Choose The Statement That Correctly Lists Equivalent Ratios.A. $1:6 = 20:120 = 5:30$B. $1:6 = 4:18 = 6:36$C. $1:8 = 20:120 = 3:24$D. $1:8 = 3:32 = 5:48$

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Equivalent ratios are a fundamental concept in mathematics, particularly in the realm of fractions and proportions. They are used to compare the relationship between two or more sets of numbers, and are essential in various mathematical operations, such as solving equations and inequalities. In this article, we will delve into the concept of equivalent ratios, explore their properties, and examine the given options to determine the correct statement.

What are Equivalent Ratios?

Equivalent ratios are ratios that have the same value, even though they may appear different. They are obtained by multiplying or dividing both terms of a ratio by the same non-zero number. For example, the ratios 2:4 and 3:6 are equivalent because they can be obtained by multiplying the first ratio by 3/2.

Properties of Equivalent Ratios

Equivalent ratios have several properties that make them useful in mathematical operations. Some of these properties include:

  • Multiplication Property: If two ratios are equivalent, then multiplying both terms of each ratio by the same non-zero number will result in equivalent ratios.
  • Division Property: If two ratios are equivalent, then dividing both terms of each ratio by the same non-zero number will result in equivalent ratios.
  • Addition Property: If two ratios are equivalent, then adding the same number to both terms of each ratio will result in equivalent ratios.
  • Subtraction Property: If two ratios are equivalent, then subtracting the same number from both terms of each ratio will result in equivalent ratios.

Analyzing the Options

Now that we have a solid understanding of equivalent ratios and their properties, let's analyze the given options to determine the correct statement.

Option A: $1:6 = 20:120 = 5:30$

To determine if this statement is correct, we need to check if the ratios are equivalent. We can do this by finding the greatest common divisor (GCD) of the terms in each ratio and dividing both terms by the GCD.

  • For the ratio 1:6, the GCD is 1. Dividing both terms by 1 gives us 1:6.
  • For the ratio 20:120, the GCD is 20. Dividing both terms by 20 gives us 1:6.
  • For the ratio 5:30, the GCD is 5. Dividing both terms by 5 gives us 1:6.

Since all three ratios have the same value, option A is correct.

Option B: $1:6 = 4:18 = 6:36$

To determine if this statement is correct, we need to check if the ratios are equivalent. We can do this by finding the greatest common divisor (GCD) of the terms in each ratio and dividing both terms by the GCD.

  • For the ratio 1:6, the GCD is 1. Dividing both terms by 1 gives us 1:6.
  • For the ratio 4:18, the GCD is 2. Dividing both terms by 2 gives us 2:9.
  • For the ratio 6:36, the GCD is 6. Dividing both terms by 6 gives us 1:6.

Since the ratios 4:18 and 6:36 are not equivalent to 1:6, option B is incorrect.

Option C: $1:8 = 20:120 = 3:24$

To determine if this statement is correct, we need to check if the ratios are equivalent. We can do this by finding the greatest common divisor (GCD) of the terms in each ratio and dividing both terms by the GCD.

  • For the ratio 1:8, the GCD is 1. Dividing both terms by 1 gives us 1:8.
  • For the ratio 20:120, the GCD is 20. Dividing both terms by 20 gives us 1:6.
  • For the ratio 3:24, the GCD is 3. Dividing both terms by 3 gives us 1:8.

Since the ratios 20:120 and 3:24 are not equivalent to 1:8, option C is incorrect.

Option D: $1:8 = 3:32 = 5:48$

To determine if this statement is correct, we need to check if the ratios are equivalent. We can do this by finding the greatest common divisor (GCD) of the terms in each ratio and dividing both terms by the GCD.

  • For the ratio 1:8, the GCD is 1. Dividing both terms by 1 gives us 1:8.
  • For the ratio 3:32, the GCD is 1. Dividing both terms by 1 gives us 3:32.
  • For the ratio 5:48, the GCD is 1. Dividing both terms by 1 gives us 5:48.

Since the ratios 3:32 and 5:48 are not equivalent to 1:8, option D is incorrect.

Conclusion

In conclusion, the correct statement that lists equivalent ratios is option A: $1:6 = 20:120 = 5:30$. This is because all three ratios have the same value, and can be obtained by multiplying or dividing both terms of each ratio by the same non-zero number.

Understanding Equivalent Ratios in Real-Life Scenarios

Equivalent ratios are not only useful in mathematical operations, but also in real-life scenarios. For example, in cooking, equivalent ratios can be used to scale up or down a recipe. In finance, equivalent ratios can be used to compare the interest rates of different investments. In science, equivalent ratios can be used to compare the concentrations of different substances.

Real-Life Examples of Equivalent Ratios

  • Cooking: If a recipe calls for 2 cups of flour and 1 cup of sugar, and you want to make half the recipe, you can use equivalent ratios to scale down the ingredients. For example, you can multiply both terms of the ratio 2:1 by 1/2 to get 1:1/2.
  • Finance: If you are considering investing in two different stocks, and you want to compare their interest rates, you can use equivalent ratios to compare the ratios of their interest rates. For example, if one stock has an interest rate of 5% and the other has an interest rate of 3%, you can use equivalent ratios to compare the ratios 5:3.
  • Science: If you are conducting an experiment and you want to compare the concentrations of two different substances, you can use equivalent ratios to compare the ratios of their concentrations. For example, if one substance has a concentration of 10% and the other has a concentration of 5%, you can use equivalent ratios to compare the ratios 10:5.

Conclusion

Equivalent ratios are a fundamental concept in mathematics that can be used to compare the relationship between two or more sets of numbers. In this article, we will answer some frequently asked questions about equivalent ratios to help you better understand this concept.

Q: What is an equivalent ratio?

A: An equivalent ratio is a ratio that has the same value as another ratio, even though they may appear different. Equivalent ratios are obtained by multiplying or dividing both terms of a ratio by the same non-zero number.

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

A: To determine if two ratios are equivalent, you can find the greatest common divisor (GCD) of the terms in each ratio and divide both terms by the GCD. If the resulting ratios are the same, then the original ratios are equivalent.

Q: What are some examples of equivalent ratios?

A: Here are some examples of equivalent ratios:

  • 2:4 and 3:6 are equivalent because they can be obtained by multiplying the first ratio by 3/2.
  • 1:8 and 20:160 are equivalent because they can be obtained by multiplying the first ratio by 20/1.
  • 3:9 and 6:18 are equivalent because they can be obtained by multiplying the first ratio by 2/1.

Q: How do I use equivalent ratios in real-life scenarios?

A: Equivalent ratios can be used in various real-life scenarios, such as:

  • Cooking: Equivalent ratios can be used to scale up or down a recipe.
  • Finance: Equivalent ratios can be used to compare the interest rates of different investments.
  • Science: Equivalent ratios can be used to compare the concentrations of different substances.

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

A: Here are some common mistakes to avoid when working with equivalent ratios:

  • Not finding the greatest common divisor (GCD) of the terms in each ratio.
  • Not dividing both terms of each ratio by the GCD.
  • Not checking if the resulting ratios are the same.

Q: How do I simplify equivalent ratios?

A: To simplify equivalent ratios, you can find the greatest common divisor (GCD) of the terms in each ratio and divide both terms by the GCD. This will result in a simplified ratio that has the same value as the original ratio.

Q: Can equivalent ratios be used to compare the values of different variables?

A: Yes, equivalent ratios can be used to compare the values of different variables. For example, if you have two variables x and y, and you want to compare their values, you can use equivalent ratios to compare the ratios of their values.

Q: How do I use equivalent ratios to solve mathematical problems?

A: Equivalent ratios can be used to solve mathematical problems in various ways, such as:

  • Finding the value of a variable by setting up an equivalent ratio.
  • Comparing the values of different variables by setting up equivalent ratios.
  • Solving equations and inequalities by using equivalent ratios.

Conclusion

In conclusion, equivalent ratios are a fundamental concept in mathematics that can be used to compare the relationship between two or more sets of numbers. By understanding equivalent ratios and their properties, you can solve mathematical problems and make informed decisions in various fields.