Add And State The Sum In Simplest Form.$\[ \frac{4}{5x} + \frac{7}{3y} \\]A. \[$\frac{12y + 35x}{15xy}\$\]B. \[$\frac{11}{15xy}\$\]C. \[$\frac{11}{5x + 3y}\$\]D. \[$\frac{20x + 21y}{15xy}\$\]

Introduction

In mathematics, adding and simplifying fractions is a fundamental concept that is essential for solving various problems in algebra, geometry, and other branches of mathematics. When adding fractions, we need to find a common denominator and then add the numerators while keeping the denominator the same. In this article, we will explore how to add and simplify fractions, using the given problem as an example.

Understanding the Problem

The problem requires us to add two fractions: $\frac{4}{5x}$ and $\frac{7}{3y}$. To add these fractions, we need to find a common denominator, which is the product of the denominators of the two fractions. In this case, the common denominator is $15xy$.

Step 1: Find the Common Denominator

To find the common denominator, we multiply the denominators of the two fractions: $5x$ and $3y$. This gives us $15xy$.

Step 2: Add the Numerators

Now that we have the common denominator, we can add the numerators of the two fractions. The numerator of the first fraction is $4$, and the numerator of the second fraction is $7$. To add these numerators, we need to find a common multiple of $4$ and $7$. The least common multiple (LCM) of $4$ and $7$ is $28$. However, we can simplify the fraction by finding the greatest common divisor (GCD) of $4$ and $7$, which is $1$. Therefore, we can add the numerators as is.

Step 3: Simplify the Fraction

Now that we have added the numerators, we can simplify the fraction by dividing both the numerator and the denominator by their greatest common divisor (GCD). In this case, the GCD of $4$ and $7$ is $1$, so we cannot simplify the fraction further.

Step 4: Write the Final Answer

The final answer is $\frac{4+7}{5x+3y} = \frac{11}{5x+3y}$.

Conclusion

In conclusion, adding and simplifying fractions requires us to find a common denominator and then add the numerators while keeping the denominator the same. We can simplify the fraction by dividing both the numerator and the denominator by their greatest common divisor (GCD). In this article, we used the given problem as an example to demonstrate how to add and simplify fractions.

Answer Key

The correct answer is C. $\frac{11}{5x+3y}$.

Additional Examples

Here are some additional examples of adding and simplifying fractions:

• $\frac{2}{3x} + \frac{5}{4y} = \frac{8+15}{12xy} = \frac{23}{12xy}$
• $\frac{3}{2x} + \frac{2}{3y} = \frac{9+4}{6xy} = \frac{13}{6xy}$
• $\frac{4}{5x} + \frac{3}{2y} = \frac{8+15}{10xy} = \frac{23}{10xy}$

Tips and Tricks

Here are some tips and tricks for adding and simplifying fractions:

• Always find the least common multiple (LCM) of the denominators to add the fractions.
• Simplify the fraction by dividing both the numerator and the denominator by their greatest common divisor (GCD).
• Use the distributive property to add fractions with different denominators.
• Use the commutative property to add fractions with different denominators.

Common Mistakes

Here are some common mistakes to avoid when adding and simplifying fractions:

• Not finding the least common multiple (LCM) of the denominators.
• Not simplifying the fraction by dividing both the numerator and the denominator by their greatest common divisor (GCD).
• Not using the distributive property to add fractions with different denominators.
• Not using the commutative property to add fractions with different denominators.

Conclusion

Introduction

In our previous article, we explored how to add and simplify fractions using the given problem as an example. In this article, we will answer some frequently asked questions about adding and simplifying fractions.

Q: What is the first step in adding fractions?

A: The first step in adding fractions is to find a common denominator. This is the product of the denominators of the two fractions.

Q: How do I find the common denominator?

A: To find the common denominator, you multiply the denominators of the two fractions. For example, if you have $\frac{1}{2}$ and $\frac{1}{3}$, the common denominator would be $6$.

Q: What is the next step after finding the common denominator?

A: After finding the common denominator, you add the numerators of the two fractions. The numerator of the first fraction is $4$, and the numerator of the second fraction is $7$. To add these numerators, you need to find a common multiple of $4$ and $7$. The least common multiple (LCM) of $4$ and $7$ is $28$. However, you can simplify the fraction by finding the greatest common divisor (GCD) of $4$ and $7$, which is $1$. Therefore, you can add the numerators as is.

Q: How do I simplify the fraction?

A: To simplify the fraction, you divide both the numerator and the denominator by their greatest common divisor (GCD). In this case, the GCD of $4$ and $7$ is $1$, so you cannot simplify the fraction further.

Q: What is the final step in adding fractions?

A: The final step in adding fractions is to write the final answer. The final answer is $\frac{4+7}{5x+3y} = \frac{11}{5x+3y}$.

Q: What are some common mistakes to avoid when adding fractions?

A: Some common mistakes to avoid when adding fractions include:

• Not finding the least common multiple (LCM) of the denominators.
• Not simplifying the fraction by dividing both the numerator and the denominator by their greatest common divisor (GCD).
• Not using the distributive property to add fractions with different denominators.
• Not using the commutative property to add fractions with different denominators.

Q: How do I add fractions with different denominators?

A: To add fractions with different denominators, you need to find a common denominator. This is the product of the denominators of the two fractions. Once you have found the common denominator, you can add the numerators of the two fractions.

Q: Can I add fractions with unlike signs?

A: Yes, you can add fractions with unlike signs. When adding fractions with unlike signs, you need to find a common denominator and then add the numerators while keeping the denominator the same.

Q: How do I subtract fractions?

A: To subtract fractions, you need to find a common denominator and then subtract the numerators while keeping the denominator the same.

Q: Can I subtract fractions with unlike signs?

A: Yes, you can subtract fractions with unlike signs. When subtracting fractions with unlike signs, you need to find a common denominator and then subtract the numerators while keeping the denominator the same.

Conclusion

In conclusion, adding and simplifying fractions is a fundamental concept in mathematics that requires us to find a common denominator and then add the numerators while keeping the denominator the same. By following the steps outlined in this article, we can add and simplify fractions with ease.