Choose The Correct Symbol To Compare The Expressions. Do Not Multiply. 3 1 7 ? 3 1 7 × 7 9 3 \frac{1}{7} \quad ? \quad 3 \frac{1}{7} \times \frac{7}{9} 3 7 1 ​ ? 3 7 1 ​ × 9 7 ​ A. >B. <C. =

Introduction

When comparing expressions, it's essential to choose the correct symbol to indicate the relationship between them. In this article, we will explore how to compare expressions and choose the correct symbol, specifically in the context of the given problem: $3 \frac{1}{7} \quad ? \quad 3 \frac{1}{7} \times \frac{7}{9}$.

Understanding the Problem

The problem asks us to compare the expression $3 \frac{1}{7}$ with the product of $3 \frac{1}{7}$ and $\frac{7}{9}$. To solve this problem, we need to understand the concept of multiplying mixed numbers and fractions.

Multiplying Mixed Numbers and Fractions

A mixed number is a combination of a whole number and a fraction. For example, $3 \frac{1}{7}$ is a mixed number that represents $3$ whole units and $\frac{1}{7}$ of a unit. When multiplying mixed numbers and fractions, we need to follow the rules of multiplying fractions and whole numbers.

Step 1: Convert the Mixed Number to an Improper Fraction

To multiply mixed numbers and fractions, we need to convert the mixed number to an improper fraction. An improper fraction is a fraction where the numerator is greater than or equal to the denominator. To convert a mixed number to an improper fraction, we multiply the whole number by the denominator and add the numerator.

For example, to convert $3 \frac{1}{7}$ to an improper fraction, we multiply $3$ by $7$ and add $1$:

$3 \times 7 = 21$ $21 + 1 = 22$

So, $3 \frac{1}{7}$ is equal to $\frac{22}{7}$.

Step 2: Multiply the Fractions

Now that we have converted the mixed number to an improper fraction, we can multiply the fractions. To multiply fractions, we multiply the numerators and denominators separately.

For example, to multiply $\frac{22}{7}$ and $\frac{7}{9}$, we multiply the numerators and denominators:

$\frac{22}{7} \times \frac{7}{9} = \frac{22 \times 7}{7 \times 9}$ $= \frac{154}{63}$

Step 3: Simplify the Result

After multiplying the fractions, we need to simplify the result. To simplify a fraction, we divide the numerator and denominator by their greatest common divisor (GCD).

For example, to simplify $\frac{154}{63}$, we find the GCD of $154$ and $63$, which is $21$. We then divide both the numerator and denominator by $21$:

$\frac{154}{63} = \frac{154 \div 21}{63 \div 21}$ $= \frac{7.33}{3}$

Comparing the Expressions

Now that we have multiplied the fractions, we can compare the expressions. The original expression is $3 \frac{1}{7}$, and the product of $3 \frac{1}{7}$ and $\frac{7}{9}$ is $\frac{7.33}{3}$.

To compare these expressions, we need to convert them to a common form. We can convert the mixed number to an improper fraction and then compare the fractions.

For example, we can convert $3 \frac{1}{7}$ to an improper fraction:

$3 \frac{1}{7} = \frac{22}{7}$

Now we can compare the fractions:

$\frac{22}{7} > \frac{7.33}{3}$

Conclusion

In conclusion, to compare expressions, we need to choose the correct symbol to indicate the relationship between them. In this article, we explored how to compare expressions and choose the correct symbol, specifically in the context of the given problem: $3 \frac{1}{7} \quad ? \quad 3 \frac{1}{7} \times \frac{7}{9}$.

We learned how to multiply mixed numbers and fractions, convert mixed numbers to improper fractions, and simplify the result. We also compared the expressions and chose the correct symbol to indicate the relationship between them.

Choosing the Correct Symbol

Based on our comparison, we can choose the correct symbol to indicate the relationship between the expressions. The correct symbol is:

A. >

This symbol indicates that the original expression $3 \frac{1}{7}$ is greater than the product of $3 \frac{1}{7}$ and $\frac{7}{9}$.

Final Answer

The final answer is:

Q: What is the correct symbol to compare the expressions $3 \frac{1}{7}$ and $3 \frac{1}{7} \times \frac{7}{9}$?

A: The correct symbol to compare the expressions is A. >. This symbol indicates that the original expression $3 \frac{1}{7}$ is greater than the product of $3 \frac{1}{7}$ and $\frac{7}{9}$.

Q: How do I convert a mixed number to an improper fraction?

A: To convert a mixed number to an improper fraction, you multiply the whole number by the denominator and add the numerator. For example, to convert $3 \frac{1}{7}$ to an improper fraction, you multiply $3$ by $7$ and add $1$:

$3 \times 7 = 21$ $21 + 1 = 22$

So, $3 \frac{1}{7}$ is equal to $\frac{22}{7}$.

Q: How do I multiply fractions?

A: To multiply fractions, you multiply the numerators and denominators separately. For example, to multiply $\frac{22}{7}$ and $\frac{7}{9}$, you multiply the numerators and denominators:

$\frac{22}{7} \times \frac{7}{9} = \frac{22 \times 7}{7 \times 9}$ $= \frac{154}{63}$

Q: How do I simplify a fraction?

A: To simplify a fraction, you divide the numerator and denominator by their greatest common divisor (GCD). For example, to simplify $\frac{154}{63}$, you find the GCD of $154$ and $63$, which is $21$. You then divide both the numerator and denominator by $21$:

$\frac{154}{63} = \frac{154 \div 21}{63 \div 21}$ $= \frac{7.33}{3}$

Q: What is the difference between a mixed number and an improper fraction?

A: A mixed number is a combination of a whole number and a fraction. For example, $3 \frac{1}{7}$ is a mixed number that represents $3$ whole units and $\frac{1}{7}$ of a unit. An improper fraction is a fraction where the numerator is greater than or equal to the denominator. For example, $\frac{22}{7}$ is an improper fraction.

Q: Why is it important to choose the correct symbol when comparing expressions?

A: Choosing the correct symbol when comparing expressions is important because it indicates the relationship between the expressions. If you choose the wrong symbol, it can lead to incorrect conclusions and misunderstandings.

Q: Can you provide more examples of comparing expressions?

A: Yes, here are a few more examples:

• Compare $2 \frac{1}{4}$ and $2 \frac{1}{4} \times \frac{4}{5}$.
• Compare $5 \frac{3}{8}$ and $5 \frac{3}{8} \times \frac{8}{9}$.
• Compare $1 \frac{1}{2}$ and $1 \frac{1}{2} \times \frac{2}{3}$.

Conclusion

In conclusion, comparing expressions is an essential skill in mathematics. By understanding how to convert mixed numbers to improper fractions, multiply fractions, and simplify the result, you can choose the correct symbol to indicate the relationship between the expressions. Remember to choose the correct symbol to avoid incorrect conclusions and misunderstandings.

Final Tips

• Always convert mixed numbers to improper fractions before comparing expressions.
• Multiply fractions by multiplying the numerators and denominators separately.
• Simplify fractions by dividing the numerator and denominator by their greatest common divisor (GCD).
• Choose the correct symbol to indicate the relationship between the expressions.

By following these tips and practicing comparing expressions, you will become more confident and proficient in mathematics.