Choose All Equations That Are True.A. 5 6 × 4 2 3 = 3 8 9 \frac{5}{6} \times 4 \frac{2}{3} = 3 \frac{8}{9} 6 5 ​ × 4 3 2 ​ = 3 9 8 ​ B. 1 1 10 × 8 3 5 = 9 1 2 1 \frac{1}{10} \times 8 \frac{3}{5} = 9 \frac{1}{2} 1 10 1 ​ × 8 5 3 ​ = 9 2 1 ​ C. 3 2 9 × 2 1 6 = 6 53 54 3 \frac{2}{9} \times 2 \frac{1}{6} = 6 \frac{53}{54} 3 9 2 ​ × 2 6 1 ​ = 6 54 53 ​ D. $2 \frac{3}{7}

Introduction

In mathematics, mixed numbers are a combination of a whole number and a fraction. When we multiply mixed numbers, we need to follow a specific procedure to ensure that the result is accurate. In this article, we will explore four mixed number multiplication equations and determine which ones are true.

Understanding Mixed Number Multiplication

To multiply mixed numbers, we need to convert them into improper fractions first. An improper fraction is a fraction where the numerator is greater than or equal to the denominator. We can convert a mixed number to an improper fraction by multiplying the whole number part by the denominator and then adding the numerator.

For example, let's convert the mixed number $4 \frac{2}{3}$ to an improper fraction:

$$4 \frac{2}{3} = \frac{(4 \times 3) + 2}{3} = \frac{12 + 2}{3} = \frac{14}{3}$$

Analyzing Equation A

Let's analyze the first equation:

$$\frac{5}{6} \times 4 \frac{2}{3} = 3 \frac{8}{9}$$

To solve this equation, we need to convert the mixed number $4 \frac{2}{3}$ to an improper fraction:

$$4 \frac{2}{3} = \frac{14}{3}$$

Now, we can multiply the two fractions:

$$\frac{5}{6} \times \frac{14}{3} = \frac{5 \times 14}{6 \times 3} = \frac{70}{18}$$

To simplify the fraction, we can divide both the numerator and the denominator by their greatest common divisor, which is 2:

$$\frac{70}{18} = \frac{35}{9}$$

Now, let's convert the mixed number $3 \frac{8}{9}$ to an improper fraction:

$$3 \frac{8}{9} = \frac{(3 \times 9) + 8}{9} = \frac{27 + 8}{9} = \frac{35}{9}$$

Since both sides of the equation are equal, we can conclude that Equation A is true.

Analyzing Equation B

Let's analyze the second equation:

$$1 \frac{1}{10} \times 8 \frac{3}{5} = 9 \frac{1}{2}$$

To solve this equation, we need to convert the mixed numbers to improper fractions:

$$1 \frac{1}{10} = \frac{11}{10}$$

$$8 \frac{3}{5} = \frac{(8 \times 5) + 3}{5} = \frac{40 + 3}{5} = \frac{43}{5}$$

Now, we can multiply the two fractions:

$$\frac{11}{10} \times \frac{43}{5} = \frac{11 \times 43}{10 \times 5} = \frac{473}{50}$$

To simplify the fraction, we can divide both the numerator and the denominator by their greatest common divisor, which is 1:

$$\frac{473}{50} = \frac{473}{50}$$

Now, let's convert the mixed number $9 \frac{1}{2}$ to an improper fraction:

$$9 \frac{1}{2} = \frac{(9 \times 2) + 1}{2} = \frac{18 + 1}{2} = \frac{19}{2}$$

Since the two sides of the equation are not equal, we can conclude that Equation B is false.

Analyzing Equation C

Let's analyze the third equation:

$$3 \frac{2}{9} \times 2 \frac{1}{6} = 6 \frac{53}{54}$$

To solve this equation, we need to convert the mixed numbers to improper fractions:

$$3 \frac{2}{9} = \frac{(3 \times 9) + 2}{9} = \frac{27 + 2}{9} = \frac{29}{9}$$

$$2 \frac{1}{6} = \frac{(2 \times 6) + 1}{6} = \frac{12 + 1}{6} = \frac{13}{6}$$

Now, we can multiply the two fractions:

$$\frac{29}{9} \times \frac{13}{6} = \frac{29 \times 13}{9 \times 6} = \frac{377}{54}$$

To simplify the fraction, we can divide both the numerator and the denominator by their greatest common divisor, which is 1:

$$\frac{377}{54} = \frac{377}{54}$$

Now, let's convert the mixed number $6 \frac{53}{54}$ to an improper fraction:

$$6 \frac{53}{54} = \frac{(6 \times 54) + 53}{54} = \frac{324 + 53}{54} = \frac{377}{54}$$

Since both sides of the equation are equal, we can conclude that Equation C is true.

Analyzing Equation D

Let's analyze the fourth equation:

$$2 \frac{3}{7} \times 5 \frac{2}{3} = 10 \frac{53}{63}$$

To solve this equation, we need to convert the mixed numbers to improper fractions:

$$2 \frac{3}{7} = \frac{(2 \times 7) + 3}{7} = \frac{14 + 3}{7} = \frac{17}{7}$$

$$5 \frac{2}{3} = \frac{(5 \times 3) + 2}{3} = \frac{15 + 2}{3} = \frac{17}{3}$$

Now, we can multiply the two fractions:

$$\frac{17}{7} \times \frac{17}{3} = \frac{17 \times 17}{7 \times 3} = \frac{289}{21}$$

To simplify the fraction, we can divide both the numerator and the denominator by their greatest common divisor, which is 1:

$$\frac{289}{21} = \frac{289}{21}$$

Now, let's convert the mixed number $10 \frac{53}{63}$ to an improper fraction:

$$10 \frac{53}{63} = \frac{(10 \times 63) + 53}{63} = \frac{630 + 53}{63} = \frac{683}{63}$$

Since the two sides of the equation are not equal, we can conclude that Equation D is false.

Conclusion

Introduction

In our previous article, we explored four mixed number multiplication equations and determined which ones are true. In this article, we will provide a Q&A guide to help you better understand mixed number multiplication and how to solve these types of equations.

Q: What is a mixed number?

A: A mixed number is a combination of a whole number and a fraction. For example, $4 \frac{2}{3}$ is a mixed number where 4 is the whole number part and $\frac{2}{3}$ is the fractional part.

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

A: To convert a mixed number to an improper fraction, you need to multiply the whole number part by the denominator and then add the numerator. For example, to convert $4 \frac{2}{3}$ to an improper fraction, you would multiply 4 by 3 and add 2, resulting in $\frac{14}{3}$.

Q: How do I multiply mixed numbers?

A: To multiply mixed numbers, you need to convert them to improper fractions first. Then, you can multiply the numerators and denominators separately and simplify the resulting 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. For example, the GCD of 12 and 18 is 6.

Q: How do I simplify a fraction?

A: To simplify a fraction, you need to divide both the numerator and the denominator by their greatest common divisor (GCD). For example, to simplify $\frac{12}{18}$, you would divide both 12 and 18 by 6, resulting in $\frac{2}{3}$.

Q: What are some common mistakes to avoid when multiplying mixed numbers?

A: Some common mistakes to avoid when multiplying mixed numbers include:

• Not converting the mixed numbers to improper fractions before multiplying
• Not multiplying the numerators and denominators separately
• Not simplifying the resulting fraction
• Not checking for common factors between the numerator and denominator

Q: How can I practice multiplying mixed numbers?

A: You can practice multiplying mixed numbers by working through examples and exercises. You can also use online resources or math software to help you practice and check your work.

Q: What are some real-world applications of mixed number multiplication?

A: Mixed number multiplication has many real-world applications, including:

• Cooking and recipe scaling
• Building and construction
• Finance and accounting
• Science and engineering

Conclusion

In this article, we provided a Q&A guide to help you better understand mixed number multiplication and how to solve these types of equations. By following the steps outlined in this guide, you can become more confident and proficient in multiplying mixed numbers. Remember to practice regularly and check your work to ensure accuracy.

Additional Resources

• Khan Academy: Mixed Numbers and Improper Fractions
• Mathway: Mixed Number Multiplication
• IXL: Mixed Number Multiplication

We hope this article has been helpful in your understanding of mixed number multiplication. If you have any further questions or need additional help, please don't hesitate to ask.