Calculate The Product:$\[ 5 \frac{2}{9} \times 2 \frac{3}{7} = \\]

Introduction

In mathematics, mixed numbers are a combination of a whole number and a fraction. When we need to multiply mixed numbers, it can be a bit challenging, but with the right approach, it becomes manageable. In this article, we will explore how to calculate the product of two mixed numbers, $5 \frac{2}{9}$ and $2 \frac{3}{7}$.

Understanding Mixed Numbers

Before we dive into the calculation, let's understand what mixed numbers are. A mixed number is a combination of a whole number and a fraction. It is written in the form $a \frac{b}{c}$, where $a$ is the whole number, $b$ is the numerator, and $c$ is the denominator. For example, $5 \frac{2}{9}$ is a mixed number where $5$ is the whole number, $2$ is the numerator, and $9$ is the denominator.

Converting Mixed Numbers to Improper Fractions

To multiply mixed numbers, we need to convert them to improper fractions first. An improper fraction is a fraction where the numerator is greater than or equal to the denominator. To convert a mixed number to an improper fraction, we multiply the whole number by the denominator and add the numerator. The result is the new numerator, and the denominator remains the same.

For example, let's convert $5 \frac{2}{9}$ to an improper fraction:

1. Multiply the whole number by the denominator: $5 \times 9 = 45$
2. Add the numerator: $45 + 2 = 47$
3. The new numerator is $47$, and the denominator remains $9$. Therefore, $5 \frac{2}{9}$ is equal to $\frac{47}{9}$.

Similarly, let's convert $2 \frac{3}{7}$ to an improper fraction:

1. Multiply the whole number by the denominator: $2 \times 7 = 14$
2. Add the numerator: $14 + 3 = 17$
3. The new numerator is $17$, and the denominator remains $7$. Therefore, $2 \frac{3}{7}$ is equal to $\frac{17}{7}$.

Multiplying Improper Fractions

Now that we have converted both mixed numbers to improper fractions, we can multiply them. To multiply improper fractions, we simply multiply the numerators and multiply the denominators.

Let's multiply $\frac{47}{9}$ and $\frac{17}{7}$:

1. Multiply the numerators: $47 \times 17 = 799$
2. Multiply the denominators: $9 \times 7 = 63$
3. The product of the two improper fractions is $\frac{799}{63}$.

Converting the Product Back to a Mixed Number

Now that we have the product of the two improper fractions, we can convert it back to a mixed number. To do this, we divide the numerator by the denominator and write the remainder as the new numerator.

Let's convert $\frac{799}{63}$ to a mixed number:

1. Divide the numerator by the denominator: $799 \div 63 = 12$ with a remainder of $47$
2. The new numerator is $47$, and the denominator remains $63$. Therefore, the product of $5 \frac{2}{9}$ and $2 \frac{3}{7}$ is $12 \frac{47}{63}$.

Conclusion

Calculating the product of mixed numbers can be a bit challenging, but with the right approach, it becomes manageable. By converting mixed numbers to improper fractions, multiplying the improper fractions, and converting the product back to a mixed number, we can find the product of two mixed numbers. In this article, we have seen how to calculate the product of $5 \frac{2}{9}$ and $2 \frac{3}{7}$.

Final Answer

Q: What is the first step in calculating the product of two mixed numbers?

A: The first step in calculating the product of two mixed numbers is to convert them to improper fractions. This involves multiplying the whole number by the denominator and adding the numerator to get the new numerator, while keeping the denominator the same.

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

A: To convert a mixed number to an improper fraction, follow these steps:

1. Multiply the whole number by the denominator.
2. Add the numerator to the result.
3. The new numerator is the result, and the denominator remains the same.

For example, to convert $5 \frac{2}{9}$ to an improper fraction, you would multiply $5$ by $9$ to get $45$, then add $2$ to get $47$. Therefore, $5 \frac{2}{9}$ is equal to $\frac{47}{9}$.

Q: Can I multiply mixed numbers directly without converting them to improper fractions?

A: No, you cannot multiply mixed numbers directly without converting them to improper fractions. Mixed numbers have both a whole number and a fraction part, which makes it difficult to multiply them directly. Converting them to improper fractions simplifies the multiplication process.

Q: What is the next step after converting the mixed numbers to improper fractions?

A: After converting the mixed numbers to improper fractions, the next step is to multiply the numerators and multiply the denominators. This will give you the product of the two improper fractions.

Q: How do I multiply improper fractions?

A: To multiply improper fractions, follow these steps:

1. Multiply the numerators.
2. Multiply the denominators.
3. The product of the two improper fractions is the result.

For example, to multiply $\frac{47}{9}$ and $\frac{17}{7}$, you would multiply $47$ by $17$ to get $799$, then multiply $9$ by $7$ to get $63$. Therefore, the product of $\frac{47}{9}$ and $\frac{17}{7}$ is $\frac{799}{63}$.

Q: Can I convert the product of improper fractions back to a mixed number?

A: Yes, you can convert the product of improper fractions back to a mixed number. To do this, divide the numerator by the denominator and write the remainder as the new numerator.

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

A: To convert an improper fraction back to a mixed number, follow these steps:

1. Divide the numerator by the denominator.
2. Write the remainder as the new numerator.
3. The denominator remains the same.

For example, to convert $\frac{799}{63}$ to a mixed number, you would divide $799$ by $63$ to get $12$ with a remainder of $47$. Therefore, the product of $5 \frac{2}{9}$ and $2 \frac{3}{7}$ is $12 \frac{47}{63}$.

Q: What are some common mistakes to avoid when calculating the product of mixed numbers?

A: Some common mistakes to avoid when calculating the product of mixed numbers include:

• Not converting mixed numbers to improper fractions before multiplying
• Multiplying the whole numbers and fractions separately instead of multiplying the improper fractions
• Not converting the product of improper fractions back to a mixed number

By avoiding these common mistakes, you can ensure accurate results when calculating the product of mixed numbers.