Calculate The Following Fractions:a. $4 \frac{1}{6} \div 3 \frac{1}{3}$b. $6 \frac{1}{2} \times 3 \frac{4}{5}$
Introduction
Fractions are an essential part of mathematics, and understanding how to operate with them is crucial for success in various mathematical disciplines. In this article, we will delve into the world of fraction operations, focusing on division and multiplication. We will explore two specific problems: calculating the result of dividing a mixed number by another mixed number, and multiplying two mixed numbers together.
Understanding Mixed Numbers
Before we dive into the problems, let's take a moment to understand what mixed numbers are. A mixed number is a combination of a whole number and a fraction. It is written in the form of a whole number followed by a fraction, with the fraction being separated from the whole number by a space or a horizontal line. For example, 4 1/6 is a mixed number, where 4 is the whole number and 1/6 is the fraction.
Problem a: Dividing Mixed Numbers
Let's start with the first problem: dividing the mixed number 4 1/6 by the mixed number 3 1/3.
Step 1: Convert Mixed Numbers to Improper Fractions
To divide 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. 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 the new numerator over the original denominator.
For the mixed number 4 1/6, we multiply 4 by 6 and add 1, which gives us 25. So, 4 1/6 is equal to 25/6.
For the mixed number 3 1/3, we multiply 3 by 3 and add 1, which gives us 10. So, 3 1/3 is equal to 10/3.
Step 2: Invert the Divisor and Multiply
Now that we have the mixed numbers converted to improper fractions, we can proceed with the division. To divide fractions, we invert the divisor (i.e., flip the numerator and denominator) and multiply.
So, to divide 25/6 by 10/3, we invert the divisor 10/3 to become 3/10, and then multiply the two fractions together.
Step 3: Multiply the Numerators and Denominators
To multiply fractions, we multiply the numerators together and the denominators together.
So, (25 × 3) / (6 × 10) = 75/60.
Step 4: Simplify the Result
To simplify the result, we divide both the numerator and the denominator by their greatest common divisor (GCD). The GCD of 75 and 60 is 15.
So, 75 ÷ 15 = 5, and 60 ÷ 15 = 4.
Therefore, the result of dividing 4 1/6 by 3 1/3 is 5/4.
Problem b: Multiplying Mixed Numbers
Now, let's move on to the second problem: multiplying the mixed number 6 1/2 by the mixed number 3 4/5.
Step 1: Convert Mixed Numbers to Improper Fractions
To multiply mixed numbers, we need to convert them into improper fractions first.
For the mixed number 6 1/2, we multiply 6 by 2 and add 1, which gives us 13. So, 6 1/2 is equal to 13/2.
For the mixed number 3 4/5, we multiply 3 by 5 and add 4, which gives us 19. So, 3 4/5 is equal to 19/5.
Step 2: Multiply the Numerators and Denominators
To multiply fractions, we multiply the numerators together and the denominators together.
So, (13 × 19) / (2 × 5) = 247/10.
Step 3: Simplify the Result
To simplify the result, we divide both the numerator and the denominator by their greatest common divisor (GCD). The GCD of 247 and 10 is 1.
Therefore, the result of multiplying 6 1/2 by 3 4/5 is 247/10.
Conclusion
In this article, we have explored two specific problems involving fraction operations: dividing a mixed number by another mixed number, and multiplying two mixed numbers together. We have seen how to convert mixed numbers to improper fractions, invert the divisor and multiply, and simplify the result. By following these steps, we can confidently tackle a wide range of fraction operations and become proficient in this essential area of mathematics.
Final Thoughts
Q: What is the difference between a mixed number and an improper fraction?
A: A mixed number is a combination of a whole number and a fraction, while an improper fraction is a fraction where the numerator is greater than or equal to the denominator.
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 by the denominator and add the numerator. Then, write the result as the new numerator over the original denominator.
Q: What is the rule for dividing fractions?
A: To divide fractions, invert the divisor (i.e., flip the numerator and denominator) and multiply.
Q: How do I simplify a fraction?
A: To simplify a fraction, divide both the numerator and the denominator by their greatest common divisor (GCD).
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: Can I multiply mixed numbers directly?
A: No, to multiply mixed numbers, you need to convert them to improper fractions first.
Q: Can I divide mixed numbers directly?
A: No, to divide mixed numbers, you need to convert them to improper fractions first and then invert the divisor and multiply.
Q: What is the rule for multiplying fractions?
A: To multiply fractions, multiply the numerators together and the denominators together.
Q: Can I add or subtract mixed numbers directly?
A: No, to add or subtract mixed numbers, you need to convert them to improper fractions first.
Q: How do I add or subtract fractions?
A: To add or subtract fractions, you need to have the same denominator. If the denominators are different, find the least common multiple (LCM) of the two denominators and convert both fractions to have the LCM as the denominator. Then, add or subtract the numerators and keep the same denominator.
Q: What is the least common multiple (LCM)?
A: The least common multiple (LCM) is the smallest number that is a multiple of both the numerator and the denominator of a fraction.
Q: Can I use a calculator to simplify fractions?
A: Yes, you can use a calculator to simplify fractions by dividing both the numerator and the denominator by their GCD.
Q: Can I use a calculator to multiply or divide fractions?
A: Yes, you can use a calculator to multiply or divide fractions by following the rules for multiplying or dividing fractions.
Conclusion
In this article, we have answered some of the most frequently asked questions about fraction operations. We hope that this article has provided you with a clear and concise guide to fraction operations and has helped you to understand the rules and procedures for working with fractions. Whether you're a student, a teacher, or simply someone looking to improve your math skills, we hope that this article has been helpful to you.