The Rules of Arithmetic Operations with Fractions

Oct 13, 2025 by miodeya.com 130 views

Introduction

Fractions are a fundamental concept in mathematics, and simplifying them is an essential skill to master. In this article, we will explore the process of simplifying fractions, focusing on the specific problem of $\frac{1}{2}-\left(-\frac{8}{7}\right)$ . We will break down the solution into manageable steps, making it easy to understand and follow.

Understanding the Problem

The given problem involves subtracting a negative fraction from a positive fraction. To simplify this expression, we need to apply the rules of arithmetic operations with fractions.

The Rules of Arithmetic Operations with Fractions

When subtracting fractions, we need to follow these rules:

If the fractions have the same denominator, we can subtract the numerators directly.
If the fractions have different denominators, we need to find the least common multiple (LCM) of the denominators and then convert both fractions to have the LCM as the denominator.

Step 1: Simplify the Expression

To simplify the expression $\frac{1}{2}-\left(-\frac{8}{7}\right)$ , we need to apply the rules of arithmetic operations with fractions.

Subtracting a Negative Fraction

When subtracting a negative fraction, we can rewrite the expression as the sum of the positive fraction and the absolute value of the negative fraction.

$\frac{1}{2}-\left(-\frac{8}{7}\right) = \frac{1}{2} + \left|\frac{8}{7}\right|$

Evaluating the Absolute Value

The absolute value of a fraction is equal to the fraction itself.

$\left|\frac{8}{7}\right| = \frac{8}{7}$

Simplifying the Expression

Now we can simplify the expression by adding the fractions.

$\frac{1}{2} + \frac{8}{7} = \frac{7}{14} + \frac{16}{14}$

Combining the Fractions

Since the fractions have the same denominator, we can combine them by adding the numerators.

$\frac{7}{14} + \frac{16}{14} = \frac{23}{14}$

Conclusion

In this article, we have simplified the expression $\frac{1}{2}-\left(-\frac{8}{7}\right)$ using the rules of arithmetic operations with fractions. We have broken down the solution into manageable steps, making it easy to understand and follow. By applying the rules of arithmetic operations with fractions, we have arrived at the simplified expression $\frac{23}{14}$ .

Final Answer

The final answer is $\boxed{\frac{23}{14}}$ .

Additional Resources

For more information on simplifying fractions, check out the following resources:

Frequently Asked Questions

Q: What is the difference between a positive fraction and a negative fraction?

A: A positive fraction is a fraction with a positive numerator and a positive denominator, while a negative fraction is a fraction with a negative numerator or a negative denominator.

Q: How do I simplify a fraction with a negative numerator?

A: To simplify a fraction with a negative numerator, you can rewrite the fraction as the sum of the absolute value of the numerator and the negative sign.

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

$A: The least common multiple (LCM) of two fractions is the smallest number that both fractions can divide into evenly.<br/> **Frequently Asked Questions: Simplifying Fractions**$

Introduction

Simplifying fractions is an essential skill in mathematics, and it can be a bit tricky at times. In this article, we will answer some of the most frequently asked questions about simplifying fractions, including questions about positive and negative fractions, simplifying fractions with negative numerators, and finding the least common multiple (LCM) of two fractions.

Q&A

Q: What is the difference between a positive fraction and a negative fraction?

A: A positive fraction is a fraction with a positive numerator and a positive denominator, while a negative fraction is a fraction with a negative numerator or a negative denominator.

Example:

Positive fraction: $\frac{1}{2}$
Negative fraction: $-\frac{1}{2}$

Q: How do I simplify a fraction with a negative numerator?

A: To simplify a fraction with a negative numerator, you can rewrite the fraction as the sum of the absolute value of the numerator and the negative sign.

Example:

Simplify the fraction: $-\frac{8}{7}$
Rewrite the fraction as the sum of the absolute value of the numerator and the negative sign: $-\frac{8}{7} = -\left|\frac{8}{7}\right|$

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

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

Example:

Find the LCM of the fractions: $\frac{1}{2}$ and $\frac{3}{4}$
The LCM of the fractions is 4, since both fractions can divide into 4 evenly.

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

A: To find the LCM of two fractions, you can follow these steps:

Find the LCM of the denominators of the fractions.
Rewrite both fractions with the LCM as the denominator.
Simplify the fractions by dividing the numerator and denominator by their greatest common divisor (GCD).

Example:

Find the LCM of the fractions: $\frac{1}{2}$ and $\frac{3}{4}$
Find the LCM of the denominators: 4
Rewrite both fractions with the LCM as the denominator: $\frac{1}{2} = \frac{2}{4}$ and $\frac{3}{4} = \frac{3}{4}$
Simplify the fractions by dividing the numerator and denominator by their GCD: $\frac{2}{4} = \frac{1}{2}$ and $\frac{3}{4} = \frac{3}{4}$

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

A: A proper fraction is a fraction with a numerator that is less than the denominator, while an improper fraction is a fraction with a numerator that is greater than or equal to the denominator.

Example:

Proper fraction: $\frac{1}{2}$
Improper fraction: $\frac{3}{2}$

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

A: To convert a mixed number to an improper fraction, you can follow these steps:

Multiply the whole number by the denominator.
Add the product to the numerator.
Write the result as the numerator of the improper fraction.
Keep the denominator the same.

Example:

Convert the mixed number: $2\frac{1}{2}$ to an improper fraction
Multiply the whole number by the denominator: $2 \times 2 = 4$
Add the product to the numerator: $4 + 1 = 5$
Write the result as the numerator of the improper fraction: $\frac{5}{2}$

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

A: To convert an improper fraction to a mixed number, you can follow these steps:

Divide the numerator by the denominator.
Write the result as the whole number.
Write the remainder as the numerator of the fraction.
Keep the denominator the same.

Example:

Convert the improper fraction: $\frac{5}{2}$ to a mixed number
Divide the numerator by the denominator: $5 \div 2 = 2$
Write the result as the whole number: $2$
Write the remainder as the numerator of the fraction: $\frac{1}{2}$

Conclusion

In this article, we have answered some of the most frequently asked questions about simplifying fractions, including questions about positive and negative fractions, simplifying fractions with negative numerators, and finding the least common multiple (LCM) of two fractions. We have also provided examples to illustrate each concept and have provided step-by-step instructions for converting mixed numbers to improper fractions and vice versa.

Final Answer

The final answer is $\boxed{\frac{23}{14}}$ .

Additional Resources

For more information on simplifying fractions, check out the following resources: