Calculate The Following Expression:$\[ 6 \frac{5}{8} - 1 \frac{2}{7} \\]

Introduction

In mathematics, mixed numbers are a combination of a whole number and a fraction. They are often used to represent quantities that are not whole, but can be expressed as a sum of a whole number and a fraction. In this article, we will focus on calculating the expression $6 \frac{5}{8} - 1 \frac{2}{7}$, which involves subtracting two mixed numbers.

Understanding Mixed Numbers

A mixed number is a combination of a whole number and a fraction. It is written in the form $a \frac{b}{c}$, where $a$ is the whole number, $b$ is the numerator, and $c$ is the denominator. For example, $3 \frac{2}{5}$ is a mixed number that represents the quantity $3 + \frac{2}{5}$.

Converting Mixed Numbers to Improper Fractions

To simplify the calculation of mixed number expressions, it is often helpful to convert the mixed numbers to improper fractions. 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. We then write the result as a fraction with the original denominator.

For example, to convert $6 \frac{5}{8}$ to an improper fraction, we multiply $6$ by $8$ and add $5$, which gives us $48 + 5 = 53$. We then write the result as a fraction with the original denominator, which gives us $\frac{53}{8}$.

Converting $1 \frac{2}{7}$ to an Improper Fraction

To convert $1 \frac{2}{7}$ to an improper fraction, we multiply $1$ by $7$ and add $2$, which gives us $7 + 2 = 9$. We then write the result as a fraction with the original denominator, which gives us $\frac{9}{7}$.

Subtracting Improper Fractions

Now that we have converted both mixed numbers to improper fractions, we can subtract them. To subtract improper fractions, we need to have the same denominator. In this case, the denominators are $8$ and $7$, so we need to find the least common multiple (LCM) of $8$ and $7$. The LCM of $8$ and $7$ is $56$, so we can rewrite both fractions with a denominator of $56$.

$\frac{53}{8} = \frac{53 \times 7}{8 \times 7} = \frac{371}{56}$

$\frac{9}{7} = \frac{9 \times 8}{7 \times 8} = \frac{72}{56}$

Now that we have the same denominator, we can subtract the fractions.

$\frac{371}{56} - \frac{72}{56} = \frac{299}{56}$

Converting the Result to a Mixed Number

To convert the improper fraction $\frac{299}{56}$ to a mixed number, we divide the numerator by the denominator. We get a quotient of $5$ and a remainder of $19$. We then write the result as a mixed number, which gives us $5 \frac{19}{56}$.

Conclusion

In this article, we have learned how to calculate the expression $6 \frac{5}{8} - 1 \frac{2}{7}$. We first converted the mixed numbers to improper fractions, then subtracted the fractions, and finally converted the result back to a mixed number. This process demonstrates the importance of converting mixed numbers to improper fractions when performing arithmetic operations with them.

Tips and Tricks

• When subtracting mixed numbers, it is often helpful to convert them to improper fractions first.
• To convert a mixed number to an improper fraction, multiply the whole number by the denominator and add the numerator.
• To subtract improper fractions, find the least common multiple (LCM) of the denominators and rewrite both fractions with the LCM as the denominator.
• To convert an improper fraction to a mixed number, divide the numerator by the denominator and write the result as a mixed number.

Common Mistakes to Avoid

• When subtracting mixed numbers, do not forget to convert them to improper fractions first.
• When converting a mixed number to an improper fraction, do not forget to multiply the whole number by the denominator and add the numerator.
• When subtracting improper fractions, do not forget to find the least common multiple (LCM) of the denominators and rewrite both fractions with the LCM as the denominator.

Real-World Applications

• Mixed numbers are often used in real-world applications, such as measuring quantities of materials or liquids.
• When working with mixed numbers, it is essential to convert them to improper fractions to simplify the calculation.
• Improper fractions are often used in algebra and other branches of mathematics.

Conclusion

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Introduction

In our previous article, we explored the concept of mixed number expressions and how to calculate them. However, we know that practice makes perfect, and the best way to learn is by asking questions and getting answers. In this article, we will address some of the most frequently asked questions about mixed number expressions and provide detailed answers to help you better understand the concept.

Q: What is a mixed number?

A: A mixed number is a combination of a whole number and a fraction. It is written in the form $a \frac{b}{c}$, where $a$ is the whole number, $b$ is the numerator, and $c$ is the denominator.

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. You then write the result as a fraction with the original denominator.

For example, to convert $6 \frac{5}{8}$ to an improper fraction, you multiply $6$ by $8$ and add $5$, which gives you $48 + 5 = 53$. You then write the result as a fraction with the original denominator, which gives you $\frac{53}{8}$.

Q: How do I subtract mixed numbers?

A: To subtract mixed numbers, you first convert them to improper fractions. You then find the least common multiple (LCM) of the denominators and rewrite both fractions with the LCM as the denominator. Finally, you subtract the fractions.

For example, to subtract $6 \frac{5}{8}$ and $1 \frac{2}{7}$, you first convert them to improper fractions. You then find the LCM of $8$ and $7$, which is $56$. You rewrite both fractions with a denominator of $56$ and then subtract the fractions.

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

A: The least common multiple (LCM) is the smallest number that is a multiple of two or more numbers. To find the LCM of two numbers, you list the multiples of each number and find the smallest number that appears in both lists.

For example, to find the LCM of $8$ and $7$, you list the multiples of each number:

Multiples of $8$: $8, 16, 24, 32, 40, 48, 56, ...$ Multiples of $7$: $7, 14, 21, 28, 35, 42, 49, 56, ...$

The smallest number that appears in both lists is $56$, so the LCM of $8$ and $7$ is $56$.

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

A: To convert an improper fraction to a mixed number, you divide the numerator by the denominator and write the result as a mixed number.

For example, to convert $\frac{53}{8}$ to a mixed number, you divide $53$ by $8$, which gives you $6$ with a remainder of $5$. You then write the result as a mixed number, which gives you $6 \frac{5}{8}$.

Q: What are some common mistakes to avoid when working with mixed numbers?

A: Some common mistakes to avoid when working with mixed numbers include:

• Forgetting to convert mixed numbers to improper fractions before performing arithmetic operations
• Forgetting to find the least common multiple (LCM) of the denominators before subtracting fractions
• Writing the result of a subtraction as a mixed number without converting the improper fraction to a mixed number first

Q: How do I apply mixed number expressions in real-world situations?

A: Mixed number expressions are often used in real-world situations, such as measuring quantities of materials or liquids. For example, you might need to calculate the amount of paint needed to cover a wall, or the amount of water needed to fill a tank.

To apply mixed number expressions in real-world situations, you need to understand how to convert mixed numbers to improper fractions and how to perform arithmetic operations with them. You also need to be able to find the least common multiple (LCM) of the denominators and rewrite fractions with the LCM as the denominator.

Conclusion

In this article, we have addressed some of the most frequently asked questions about mixed number expressions and provided detailed answers to help you better understand the concept. By following the tips and tricks outlined in this article, you can avoid common mistakes and apply mixed number expressions in real-world situations.