Setting Up the Equivalent Ratios

Oct 13, 2025 by miodeya.com 133 views

Introduction to Equivalent Ratios

Equivalent ratios are a fundamental concept in mathematics that allows us to compare and solve problems involving proportions. In this article, we will explore how equivalent ratios can be used to find the cost of a certain quantity of grapes, given the cost of a smaller quantity.

The Problem: Finding the Cost of 3.5 Pounds of Grapes

We are given that one pound of grapes costs $\$1.55$ . We need to find the cost of 3.5 pounds of grapes. To do this, we can use equivalent ratios to set up a proportion that relates the cost of 1 pound of grapes to the cost of 3.5 pounds of grapes.

Setting Up the Equivalent Ratios

Let's examine the options given to us:

A. $\frac{1.55}{1} = \frac{x}{3.5}$

B. $\frac{1.55}{1} = \frac{3.5}{x}$

C. (no option provided)

To determine which option is correct, we need to understand the concept of equivalent ratios. Equivalent ratios are ratios that have the same value, but may have different numbers. In this case, we want to find a ratio that relates the cost of 1 pound of grapes to the cost of 3.5 pounds of grapes.

Analyzing Option A

Option A is $\frac{1.55}{1} = \frac{x}{3.5}$ . This option sets up a proportion where the cost of 1 pound of grapes is equal to the cost of x pounds of grapes. However, this option does not provide a clear relationship between the cost of 1 pound of grapes and the cost of 3.5 pounds of grapes.

Analyzing Option B

Option B is $\frac{1.55}{1} = \frac{3.5}{x}$ . This option sets up a proportion where the cost of 1 pound of grapes is equal to the cost of 3.5 pounds of grapes. However, this option does not provide a clear relationship between the cost of 1 pound of grapes and the cost of x pounds of grapes.

The Correct Option

After analyzing both options, we can see that neither option A nor option B provides a clear relationship between the cost of 1 pound of grapes and the cost of 3.5 pounds of grapes. However, we can see that option B is closer to the correct solution. To find the correct solution, we need to set up a proportion that relates the cost of 1 pound of grapes to the cost of 3.5 pounds of grapes.

The Correct Equation

The correct equation is $\frac{1.55}{1} = \frac{1.55 \times 3.5}{x}$ . This equation sets up a proportion where the cost of 1 pound of grapes is equal to the cost of 3.5 pounds of grapes. To find the cost of 3.5 pounds of grapes, we can multiply the cost of 1 pound of grapes by 3.5.

Solving for x

To solve for x, we can multiply the cost of 1 pound of grapes by 3.5:

$x = \frac{1.55 \times 3.5}{1}$

$x = 5.425$

Conclusion

In conclusion, the correct equation that shows a pair of equivalent ratios that can be used to find the cost of 3.5 pounds of grapes is $\frac{1.55}{1} = \frac{1.55 \times 3.5}{x}$ . This equation sets up a proportion that relates the cost of 1 pound of grapes to the cost of 3.5 pounds of grapes. To find the cost of 3.5 pounds of grapes, we can multiply the cost of 1 pound of grapes by 3.5.

Understanding Equivalent Ratios in Real-World Applications

Equivalent ratios are a fundamental concept in mathematics that has many real-world applications. In this article, we have seen how equivalent ratios can be used to find the cost of a certain quantity of grapes, given the cost of a smaller quantity. Equivalent ratios can also be used to solve problems involving proportions, such as finding the cost of a certain quantity of goods, given the cost of a smaller quantity.

Real-World Applications of Equivalent Ratios

Equivalent ratios have many real-world applications, including:

Cost calculations: Equivalent ratios can be used to find the cost of a certain quantity of goods, given the cost of a smaller quantity.
Proportion problems: Equivalent ratios can be used to solve problems involving proportions, such as finding the cost of a certain quantity of goods, given the cost of a smaller quantity.
Measurement conversions: Equivalent ratios can be used to convert between different units of measurement, such as converting between inches and feet.

Conclusion

In conclusion, equivalent ratios are a fundamental concept in mathematics that has many real-world applications. In this article, we have seen how equivalent ratios can be used to find the cost of a certain quantity of grapes, given the cost of a smaller quantity. Equivalent ratios can also be used to solve problems involving proportions, such as finding the cost of a certain quantity of goods, given the cost of a smaller quantity.

Q: What is an equivalent ratio?

A: An equivalent ratio is a ratio that has the same value, but may have different numbers. In other words, equivalent ratios are ratios that are equal in value, but may have different numbers.

Q: How do I set up an equivalent ratio?

A: To set up an equivalent ratio, you need to identify the relationship between two quantities. For example, if you want to find the cost of 3.5 pounds of grapes, given the cost of 1 pound of grapes, you can set up an equivalent ratio as follows:

$\frac{1.55}{1} = \frac{x}{3.5}$

Q: What is the difference between a proportion and an equivalent ratio?

A: A proportion is a statement that two ratios are equal, while an equivalent ratio is a ratio that has the same value, but may have different numbers. In other words, a proportion is a statement that two ratios are equal, while an equivalent ratio is a ratio that is equal in value, but may have different numbers.

Q: How do I solve for x in an equivalent ratio?

A: To solve for x in an equivalent ratio, you need to isolate x on one side of the equation. For example, if you have the following equivalent ratio:

$\frac{1.55}{1} = \frac{x}{3.5}$

You can solve for x by multiplying both sides of the equation by 3.5:

$x = \frac{1.55 \times 3.5}{1}$

$x = 5.425$

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

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

Cost calculations: Equivalent ratios can be used to find the cost of a certain quantity of goods, given the cost of a smaller quantity.
Proportion problems: Equivalent ratios can be used to solve problems involving proportions, such as finding the cost of a certain quantity of goods, given the cost of a smaller quantity.
Measurement conversions: Equivalent ratios can be used to convert between different units of measurement, such as converting between inches and feet.

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

A: To determine if two ratios are equivalent, you need to check if they have the same value. You can do this by multiplying both ratios by a common factor, or by dividing both ratios by a common factor.

Q: What is the importance of equivalent ratios in mathematics?

A: Equivalent ratios are an important concept in mathematics because they allow us to compare and solve problems involving proportions. They are used in a wide range of mathematical applications, including algebra, geometry, and trigonometry.

Q: Can equivalent ratios be used to solve problems involving fractions?

A: Yes, equivalent ratios can be used to solve problems involving fractions. For example, if you have the following fraction:

$\frac{1}{2}$

You can use an equivalent ratio to convert it to a decimal:

$\frac{1}{2} = \frac{0.5}{1}$

Q: Can equivalent ratios be used to solve problems involving percentages?

A: Yes, equivalent ratios can be used to solve problems involving percentages. For example, if you have the following percentage:

$25\%$

You can use an equivalent ratio to convert it to a decimal:

$25\% = \frac{0.25}{1}$

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 ratios are equivalent: Make sure to check if the ratios are equivalent before using them to solve a problem.
Not using a common factor: Make sure to use a common factor when multiplying or dividing both ratios.
Not simplifying the ratios: Make sure to simplify the ratios before using them to solve a problem.

Conclusion

In conclusion, equivalent ratios are an important concept in mathematics that has many real-world applications. They allow us to compare and solve problems involving proportions, and are used in a wide range of mathematical applications. By understanding equivalent ratios, you can solve problems involving proportions, fractions, and percentages, and make informed decisions in your personal and professional life.