Express The Following As A Mixed Number In Simplest Form.${ 3 \frac{4}{9} \times 4 \frac{1}{3} = }$

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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 simplify the result. In this article, we will explore how to express the product of two mixed numbers in simplest form.

Understanding Mixed Numbers

A mixed number is a combination of a whole number and a fraction. It is written in the form of a + b/c, where a is the whole number and b/c is the fraction. For example, 3 4/9 is a mixed number where 3 is the whole number and 4/9 is the fraction.

Multiplying Mixed Numbers

To multiply mixed numbers, we need to follow the same procedure as multiplying fractions. We multiply the numerators (the numbers on top) and the denominators (the numbers on the bottom) separately. Then, we simplify the result to its simplest form.

Step 1: Convert 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. Then, we write the result as a fraction with the same denominator.

For example, let's convert 3 4/9 to an improper fraction:

3 4/9 = (3 × 9) + 4/9 = 27 + 4/9 = 31/9

Similarly, let's convert 4 1/3 to an improper fraction:

4 1/3 = (4 × 3) + 1/3 = 12 + 1/3 = 37/3

Step 2: Multiply the Numerators and Denominators

Now that we have converted the mixed numbers to improper fractions, we can multiply them. We multiply the numerators (31 and 37) and the denominators (9 and 3) separately.

(31 × 37) = 1147 (9 × 3) = 27

Step 3: Simplify the Result

Now that we have multiplied the numerators and denominators, we need to simplify the result. To simplify a fraction, we divide the numerator by the denominator and write the result as a fraction.

1147 ÷ 27 = 42 7/27

Therefore, the product of 3 4/9 and 4 1/3 is 42 7/27.

Conclusion

In this article, we have explored how to express the product of two mixed numbers in simplest form. We have followed a step-by-step procedure to convert the mixed numbers to improper fractions, multiply the numerators and denominators, and simplify the result. By following these steps, we can simplify the product of mixed numbers and express it in its simplest form.

Example Problems

  1. Express the product of 2 1/4 and 3 2/5 as a mixed number in simplest form.
  2. Express the product of 5 3/8 and 2 1/2 as a mixed number in simplest form.
  3. Express the product of 4 2/3 and 1 3/4 as a mixed number in simplest form.

Solutions

  1. To express the product of 2 1/4 and 3 2/5 as a mixed number in simplest form, we need to follow the same procedure as before.

First, we convert the mixed numbers to improper fractions:

2 1/4 = (2 × 4) + 1/4 = 9 + 1/4 = 37/4

3 2/5 = (3 × 5) + 2/5 = 17 + 2/5 = 83/5

Next, we multiply the numerators and denominators:

(37 × 83) = 3061 (4 × 5) = 20

Then, we simplify the result:

3061 ÷ 20 = 153 1/20

Therefore, the product of 2 1/4 and 3 2/5 is 153 1/20.

  1. To express the product of 5 3/8 and 2 1/2 as a mixed number in simplest form, we need to follow the same procedure as before.

First, we convert the mixed numbers to improper fractions:

5 3/8 = (5 × 8) + 3/8 = 41 + 3/8 = 329/8

2 1/2 = (2 × 2) + 1/2 = 5 + 1/2 = 11/2

Next, we multiply the numerators and denominators:

(329 × 11) = 3619 (8 × 2) = 16

Then, we simplify the result:

3619 ÷ 16 = 226 3/16

Therefore, the product of 5 3/8 and 2 1/2 is 226 3/16.

  1. To express the product of 4 2/3 and 1 3/4 as a mixed number in simplest form, we need to follow the same procedure as before.

First, we convert the mixed numbers to improper fractions:

4 2/3 = (4 × 3) + 2/3 = 14 + 2/3 = 44/3

1 3/4 = (1 × 4) + 3/4 = 7 + 3/4 = 31/4

Next, we multiply the numerators and denominators:

(44 × 31) = 1364 (3 × 4) = 12

Then, we simplify the result:

1364 ÷ 12 = 113 4/12 = 113 1/3

Q: What is the first step in multiplying mixed numbers?

A: The first step in multiplying mixed numbers is to convert them to improper fractions. This involves multiplying the whole number by the denominator and adding the numerator.

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

A: To convert a mixed number to an improper fraction, you multiply the whole number by the denominator and add the numerator. For example, to convert 3 4/9 to an improper fraction, you would multiply 3 by 9 and add 4, resulting in 31/9.

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

A: After converting the mixed numbers to improper fractions, you multiply the numerators and denominators separately. This involves multiplying the numerators (the numbers on top) and the denominators (the numbers on the bottom) separately.

Q: How do I simplify the result of multiplying mixed numbers?

A: To simplify the result of multiplying mixed numbers, you divide the numerator by the denominator and write the result as a fraction. If the result is a whole number, you can write it as a whole number. If the result is a fraction, you can simplify it by dividing the numerator and denominator by their greatest common divisor.

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 result after multiplying
  • Not following the order of operations (PEMDAS)

Q: Can I use a calculator to multiply mixed numbers?

A: Yes, you can use a calculator to multiply mixed numbers. However, it's often more helpful to practice multiplying mixed numbers by hand to develop your understanding of the concept.

Q: How do I apply the concept of multiplying mixed numbers to real-world problems?

A: The concept of multiplying mixed numbers can be applied to a variety of real-world problems, such as:

  • Calculating the area of a rectangle with mixed number dimensions
  • Finding the volume of a rectangular prism with mixed number dimensions
  • Determining the cost of a mixed number quantity of items

Q: What are some tips for mastering the concept of multiplying mixed numbers?

A: Some tips for mastering the concept of multiplying mixed numbers include:

  • Practicing multiplying mixed numbers by hand to develop your understanding of the concept
  • Using visual aids, such as diagrams or charts, to help you understand the concept
  • Breaking down complex problems into simpler steps
  • Checking your work to ensure accuracy

Q: Can I use the concept of multiplying mixed numbers to solve other types of problems?

A: Yes, the concept of multiplying mixed numbers can be applied to a variety of other types of problems, such as:

  • Adding and subtracting mixed numbers
  • Dividing mixed numbers
  • Converting mixed numbers to decimals or percents

Q: How do I know if I have mastered the concept of multiplying mixed numbers?

A: You have mastered the concept of multiplying mixed numbers when you can:

  • Convert mixed numbers to improper fractions with ease
  • Multiply mixed numbers accurately and efficiently
  • Simplify the result of multiplying mixed numbers correctly
  • Apply the concept of multiplying mixed numbers to real-world problems with confidence

Q: What are some common applications of the concept of multiplying mixed numbers?

A: Some common applications of the concept of multiplying mixed numbers include:

  • Calculating the area of a rectangle with mixed number dimensions
  • Finding the volume of a rectangular prism with mixed number dimensions
  • Determining the cost of a mixed number quantity of items
  • Solving problems involving mixed number rates or ratios

Q: Can I use the concept of multiplying mixed numbers to solve problems in other subjects?

A: Yes, the concept of multiplying mixed numbers can be applied to a variety of other subjects, such as:

  • Science: calculating the area of a rectangle with mixed number dimensions
  • Engineering: finding the volume of a rectangular prism with mixed number dimensions
  • Finance: determining the cost of a mixed number quantity of items
  • Architecture: calculating the area of a building with mixed number dimensions