4:16, we can multiply both numbers by 1, 2, 3, 4, or 5. However, we are looking for a ratio in the form A:B, where A and B are integers. Therefore, we can multiply both numbers by 4 to get 16:64, which is equivalent to the ratio 16:64 in the table.

Ratio 4: 5:20

To find the equivalent ratio of 5:20, we can multiply both numbers by 1, 2, 3, 4, or 5. However, we are looking for a ratio in the form A:B, where A and B are integers. Therefore, we can multiply both numbers by 4 to get 20:80, which is equivalent to the ratio 20:80 in the table.

In conclusion, the equivalent ratios in the form A:B are:

  • 4:16
  • 12:48
  • 16:64
  • 20:80

These ratios are equivalent to the ratios in the table because they have the same value, even if they are expressed differently.

The concept of equivalent ratios is an important one in mathematics, as it allows us to compare and contrast different ratios. By understanding how to determine equivalent ratios, we can solve a wide range of problems in mathematics, science, and engineering.

Real-World Applications

Equivalent ratios have many real-world applications. For example, in cooking, a recipe may call for a certain ratio of ingredients. If we want to make a larger or smaller batch of the recipe, we can use equivalent ratios to determine the correct amount of each ingredient.

In finance, equivalent ratios are used to compare the performance of different investments. For example, a stock may have a ratio of earnings to price that is equivalent to the ratio of earnings to price of another stock.

In science, equivalent ratios are used to compare the properties of different materials. For example, a material may have a ratio of strength to weight that is equivalent to the ratio of strength to weight of another material.

In conclusion, equivalent ratios are an important concept in mathematics that has many real-world applications. By understanding how to determine equivalent ratios, we can solve a wide range of problems in mathematics, science, and engineering. The equivalent ratios in the form A:B are:

  • 4:16
  • 12:48
  • 16:64
  • 20:80

Q: What is an equivalent ratio?

A: An equivalent ratio is a ratio that has the same value as another ratio, even if they are expressed differently. For example, the ratios 2:4 and 3:6 are equivalent because they both represent the same relationship between two quantities.

Q: How do I determine if two ratios are equivalent?

A: To determine if two ratios are equivalent, you can use the following method:

  • Multiply or divide both numbers in the first ratio by the same number.
  • Check if the resulting ratio is the same as the second ratio.

Q: What are some real-world applications of equivalent ratios?

A: Equivalent ratios have many real-world applications, including:

  • Cooking: Equivalent ratios are used to compare the amount of ingredients in a recipe.
  • Finance: Equivalent ratios are used to compare the performance of different investments.
  • Science: Equivalent ratios are used to compare the properties of different materials.

Q: How do I find the equivalent ratio of a given ratio?

A: To find the equivalent ratio of a given ratio, you can multiply or divide both numbers in the ratio by the same number. For example, to find the equivalent ratio of 2:8, you can multiply both numbers by 2 to get 4:16.

Q: What are some common mistakes to avoid when working with equivalent ratios?

A: Some common mistakes to avoid when working with equivalent ratios include:

  • Not checking if the resulting ratio is the same as the original ratio.
  • Not using the same number to multiply or divide both numbers in the ratio.
  • Not simplifying the ratio to its simplest form.

Q: How do I simplify a ratio to its simplest form?

A: To simplify a ratio to its simplest form, you can divide both numbers in the ratio by their greatest common divisor (GCD). For example, to simplify the ratio 6:12, you can divide both numbers by 6 to get 1:2.

Q: What is the greatest common divisor (GCD) of two numbers?

A: The greatest common divisor (GCD) of two numbers is the largest number that divides both numbers without leaving a remainder. For example, the GCD of 6 and 12 is 6.

Q: How do I find the GCD of two numbers?

A: To find the GCD of two numbers, you can use the following method:

  • List the factors of each number.
  • Identify the largest factor that is common to both numbers.
  • The GCD is the largest common factor.

Q: What are some common uses of equivalent ratios in everyday life?

A: Equivalent ratios are used in many everyday situations, including:

  • Cooking: Equivalent ratios are used to compare the amount of ingredients in a recipe.
  • Finance: Equivalent ratios are used to compare the performance of different investments.
  • Science: Equivalent ratios are used to compare the properties of different materials.

Q: How do I apply equivalent ratios to solve real-world problems?

A: To apply equivalent ratios to solve real-world problems, you can follow these steps:

  • Identify the problem and the information given.
  • Determine the equivalent ratio that is relevant to the problem.
  • Use the equivalent ratio to solve the problem.

In conclusion, equivalent ratios are an important concept in mathematics that has many real-world applications. By understanding how to determine equivalent ratios, you can solve a wide range of problems in mathematics, science, and engineering.