What Is The Product Of $\frac{4f}{5g}$ And $\frac{f}{g}$ When $g \neq 0$?A. $\frac{16}{25}$ B. $\frac{4}{5}$ C. $\frac{4f^2}{5g^2}$ D. $\frac{16f^2}{25g^2}$

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Understanding the Problem

When dealing with fractions, it's essential to understand the rules of multiplication and division. In this problem, we are given two fractions: $\frac{4f}{5g}$ and $\frac{f}{g}$. We need to find the product of these two fractions when $g \neq 0$.

The Rules of Multiplication

To find the product of two fractions, we need to multiply the numerators (the numbers on top) and multiply the denominators (the numbers on the bottom). This is a fundamental rule in mathematics, and it applies to all fractions.

Applying the Rules of Multiplication

Let's apply the rules of multiplication to the given fractions. We have:

$\frac{4f}{5g} \cdot \frac{f}{g}$

To find the product, we multiply the numerators and multiply the denominators:

$\frac{(4f) \cdot (f)}{(5g) \cdot (g)}$

Simplifying the Expression

Now, let's simplify the expression by multiplying the numbers and combining like terms:

$\frac{4f^2}{5g^2}$

The Final Answer

The product of $\frac{4f}{5g}$ and $\frac{f}{g}$ when $g \neq 0$ is $\frac{4f^2}{5g^2}$.

Understanding the Options

Now, let's look at the options provided:

A. $\frac{16}{25}$ B. $\frac{4}{5}$ C. $\frac{4f^2}{5g^2}$ D. $\frac{16f^2}{25g^2}$

Eliminating Incorrect Options

We can eliminate options A and B because they do not match the product we found. Option A is a fraction with no variables, and option B is a fraction with a different numerator and denominator.

Choosing the Correct Option

The correct option is C. $\frac{4f^2}{5g^2}$. This option matches the product we found, and it is the only option that includes the variables $f$ and $g$.

Conclusion

In conclusion, the product of $\frac{4f}{5g}$ and $\frac{f}{g}$ when $g \neq 0$ is $\frac{4f^2}{5g^2}$. This is a fundamental rule in mathematics, and it applies to all fractions.

Frequently Asked Questions

• What is the product of two fractions?
• How do you multiply fractions?
• What is the rule for multiplying fractions?

Answers

• The product of two fractions is found by multiplying the numerators and multiplying the denominators.
• To multiply fractions, you multiply the numerators and multiply the denominators.
• The rule for multiplying fractions is to multiply the numerators and multiply the denominators.

Final Thoughts

In this article, we discussed the product of two fractions when $g \neq 0$. We applied the rules of multiplication and simplified the expression to find the final answer. We also eliminated incorrect options and chose the correct option. This article provides a clear understanding of the product of two fractions and the rules of multiplication.

Understanding the Basics

When dealing with fractions, it's essential to understand the rules of multiplication and division. In this article, we'll answer some frequently asked questions about the product of two fractions.

Q: What is the product of two fractions?

A: The product of two fractions is found by multiplying the numerators (the numbers on top) and multiplying the denominators (the numbers on the bottom).

Q: How do you multiply fractions?

A: To multiply fractions, you multiply the numerators and multiply the denominators. For example, if you have $\frac{a}{b} \cdot \frac{c}{d}$, the product is $\frac{ac}{bd}$.

Q: What is the rule for multiplying fractions?

A: The rule for multiplying fractions is to multiply the numerators and multiply the denominators. This is a fundamental rule in mathematics, and it applies to all fractions.

Q: Can you give an example of multiplying fractions?

A: Yes, let's consider the example $\frac{2}{3} \cdot \frac{4}{5}$. To find the product, we multiply the numerators and multiply the denominators:

$\frac{(2) \cdot (4)}{(3) \cdot (5)} = \frac{8}{15}$

Q: What if the fractions have variables?

A: If the fractions have variables, we can still multiply them by multiplying the numerators and multiplying the denominators. For example, if we have $\frac{4f}{5g} \cdot \frac{f}{g}$, the product is $\frac{(4f) \cdot (f)}{(5g) \cdot (g)} = \frac{4f^2}{5g^2}$.

Q: Can you give an example of multiplying fractions with variables?

A: Yes, let's consider the example $\frac{4f}{5g} \cdot \frac{f}{g}$. To find the product, we multiply the numerators and multiply the denominators:

$\frac{(4f) \cdot (f)}{(5g) \cdot (g)} = \frac{4f^2}{5g^2}$

Q: What if the fractions have different signs?

A: If the fractions have different signs, we can still multiply them by multiplying the numerators and multiplying the denominators. However, we need to remember that a negative times a negative is a positive, and a negative times a positive is a negative.

Q: Can you give an example of multiplying fractions with different signs?

A: Yes, let's consider the example $\frac{2}{3} \cdot \frac{-4}{5}$. To find the product, we multiply the numerators and multiply the denominators:

$\frac{(2) \cdot (-4)}{(3) \cdot (5)} = \frac{-8}{15}$

Q: What if the fractions have a common factor?

A: If the fractions have a common factor, we can simplify the product by canceling out the common factor. For example, if we have $\frac{4}{6} \cdot \frac{3}{4}$, we can simplify the product by canceling out the common factor of 2:

$\frac{4}{6} \cdot \frac{3}{4} = \frac{1}{1} \cdot \frac{3}{2} = \frac{3}{2}$

Q: Can you give an example of multiplying fractions with a common factor?

A: Yes, let's consider the example $\frac{4}{6} \cdot \frac{3}{4}$. To simplify the product, we can cancel out the common factor of 2:

$\frac{4}{6} \cdot \frac{3}{4} = \frac{1}{1} \cdot \frac{3}{2} = \frac{3}{2}$

Conclusion

In this article, we answered some frequently asked questions about the product of two fractions. We discussed the rules of multiplication, how to multiply fractions, and how to simplify the product. We also provided examples of multiplying fractions with variables, different signs, and common factors.