Choose The Correct Symbol To Compare: 3 5 6 × 3 5 3 \frac{5}{6} \times \frac{3}{5} 3 6 5 ​ × 5 3 ​ ? 3 5 6 3 \frac{5}{6} 3 6 5 ​ A. $\ \textgreater \ $ B. $\ \textless \ $ C. = = =

Introduction

When dealing with mixed numbers and fractions, it's essential to understand how to compare them accurately. In this article, we will explore the concept of comparing mixed numbers and fractions, focusing on the given problem: $3 \frac{5}{6} \times \frac{3}{5}$ ? $3 \frac{5}{6}$. We will break down the solution step by step, using real-world examples and explanations to make the concept more accessible.

Understanding Mixed Numbers and Fractions

Before diving into the problem, let's clarify the concept of mixed numbers and fractions. A mixed number is a combination of a whole number and a fraction, while a fraction represents a part of a whole. For example, $3 \frac{5}{6}$ is a mixed number, where 3 is the whole number and $\frac{5}{6}$ is the fraction.

The Problem: Comparing Mixed Numbers and Fractions

The given problem is to compare $3 \frac{5}{6} \times \frac{3}{5}$ and $3 \frac{5}{6}$. To solve this problem, we need to follow the order of operations (PEMDAS):

1. Multiply the mixed number $3 \frac{5}{6}$ by the fraction $\frac{3}{5}$.
2. Compare the result with the mixed number $3 \frac{5}{6}$.

Step 1: Multiplying the Mixed Number and Fraction

To multiply a mixed number by a fraction, we need to multiply the whole number part by the fraction, and then multiply the fraction part by the fraction.

$3 \frac{5}{6} \times \frac{3}{5} = (3 \times \frac{3}{5}) + (\frac{5}{6} \times \frac{3}{5})$

$= \frac{9}{5} + \frac{5}{30}$

$= \frac{9}{5} + \frac{1}{6}$

Step 2: Converting the Result to a Common Denominator

To compare the result with the mixed number $3 \frac{5}{6}$, we need to convert the result to a common denominator. The least common multiple (LCM) of 5 and 6 is 30.

$\frac{9}{5} + \frac{1}{6} = \frac{54}{30} + \frac{5}{30}$

$= \frac{59}{30}$

Step 3: Comparing the Result with the Mixed Number

Now that we have the result in the form of a fraction, we can compare it with the mixed number $3 \frac{5}{6}$.

$3 \frac{5}{6} = \frac{23}{6}$

$\frac{59}{30} = \frac{177}{90}$

Since $\frac{177}{90}$ is greater than $\frac{23}{6}$, we can conclude that $3 \frac{5}{6} \times \frac{3}{5}$ is greater than $3 \frac{5}{6}$.

Conclusion

In conclusion, comparing mixed numbers and fractions requires a step-by-step approach. By following the order of operations and converting the result to a common denominator, we can accurately compare mixed numbers and fractions. In this article, we solved the problem $3 \frac{5}{6} \times \frac{3}{5}$ ? $3 \frac{5}{6}$ and found that the result is greater than the mixed number $3 \frac{5}{6}$.

Comparison of Options

Option	Description
A	$3 \frac{5}{6} \times \frac{3}{5}$ is greater than $3 \frac{5}{6}$
B	$3 \frac{5}{6} \times \frac{3}{5}$ is less than $3 \frac{5}{6}$
C	$3 \frac{5}{6} \times \frac{3}{5}$ is equal to $3 \frac{5}{6}$

Based on our solution, the correct answer is:

A. $\ \textgreater \ $

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

A: A mixed number is a combination of a whole number and a fraction, while a fraction represents a part of a whole. For example, $3 \frac{5}{6}$ is a mixed number, where 3 is the whole number and $\frac{5}{6}$ is the fraction.

Q: How do I compare two mixed numbers?

A: To compare two mixed numbers, you need to compare the whole number parts first, and then the fraction parts. If the whole number parts are equal, you can compare the fraction parts.

Q: How do I compare a mixed number and a fraction?

A: To compare a mixed number and a fraction, you need to convert the mixed number to an improper fraction, and then compare the two fractions.

Q: What is the order of operations when comparing mixed numbers and fractions?

A: The order of operations is:

1. Multiply the mixed number by the fraction (if necessary)
2. Convert the result to a common denominator
3. Compare the result with the mixed number or fraction

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

A: To convert a mixed number to an improper fraction, you need to multiply the whole number part by the denominator, and then add the numerator.

For example, to convert $3 \frac{5}{6}$ to an improper fraction:

$3 \frac{5}{6} = \frac{(3 \times 6) + 5}{6}$

$= \frac{18 + 5}{6}$

$= \frac{23}{6}$

Q: What is the least common multiple (LCM) of two numbers?

A: The least common multiple (LCM) of two numbers is the smallest number that both numbers can divide into evenly.

For example, the LCM of 5 and 6 is 30.

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

A: To find the LCM of two numbers, you can list the multiples of each number and find the smallest number that appears in both lists.

Alternatively, you can use the following formula:

LCM(a, b) = (a × b) / GCD(a, b)

where GCD(a, b) is the greatest common divisor of a and b.

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 both numbers can divide into evenly.

For example, the GCD of 5 and 6 is 1.

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

A: To find the GCD of two numbers, you can list the factors of each number and find the largest number that appears in both lists.

Alternatively, you can use the following formula:

GCD(a, b) = (a × b) / LCM(a, b)

Conclusion

Comparing mixed numbers and fractions can be a challenging task, but by following the order of operations and using the correct techniques, you can accurately compare these types of numbers. We hope this article has provided you with a better understanding of how to compare mixed numbers and fractions, and has answered some of the most frequently asked questions in this area.