Simplify The Expression: ${ \frac{16}{10} + \frac{18}{19} = }$

Introduction

When it comes to simplifying expressions involving fractions and algebra, it's essential to have a solid understanding of the underlying concepts. In this article, we'll delve into the world of fractions and algebra, exploring the steps required to simplify the expression $\frac{16}{10} + \frac{18}{19}$. By the end of this guide, you'll be equipped with the knowledge and skills necessary to tackle even the most complex expressions.

Understanding Fractions

Fractions are a fundamental concept in mathematics, representing a part of a whole. They consist of two parts: the numerator (the top number) and the denominator (the bottom number). The numerator indicates the number of equal parts, while the denominator represents the total number of parts. For example, in the fraction $\frac{2}{3}$, the numerator is 2, and the denominator is 3.

Types of Fractions

There are several types of fractions, including:

• Proper fractions: These are fractions where the numerator is less than the denominator, such as $\frac{1}{2}$.
• Improper fractions: These are fractions where the numerator is greater than or equal to the denominator, such as $\frac{3}{2}$.
• Mixed numbers: These are combinations of a whole number and a proper fraction, such as $2\frac{1}{2}$.

Simplifying Fractions

Simplifying fractions involves reducing them to their lowest terms, which means finding the greatest common divisor (GCD) of the numerator and denominator and dividing both numbers by the GCD. For example, to simplify the fraction $\frac{12}{18}$, we find the GCD of 12 and 18, which is 6. We then divide both numbers by 6, resulting in the simplified fraction $\frac{2}{3}$.

Steps to Simplify Fractions

To simplify a fraction, follow these steps:

1. Find the GCD: Determine the greatest common divisor of the numerator and denominator.
2. Divide both numbers: Divide both the numerator and denominator by the GCD.
3. Write the simplified fraction: The resulting fraction is the simplified form.

Adding Fractions with Different Denominators

When adding fractions with different denominators, we need to find a common denominator. The common denominator is the least common multiple (LCM) of the two denominators. Once we have the common denominator, we can add the fractions by adding the numerators and keeping the common denominator.

Steps to Add Fractions with Different Denominators

To add fractions with different denominators, follow these steps:

1. Find the LCM: Determine the least common multiple of the two denominators.
2. Write the fractions with the common denominator: Multiply the numerator and denominator of each fraction by the necessary factor to obtain the common denominator.
3. Add the fractions: Add the numerators and keep the common denominator.
4. Simplify the result: Simplify the resulting fraction, if possible.

Simplifying the Expression

Now that we've covered the basics of fractions and algebra, let's apply these concepts to simplify the expression $\frac{16}{10} + \frac{18}{19}$. To add these fractions, we need to find a common denominator, which is the LCM of 10 and 19.

Finding the LCM

To find the LCM of 10 and 19, we can list the multiples of each number:

• Multiples of 10: 10, 20, 30, 40, 50, 60, 70, 80, 90, 100, 110, 120, 130, 140, 150, 160, 170, 180, 190, ...
• Multiples of 19: 19, 38, 57, 76, 95, 114, 133, 152, 171, 190, ...

The first number that appears in both lists is 190, which is the LCM of 10 and 19.

Writing the Fractions with the Common Denominator

Now that we have the common denominator, we can write the fractions with the common denominator:

$\frac{16}{10} = \frac{16 \times 19}{10 \times 19} = \frac{304}{190}$

$\frac{18}{19} = \frac{18 \times 10}{19 \times 10} = \frac{180}{190}$

Adding the Fractions

Now that we have the fractions with the common denominator, we can add them:

$\frac{304}{190} + \frac{180}{190} = \frac{484}{190}$

Simplifying the Result

The resulting fraction is $\frac{484}{190}$. To simplify this fraction, we need to find the GCD of 484 and 190.

Finding the GCD

To find the GCD of 484 and 190, we can list the factors of each number:

• Factors of 484: 1, 2, 4, 11, 22, 44, 121, 242, 484
• Factors of 190: 1, 2, 5, 10, 19, 38, 95, 190

The greatest common factor of 484 and 190 is 2.

Dividing Both Numbers

Now that we have the GCD, we can divide both numbers by the GCD:

$\frac{484}{190} = \frac{484 \div 2}{190 \div 2} = \frac{242}{95}$

Simplifying the Final Result

The resulting fraction is $\frac{242}{95}$. This fraction cannot be simplified further, so the final result is $\frac{242}{95}$.

Conclusion

In this article, we've explored the world of fractions and algebra, learning how to simplify expressions involving fractions and algebra. We've covered the basics of fractions, including types of fractions and steps to simplify fractions. We've also learned how to add fractions with different denominators and simplify the resulting fraction. Finally, we've applied these concepts to simplify the expression $\frac{16}{10} + \frac{18}{19}$, resulting in the final answer of $\frac{242}{95}$. With this knowledge and skills, you'll be well-equipped to tackle even the most complex expressions involving fractions and algebra.

Introduction

In our previous article, we explored the world of fractions and algebra, learning how to simplify expressions involving fractions and algebra. We covered the basics of fractions, including types of fractions and steps to simplify fractions. We also learned how to add fractions with different denominators and simplify the resulting fraction. In this article, we'll answer some of the most frequently asked questions related to simplifying expressions involving fractions and algebra.

Q&A

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

A: A proper fraction is a fraction where the numerator is less than the denominator, such as $\frac{1}{2}$. An improper fraction is a fraction where the numerator is greater than or equal to the denominator, such as $\frac{3}{2}$.

Q: How do I simplify a fraction?

A: To simplify a fraction, follow these steps:

1. Find the GCD: Determine the greatest common divisor of the numerator and denominator.
2. Divide both numbers: Divide both the numerator and denominator by the GCD.
3. Write the simplified fraction: The resulting fraction is the simplified form.

Q: How do I add fractions with different denominators?

A: To add fractions with different denominators, follow these steps:

1. Find the LCM: Determine the least common multiple of the two denominators.
2. Write the fractions with the common denominator: Multiply the numerator and denominator of each fraction by the necessary factor to obtain the common denominator.
3. Add the fractions: Add the numerators and keep the common denominator.
4. Simplify the result: Simplify the resulting fraction, if possible.

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

A: The least common multiple (LCM) of two numbers is the smallest number that is a multiple of both numbers. For example, the LCM of 10 and 19 is 190.

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

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

Q: Can I simplify a fraction with a variable in the numerator or denominator?

A: Yes, you can simplify a fraction with a variable in the numerator or denominator. However, you need to follow the same steps as simplifying a fraction with numerical values.

Q: How do I add fractions with variables in the numerator or denominator?

A: To add fractions with variables in the numerator or denominator, follow the same steps as adding fractions with numerical values. However, you need to find the LCM of the variables and simplify the resulting fraction.

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 proper fraction, such as $2\frac{1}{2}$. An improper fraction is a fraction where the numerator is greater than or equal to the denominator, such as $\frac{3}{2}$.

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

A: To convert a mixed number to an improper fraction, multiply the whole number by the denominator and add the numerator. Then, write the result as an improper fraction.

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

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

Conclusion

In this article, we've answered some of the most frequently asked questions related to simplifying expressions involving fractions and algebra. We've covered topics such as simplifying fractions, adding fractions with different denominators, and converting between mixed numbers and improper fractions. With this knowledge and skills, you'll be well-equipped to tackle even the most complex expressions involving fractions and algebra.

Additional Resources

For more information on simplifying expressions involving fractions and algebra, check out the following resources:

• Khan Academy: Simplifying Fractions and Algebra
• Mathway: Simplifying Fractions and Algebra
• Wolfram Alpha: Simplifying Fractions and Algebra

Final Thoughts

Simplifying expressions involving fractions and algebra can be a challenging task, but with practice and patience, you'll become proficient in no time. Remember to follow the steps outlined in this article, and don't be afraid to ask for help if you're unsure about a particular concept. With this knowledge and skills, you'll be able to tackle even the most complex expressions involving fractions and algebra.