Multiply The Fractions And Write Your Answer In Simplified Form:$\[ \frac{5}{7} \times \frac{10}{3} \\]

Introduction to Multiplying Fractions

Multiplying fractions is a fundamental concept in mathematics that involves multiplying two or more fractions together to obtain a product. In this article, we will focus on multiplying two fractions, $\frac{5}{7}$ and $\frac{10}{3}$, and write the answer in simplified form. To begin with, let's understand the basics of multiplying fractions.

What are Fractions?

A fraction is a way of expressing a part of a whole as a ratio of two numbers. It consists of a numerator and a denominator, which are separated by a horizontal line or a slash. The numerator represents the number of equal parts being considered, while the denominator represents the total number of parts in the whole.

Multiplying Fractions: A Step-by-Step Guide

To multiply fractions, we simply multiply the numerators together and multiply the denominators together. This is a straightforward process that can be applied to any two fractions. Let's apply this rule to the given fractions, $\frac{5}{7}$ and $\frac{10}{3}$.

Multiplying the Numerators and Denominators

To multiply the fractions, we multiply the numerators together and multiply the denominators together.

$\frac{5}{7} \times \frac{10}{3} = \frac{5 \times 10}{7 \times 3}$

Now, let's calculate the product of the numerators and the denominators.

Calculating the Product of the Numerators

The product of the numerators is $5 \times 10 = 50$.

Calculating the Product of the Denominators

The product of the denominators is $7 \times 3 = 21$.

Writing the Answer in Simplified Form

Now that we have the product of the numerators and the denominators, we can write the answer in simplified form.

$\frac{5 \times 10}{7 \times 3} = \frac{50}{21}$

However, we can simplify this fraction further by dividing both the numerator and the denominator by their greatest common divisor (GCD).

Finding the Greatest Common Divisor (GCD)

To find the GCD of 50 and 21, we can use the Euclidean algorithm or list the factors of each number.

Factors of 50

The factors of 50 are 1, 2, 5, 10, 25, and 50.

Factors of 21

The factors of 21 are 1, 3, 7, and 21.

Simplifying the Fraction

Now that we have the factors of both numbers, we can find the GCD by identifying the largest factor that is common to both numbers.

The GCD of 50 and 21 is 1.

Since the GCD is 1, we cannot simplify the fraction further. Therefore, the simplified form of the product of the fractions is $\frac{50}{21}$.

Conclusion

In this article, we learned how to multiply fractions and write the answer in simplified form. We applied the rule of multiplying the numerators together and multiplying the denominators together to obtain the product of the fractions. We then simplified the fraction by dividing both the numerator and the denominator by their greatest common divisor. The final answer is $\frac{50}{21}$.

Frequently Asked Questions

• What is the product of the fractions $\frac{5}{7}$ and $\frac{10}{3}$?
  • The product of the fractions is $\frac{50}{21}$.
• Can the fraction $\frac{50}{21}$ be simplified further?
  • No, the fraction $\frac{50}{21}$ cannot be simplified further since the GCD of 50 and 21 is 1.

Final Answer

The final answer is $\boxed{\frac{50}{21}}$.

Introduction to Multiplying Fractions Q&A

In our previous article, we discussed how to multiply fractions and write the answer in simplified form. However, we understand that some readers may still have questions or doubts about the process. In this article, we will address some of the frequently asked questions (FAQs) related to multiplying fractions.

Q&A: Multiplying Fractions

Q: What is the product of the fractions $\frac{5}{7}$ and $\frac{10}{3}$?

A: The product of the fractions is $\frac{50}{21}$.

Q: Can the fraction $\frac{50}{21}$ be simplified further?

A: No, the fraction $\frac{50}{21}$ cannot be simplified further since the GCD of 50 and 21 is 1.

Q: How do I multiply fractions with different denominators?

A: To multiply fractions with different denominators, you simply multiply the numerators together and multiply the denominators together. For example, to multiply $\frac{5}{7}$ and $\frac{10}{3}$, you would multiply the numerators together to get $5 \times 10 = 50$, and multiply the denominators together to get $7 \times 3 = 21$.

Q: What is the rule for multiplying fractions?

A: The rule for multiplying fractions is to multiply the numerators together and multiply the denominators together. This is a straightforward process that can be applied to any two fractions.

Q: Can I multiply a fraction by a whole number?

A: Yes, you can multiply a fraction by a whole number. To do this, you simply multiply the numerator of the fraction by the whole number. For example, to multiply $\frac{5}{7}$ by 3, you would multiply the numerator by 3 to get $\frac{15}{7}$.

Q: How do I simplify a fraction after multiplying it by a whole number?

A: To simplify a fraction after multiplying it by a whole number, you need to find the greatest common divisor (GCD) of the numerator and the denominator. You can then divide both the numerator and the denominator by the GCD to simplify the fraction.

Q: What is the greatest common divisor (GCD)?

A: The greatest common divisor (GCD) is the largest number that divides both the numerator and the denominator of a fraction without leaving a remainder.

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

A: There are several ways to find the GCD of two numbers. One way is to list the factors of each number and identify the largest factor that is common to both numbers. Another way is to use the Euclidean algorithm.

Q: Can I use a calculator to find the GCD of two numbers?

A: Yes, you can use a calculator to find the GCD of two numbers. Most calculators have a built-in function for finding the GCD.

Conclusion

In this article, we addressed some of the frequently asked questions related to multiplying fractions. We provided step-by-step explanations and examples to help readers understand the process. We also discussed the importance of simplifying fractions after multiplying them by whole numbers. By following these steps and using the right tools, readers can confidently multiply fractions and write their answers in simplified form.

Final Answer

The final answer is $\boxed{\frac{50}{21}}$.