Multiply: ${ 3 \frac{5}{6} \times 2 \frac{1}{2} }$

Introduction to Multiplying Mixed Numbers

Multiplying mixed numbers is a fundamental concept in mathematics that involves multiplying two or more numbers that have both a whole number part and a fractional part. In this article, we will focus on multiplying the mixed numbers $3 \frac{5}{6}$ and $2 \frac{1}{2}$. To begin with, let's understand the concept of mixed numbers and how to multiply them.

What are Mixed Numbers?

A mixed number is a combination of a whole number and a fraction. It is written in the form of $a \frac{b}{c}$, where $a$ is the whole number part, $b$ is the numerator of the fraction, and $c$ is the denominator of the fraction. For example, $3 \frac{5}{6}$ is a mixed number where $3$ is the whole number part and $\frac{5}{6}$ is the fractional part.

How to Multiply Mixed Numbers

To multiply mixed numbers, we need to follow a step-by-step process. Here's how to do it:

1. Convert the mixed numbers to improper fractions: The first step is to convert the mixed numbers to improper fractions. To do this, we multiply the whole number part by the denominator and then add the numerator. For example, $3 \frac{5}{6}$ can be converted to an improper fraction as follows:

$$3 \frac{5}{6} = \frac{(3 \times 6) + 5}{6} = \frac{18 + 5}{6} = \frac{23}{6}$$

Similarly, $2 \frac{1}{2}$ can be converted to an improper fraction as follows:

$$2 \frac{1}{2} = \frac{(2 \times 2) + 1}{2} = \frac{4 + 1}{2} = \frac{5}{2}$$

2. Multiply the improper fractions: Once we have converted the mixed numbers to improper fractions, we can multiply them together. To do this, we multiply the numerators together and the denominators together.

$$\frac{23}{6} \times \frac{5}{2} = \frac{23 \times 5}{6 \times 2} = \frac{115}{12}$$

3. Simplify the result: Finally, we need to simplify the result by dividing the numerator and denominator by their greatest common divisor (GCD). In this case, the GCD of $115$ and $12$ is $1$, so the result is already simplified.

$$\frac{115}{12} = 9 \frac{7}{12}$$

Therefore, the product of $3 \frac{5}{6}$ and $2 \frac{1}{2}$ is $9 \frac{7}{12}$.

Real-World Applications of Multiplying Mixed Numbers

Multiplying mixed numbers has many real-world applications in fields such as engineering, architecture, and finance. For example, in engineering, mixed numbers are used to represent measurements of length, area, and volume. In architecture, mixed numbers are used to represent measurements of building materials, such as the number of bricks or tiles required for a particular project. In finance, mixed numbers are used to represent interest rates and investment returns.

Example 1: Calculating the Area of a Rectangle

Suppose we want to calculate the area of a rectangle that measures $3 \frac{5}{6}$ feet in length and $2 \frac{1}{2}$ feet in width. To do this, we need to multiply the length and width together.

$$\text{Area} = \text{length} \times \text{width} = 3 \frac{5}{6} \times 2 \frac{1}{2} = 9 \frac{7}{12} \text{ square feet}$$

Example 2: Calculating the Volume of a Box

Suppose we want to calculate the volume of a box that measures $3 \frac{5}{6}$ feet in length, $2 \frac{1}{2}$ feet in width, and $1 \frac{1}{3}$ feet in height. To do this, we need to multiply the length, width, and height together.

$$\text{Volume} = \text{length} \times \text{width} \times \text{height} = 3 \frac{5}{6} \times 2 \frac{1}{2} \times 1 \frac{1}{3} = 12 \frac{5}{9} \text{ cubic feet}$$

Conclusion

In conclusion, multiplying mixed numbers is a fundamental concept in mathematics that involves multiplying two or more numbers that have both a whole number part and a fractional part. To multiply mixed numbers, we need to convert them to improper fractions, multiply the improper fractions together, and simplify the result. Multiplying mixed numbers has many real-world applications in fields such as engineering, architecture, and finance. By understanding how to multiply mixed numbers, we can solve a wide range of problems in these fields and beyond.

Frequently Asked Questions

Q: What is the product of $3 \frac{5}{6}$ and $2 \frac{1}{2}$?

A: The product of $3 \frac{5}{6}$ and $2 \frac{1}{2}$ is $9 \frac{7}{12}$.

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

A: To convert a mixed number to an improper fraction, multiply the whole number part by the denominator and then add the numerator.

Q: How do I multiply improper fractions?

A: To multiply improper fractions, multiply the numerators together and the denominators together.

Q: How do I simplify a result?

A: To simplify a result, divide the numerator and denominator by their greatest common divisor (GCD).

References

• [1] "Multiplying Mixed Numbers" by Math Open Reference
• [2] "Mixed Numbers" by Math Is Fun
• [3] "Improper Fractions" by Khan Academy
  

Q&A: Multiplying Mixed Numbers

In this article, we will continue to explore the concept of multiplying mixed numbers. We will answer some frequently asked questions about multiplying mixed numbers and provide additional examples to help you understand the concept.

Q: What is the product of $3 \frac{5}{6}$ and $2 \frac{1}{2}$?

A: The product of $3 \frac{5}{6}$ and $2 \frac{1}{2}$ is $9 \frac{7}{12}$.

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

A: To convert a mixed number to an improper fraction, multiply the whole number part by the denominator and then add the numerator. For example, to convert $3 \frac{5}{6}$ to an improper fraction, we multiply the whole number part by the denominator and then add the numerator:

$$3 \frac{5}{6} = \frac{(3 \times 6) + 5}{6} = \frac{18 + 5}{6} = \frac{23}{6}$$

Q: How do I multiply improper fractions?

A: To multiply improper fractions, multiply the numerators together and the denominators together. For example, to multiply $\frac{23}{6}$ and $\frac{5}{2}$, we multiply the numerators together and the denominators together:

$$\frac{23}{6} \times \frac{5}{2} = \frac{23 \times 5}{6 \times 2} = \frac{115}{12}$$

Q: How do I simplify a result?

A: To simplify a result, divide the numerator and denominator by their greatest common divisor (GCD). In this case, the GCD of $115$ and $12$ is $1$, so the result is already simplified:

$$\frac{115}{12} = 9 \frac{7}{12}$$

Q: What are some real-world applications of multiplying mixed numbers?

A: Multiplying mixed numbers has many real-world applications in fields such as engineering, architecture, and finance. For example, in engineering, mixed numbers are used to represent measurements of length, area, and volume. In architecture, mixed numbers are used to represent measurements of building materials, such as the number of bricks or tiles required for a particular project. In finance, mixed numbers are used to represent interest rates and investment returns.

Q: How do I calculate the area of a rectangle using mixed numbers?

A: To calculate the area of a rectangle using mixed numbers, multiply the length and width together. For example, if the length of a rectangle is $3 \frac{5}{6}$ feet and the width is $2 \frac{1}{2}$ feet, the area would be:

$$\text{Area} = \text{length} \times \text{width} = 3 \frac{5}{6} \times 2 \frac{1}{2} = 9 \frac{7}{12} \text{ square feet}$$

Q: How do I calculate the volume of a box using mixed numbers?

A: To calculate the volume of a box using mixed numbers, multiply the length, width, and height together. For example, if the length of a box is $3 \frac{5}{6}$ feet, the width is $2 \frac{1}{2}$ feet, and the height is $1 \frac{1}{3}$ feet, the volume would be:

$$\text{Volume} = \text{length} \times \text{width} \times \text{height} = 3 \frac{5}{6} \times 2 \frac{1}{2} \times 1 \frac{1}{3} = 12 \frac{5}{9} \text{ cubic feet}$$

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 together correctly
• Not simplifying the result by dividing the numerator and denominator by their GCD

Q: How can I practice multiplying mixed numbers?

A: You can practice multiplying mixed numbers by using online resources, such as math worksheets and online calculators. You can also practice by working through examples and exercises in a math textbook or online resource.

Q: What are some additional resources for learning about multiplying mixed numbers?

A: Some additional resources for learning about multiplying mixed numbers include:

• Math Open Reference: A free online math reference book that includes information on multiplying mixed numbers.
• Math Is Fun: A website that provides information and examples on multiplying mixed numbers.
• Khan Academy: A website that provides video lessons and practice exercises on multiplying mixed numbers.

Conclusion

In conclusion, multiplying mixed numbers is a fundamental concept in mathematics that involves multiplying two or more numbers that have both a whole number part and a fractional part. By understanding how to multiply mixed numbers, you can solve a wide range of problems in fields such as engineering, architecture, and finance. We hope that this article has provided you with a better understanding of multiplying mixed numbers and has helped you to practice and improve your skills.