Which Set Of Fractions Is Ordered From Least To Greatest?A. $\frac{5}{8}, \frac{8}{12}, \frac{3}{4}$ B. $\frac{8}{12}, \frac{5}{8}, \frac{3}{4}$ C. $\frac{3}{4}, \frac{5}{8}, \frac{8}{12}$ D. $\frac{5}{8}, \frac{3}{4},

Introduction

Comparing fractions is an essential skill in mathematics, and it's crucial to understand how to order them from least to greatest. In this article, we will explore the different methods for comparing fractions and provide a step-by-step guide on how to determine which set of fractions is ordered from least to greatest.

Understanding Fractions

Before we dive into comparing fractions, let's quickly review what fractions are. A fraction is a way to represent a part of a whole. It consists of two numbers: a numerator (the top number) and a denominator (the bottom number). The numerator tells us how many equal parts we have, and the denominator tells us how many parts the whole is divided into.

For example, the fraction $\frac{1}{2}$ represents one half of a whole. The numerator is 1, and the denominator is 2.

Comparing Fractions: Methods and Techniques

There are several methods and techniques for comparing fractions, including:

• Cross-multiplication: This method involves multiplying the numerator of one fraction by the denominator of the other fraction and comparing the results.
• Finding a common denominator: This method involves finding a common denominator for both fractions and then comparing the numerators.
• Comparing the size of the numerators and denominators: This method involves comparing the size of the numerators and denominators of both fractions to determine which one is larger.

Method 1: Cross-Multiplication

Cross-multiplication is a simple and effective method for comparing fractions. To use this method, we multiply the numerator of one fraction by the denominator of the other fraction and compare the results.

For example, let's compare the fractions $\frac{1}{2}$ and $\frac{1}{3}$. To use cross-multiplication, we multiply the numerator of the first fraction (1) by the denominator of the second fraction (3), and we multiply the numerator of the second fraction (1) by the denominator of the first fraction (2).

$\frac{1}{2} \times \frac{3}{1} = \frac{3}{2}$

$\frac{1}{3} \times \frac{2}{1} = \frac{2}{3}$

Since $\frac{3}{2}$ is greater than $\frac{2}{3}$, we can conclude that $\frac{1}{2}$ is greater than $\frac{1}{3}$.

Method 2: Finding a Common Denominator

Finding a common denominator is another method for comparing fractions. To use this method, we find a common denominator for both fractions and then compare the numerators.

For example, let's compare the fractions $\frac{1}{2}$ and $\frac{1}{3}$. To find a common denominator, we can multiply the denominators together.

$2 \times 3 = 6$

Now that we have a common denominator, we can compare the numerators.

$\frac{1}{2} = \frac{3}{6}$

$\frac{1}{3} = \frac{2}{6}$

Since 3 is greater than 2, we can conclude that $\frac{1}{2}$ is greater than $\frac{1}{3}$.

Method 3: Comparing the Size of the Numerators and Denominators

Comparing the size of the numerators and denominators is another method for comparing fractions. To use this method, we compare the size of the numerators and denominators of both fractions to determine which one is larger.

For example, let's compare the fractions $\frac{1}{2}$ and $\frac{1}{3}$. To compare the size of the numerators and denominators, we can look at the numerators and denominators separately.

The numerator of the first fraction is 1, and the numerator of the second fraction is 1. Since the numerators are equal, we need to look at the denominators.

The denominator of the first fraction is 2, and the denominator of the second fraction is 3. Since 3 is greater than 2, we can conclude that $\frac{1}{3}$ is greater than $\frac{1}{2}$.

Which Set of Fractions is Ordered from Least to Greatest?

Now that we have reviewed the different methods for comparing fractions, let's apply these methods to the sets of fractions provided in the discussion category.

A. $\frac{5}{8}, \frac{8}{12}, \frac{3}{4}$

B. $\frac{8}{12}, \frac{5}{8}, \frac{3}{4}$

C. $\frac{3}{4}, \frac{5}{8}, \frac{8}{12}$

D. $\frac{5}{8}, \frac{3}{4}, \frac{8}{12}$

To determine which set of fractions is ordered from least to greatest, we can use the methods we reviewed earlier.

Let's start by comparing the fractions in each set.

Set A: $\frac{5}{8}, \frac{8}{12}, \frac{3}{4}$

To compare these fractions, we can use the method of finding a common denominator.

$8 \times 3 = 24$

$12 \times 2 = 24$

Now that we have a common denominator, we can compare the numerators.

$\frac{5}{8} = \frac{15}{24}$

$\frac{8}{12} = \frac{16}{24}$

$\frac{3}{4} = \frac{18}{24}$

Since 15 is less than 16, which is less than 18, we can conclude that the fractions in Set A are ordered from least to greatest.

Set B: $\frac{8}{12}, \frac{5}{8}, \frac{3}{4}$

To compare these fractions, we can use the method of finding a common denominator.

$12 \times 2 = 24$

$8 \times 3 = 24$

Now that we have a common denominator, we can compare the numerators.

$\frac{8}{12} = \frac{16}{24}$

$\frac{5}{8} = \frac{15}{24}$

$\frac{3}{4} = \frac{18}{24}$

Since 15 is less than 16, which is less than 18, we can conclude that the fractions in Set B are not ordered from least to greatest.

Set C: $\frac{3}{4}, \frac{5}{8}, \frac{8}{12}$

To compare these fractions, we can use the method of finding a common denominator.

$4 \times 3 = 12$

$8 \times 3 = 24$

$12 \times 2 = 24$

Now that we have a common denominator, we can compare the numerators.

$\frac{3}{4} = \frac{9}{12}$

$\frac{5}{8} = \frac{15}{24}$

$\frac{8}{12} = \frac{16}{24}$

Since 9 is less than 15, which is less than 16, we can conclude that the fractions in Set C are not ordered from least to greatest.

Set D: $\frac{5}{8}, \frac{3}{4}, \frac{8}{12}$

To compare these fractions, we can use the method of finding a common denominator.

$8 \times 3 = 24$

$4 \times 6 = 24$

$12 \times 2 = 24$

Now that we have a common denominator, we can compare the numerators.

$\frac{5}{8} = \frac{15}{24}$

$\frac{3}{4} = \frac{18}{24}$

$\frac{8}{12} = \frac{16}{24}$

Since 15 is less than 16, which is less than 18, we can conclude that the fractions in Set D are not ordered from least to greatest.

Conclusion

In conclusion, the set of fractions that is ordered from least to greatest is Set A: $\frac{5}{8}, \frac{8}{12}, \frac{3}{4}$. This set of fractions is ordered from least to greatest because the numerators are in the correct order, with 5 being less than 8, which is less than 3.

Final Answer

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Q: What is the best method for comparing fractions?

A: The best method for comparing fractions depends on the specific fractions being compared. However, the method of finding a common denominator is often the most effective method.

Q: How do I find a common denominator for two fractions?

A: To find a common denominator for two fractions, you can multiply the denominators together. For example, if you have the fractions $\frac{1}{2}$ and $\frac{1}{3}$, you can multiply the denominators together to get a common denominator of 6.

Q: What if the fractions have different denominators and numerators?

A: If the fractions have different denominators and numerators, you can use the method of cross-multiplication to compare them. This involves multiplying the numerator of one fraction by the denominator of the other fraction and comparing the results.

Q: Can I compare fractions with unlike denominators?

A: Yes, you can compare fractions with unlike denominators. To do this, you can use the method of finding a common denominator or the method of cross-multiplication.

Q: How do I know if a fraction is greater than or less than another fraction?

A: To determine if a fraction is greater than or less than another fraction, you can compare the numerators and denominators. If the numerator of one fraction is greater than the numerator of the other fraction, then the first fraction is greater. If the denominator of one fraction is greater than the denominator of the other fraction, then the first fraction is less.

Q: Can I compare fractions with negative numbers?

A: Yes, you can compare fractions with negative numbers. To do this, you can use the same methods as you would for comparing fractions with positive numbers.

Q: How do I compare fractions with decimals?

A: To compare fractions with decimals, you can convert the decimals to fractions and then compare them using the methods described above.

Q: Can I compare fractions with mixed numbers?

A: Yes, you can compare fractions with mixed numbers. To do this, you can convert the mixed numbers to improper fractions and then compare them using the methods described above.

Q: How do I compare fractions with unlike signs?

A: To compare fractions with unlike signs, you can use the same methods as you would for comparing fractions with like signs. However, you may need to consider the signs of the fractions when comparing them.

Q: Can I compare fractions with variables?

A: Yes, you can compare fractions with variables. To do this, you can use the same methods as you would for comparing fractions with constants.

Q: How do I compare fractions with exponents?

A: To compare fractions with exponents, you can use the same methods as you would for comparing fractions with constants. However, you may need to consider the exponents when comparing the fractions.

Q: Can I compare fractions with radicals?

A: Yes, you can compare fractions with radicals. To do this, you can use the same methods as you would for comparing fractions with constants. However, you may need to consider the radicals when comparing the fractions.

Conclusion

In conclusion, comparing fractions is an essential skill in mathematics, and there are several methods and techniques that can be used to compare fractions. By understanding these methods and techniques, you can effectively compare fractions and make informed decisions in a variety of mathematical contexts.

Final Answer

The final answer is that comparing fractions is a complex process that requires a deep understanding of mathematical concepts and techniques. By using the methods and techniques described in this article, you can effectively compare fractions and make informed decisions in a variety of mathematical contexts.