Add The Following Fractions And State The Sum In Simplest Form:$ \frac{1}{3x} + \frac{5}{7y} }$Options A. { \frac{7y + 15x {21xy}$}$B. { \frac{3x + 35y}{21xy}$}$C. { \frac{2}{7xy}$} D . \[ D. \[ D . \[ \frac{6}{3x +

Introduction

Adding fractions with different variables can be a challenging task, especially when the variables are not the same. In this article, we will explore how to add fractions with different variables and state the sum in simplest form. We will use a step-by-step approach to make it easy to understand and follow.

Understanding the Problem

The problem requires us to add two fractions with different variables: $\frac{1}{3x}$ and $\frac{5}{7y}$. The variables $x$ and $y$ are not the same, which makes it difficult to add the fractions directly.

Step 1: Find the Least Common Multiple (LCM) of the Denominators

To add fractions with different variables, we need to find the least common multiple (LCM) of the denominators. The denominators are $3x$ and $7y$. To find the LCM, we need to list the multiples of each denominator:

• Multiples of $3x$: $3x, 6x, 9x, 12x, ...$
• Multiples of $7y$: $7y, 14y, 21y, 28y, ...$

The smallest multiple that appears in both lists is $21xy$. Therefore, the LCM of the denominators is $21xy$.

Step 2: Rewrite Each Fraction with the LCM as the Denominator

Now that we have found the LCM, we can rewrite each fraction with the LCM as the denominator. To do this, we need to multiply the numerator and denominator of each fraction by the necessary factor:

• $\frac{1}{3x} = \frac{1 \cdot 7y}{3x \cdot 7y} = \frac{7y}{21xy}$
• $\frac{5}{7y} = \frac{5 \cdot 3x}{7y \cdot 3x} = \frac{15x}{21xy}$

Step 3: Add the Fractions

Now that we have rewritten each fraction with the LCM as the denominator, we can add them:

$\frac{7y}{21xy} + \frac{15x}{21xy} = \frac{7y + 15x}{21xy}$

Step 4: Simplify the Sum

The sum is already in simplest form, so we don't need to simplify it further.

Conclusion

In this article, we have learned how to add fractions with different variables and state the sum in simplest form. We used a step-by-step approach to make it easy to understand and follow. The key steps were to find the least common multiple (LCM) of the denominators, rewrite each fraction with the LCM as the denominator, add the fractions, and simplify the sum.

Answer

The correct answer is:

$\boxed{\frac{7y + 15x}{21xy}}$

Discussion

This problem requires a good understanding of fractions and variables. The key concept is to find the least common multiple (LCM) of the denominators and rewrite each fraction with the LCM as the denominator. This makes it easy to add the fractions and state the sum in simplest form.

Related Problems

• Adding fractions with the same variable
• Subtracting fractions with different variables
• Multiplying fractions with different variables
• Dividing fractions with different variables

Practice Problems

• Add the fractions: $\frac{2}{5x} + \frac{3}{7y}$
• Add the fractions: $\frac{4}{3x} + \frac{2}{5y}$
• Add the fractions: $\frac{1}{2x} + \frac{3}{4y}$

References

• [1] "Adding Fractions with Different Variables" by Math Open Reference
• [2] "Fractions with Variables" by Khan Academy
• [3] "Adding and Subtracting Fractions with Variables" by Purplemath
  Adding Fractions with Different Variables: A Q&A Guide ===========================================================

Introduction

In our previous article, we explored how to add fractions with different variables and state the sum in simplest form. In this article, we will answer some frequently asked questions (FAQs) related to adding fractions with different variables.

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

A: The least common multiple (LCM) of the denominators is the smallest multiple that appears in both lists of multiples of each denominator. For example, if we have two fractions with denominators $3x$ and $7y$, the LCM of the denominators is $21xy$.

Q: How do I find the LCM of the denominators?

A: To find the LCM of the denominators, you need to list the multiples of each denominator and find the smallest multiple that appears in both lists. You can use a calculator or a table to help you find the LCM.

Q: What if the denominators have different variables?

A: If the denominators have different variables, you need to find the LCM of the variables as well as the LCM of the coefficients. For example, if we have two fractions with denominators $3x$ and $7y$, the LCM of the variables is $xy$ and the LCM of the coefficients is $21$.

Q: Can I add fractions with different variables if the variables are not the same?

A: Yes, you can add fractions with different variables if the variables are not the same. However, you need to find the LCM of the denominators and rewrite each fraction with the LCM as the denominator.

Q: How do I rewrite each fraction with the LCM as the denominator?

A: To rewrite each fraction with the LCM as the denominator, you need to multiply the numerator and denominator of each fraction by the necessary factor. For example, if we have two fractions with denominators $3x$ and $7y$, we can rewrite each fraction with the LCM $21xy$ as the denominator.

Q: Can I simplify the sum after adding the fractions?

A: Yes, you can simplify the sum after adding the fractions. However, you need to make sure that the sum is in simplest form.

Q: What if the sum is not in simplest form?

A: If the sum is not in simplest form, you need to simplify it by dividing both the numerator and denominator by their greatest common divisor (GCD).

Q: Can I use a calculator to add fractions with different variables?

A: Yes, you can use a calculator to add fractions with different variables. However, you need to make sure that the calculator can handle fractions with variables.

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

A: Some common mistakes to avoid when adding fractions with different variables include:

• Not finding the LCM of the denominators
• Not rewriting each fraction with the LCM as the denominator
• Not simplifying the sum after adding the fractions
• Not using the correct method to add fractions with different variables

Conclusion

In this article, we have answered some frequently asked questions (FAQs) related to adding fractions with different variables. We have also provided some tips and common mistakes to avoid when adding fractions with different variables.

Practice Problems

• Add the fractions: $\frac{2}{5x} + \frac{3}{7y}$
• Add the fractions: $\frac{4}{3x} + \frac{2}{5y}$
• Add the fractions: $\frac{1}{2x} + \frac{3}{4y}$

References

• [1] "Adding Fractions with Different Variables" by Math Open Reference
• [2] "Fractions with Variables" by Khan Academy
• [3] "Adding and Subtracting Fractions with Variables" by Purplemath