Find The Quotients.a. $ \frac{1}{7} \div \frac{1}{8} $b. $ \frac{12}{5} \div \frac{6}{5} $c. $ \frac{1}{10} \div 10 $d. $ \frac{9}{10} \div \frac{10}{9} $
Introduction
Division of fractions is a fundamental concept in mathematics that can be a bit tricky to grasp at first, but with practice and understanding, it becomes second nature. In this article, we will delve into the world of division of fractions, exploring the rules and procedures that govern this operation. We will also work through several examples to illustrate the concept and provide a deeper understanding of the subject.
What is Division of Fractions?
Division of fractions is the process of dividing one fraction by another. It involves finding the quotient of two fractions, which is the result of dividing the numerator of the first fraction by the denominator of the second fraction. Division of fractions is denoted by the symbol ÷ and is used to find the result of dividing one fraction by another.
Rules for Division of Fractions
There are several rules that govern the division of fractions. These rules are essential to understanding and performing division of fractions correctly.
Rule 1: Invert and Multiply
The first rule for division of fractions is to invert the second fraction and multiply. This means that when dividing one fraction by another, we need to flip the second fraction and then multiply the two fractions together. This rule is often represented by the equation:
a ÷ b = a × (1/b)
Rule 2: Multiply the Numerators and Denominators
The second rule for division of fractions is to multiply the numerators and denominators of the two fractions together. This means that when dividing one fraction by another, we need to multiply the numerators together and the denominators together.
Examples of Division of Fractions
Now that we have covered the rules for division of fractions, let's work through some examples to illustrate the concept.
Example 1: a. $ \frac{1}{7} \div \frac{1}{8} $
To divide $ \frac{1}{7} $ by $ \frac{1}{8} $, we need to invert the second fraction and multiply.
$ \frac{1}{7} \div \frac{1}{8} = \frac{1}{7} \times \frac{8}{1} $
Multiplying the numerators and denominators together, we get:
$ \frac{1}{7} \times \frac{8}{1} = \frac{8}{7} $
Therefore, the quotient of $ \frac{1}{7} \div \frac{1}{8} $ is $ \frac{8}{7} $.
Example 2: b. $ \frac{12}{5} \div \frac{6}{5} $
To divide $ \frac{12}{5} $ by $ \frac{6}{5} $, we need to invert the second fraction and multiply.
$ \frac{12}{5} \div \frac{6}{5} = \frac{12}{5} \times \frac{5}{6} $
Multiplying the numerators and denominators together, we get:
$ \frac{12}{5} \times \frac{5}{6} = \frac{60}{30} $
Simplifying the fraction, we get:
$ \frac{60}{30} = \frac{2}{1} $
Therefore, the quotient of $ \frac{12}{5} \div \frac{6}{5} $ is $ \frac{2}{1} $.
Example 3: c. $ \frac{1}{10} \div 10 $
To divide $ \frac{1}{10} $ by 10, we need to invert the second fraction and multiply.
$ \frac{1}{10} \div 10 = \frac{1}{10} \times \frac{1}{10} $
Multiplying the numerators and denominators together, we get:
$ \frac{1}{10} \times \frac{1}{10} = \frac{1}{100} $
Therefore, the quotient of $ \frac{1}{10} \div 10 $ is $ \frac{1}{100} $.
Example 4: d. $ \frac{9}{10} \div \frac{10}{9} $
To divide $ \frac{9}{10} $ by $ \frac{10}{9} $, we need to invert the second fraction and multiply.
$ \frac{9}{10} \div \frac{10}{9} = \frac{9}{10} \times \frac{9}{10} $
Multiplying the numerators and denominators together, we get:
$ \frac{9}{10} \times \frac{9}{10} = \frac{81}{100} $
Therefore, the quotient of $ \frac{9}{10} \div \frac{10}{9} $ is $ \frac{81}{100} $.
Conclusion
Division of fractions is a fundamental concept in mathematics that can be a bit tricky to grasp at first, but with practice and understanding, it becomes second nature. By following the rules for division of fractions, including inverting and multiplying, and multiplying the numerators and denominators together, we can perform division of fractions with ease. We have worked through several examples to illustrate the concept and provide a deeper understanding of the subject. With this knowledge, you will be able to tackle division of fractions with confidence and accuracy.
Introduction
Division of fractions can be a bit tricky to understand, and it's common to have questions and doubts. In this article, we will address some of the most frequently asked questions about division of fractions, providing clear and concise answers to help you better understand the concept.
Q1: What is the rule for dividing fractions?
A1: The rule for dividing fractions is to invert the second fraction and multiply. This means that when dividing one fraction by another, we need to flip the second fraction and then multiply the two fractions together.
Q2: How do I invert a fraction?
A2: To invert a fraction, we need to flip the numerator and denominator. For example, if we have the fraction $ \frac{1}{2} $, the inverted fraction would be $ \frac{2}{1} $.
Q3: What is the difference between dividing fractions and multiplying fractions?
A3: Dividing fractions involves inverting the second fraction and multiplying, whereas multiplying fractions involves multiplying the numerators and denominators together. For example, if we have the fractions $ \frac{1}{2} $ and $ \frac{2}{3} $, dividing them would involve inverting the second fraction and multiplying, whereas multiplying them would involve multiplying the numerators and denominators together.
Q4: Can I divide a fraction by a whole number?
A4: Yes, you can divide a fraction by a whole number. To do this, you need to invert the fraction and multiply by the whole number. For example, if we have the fraction $ \frac{1}{2} $ and we want to divide it by 3, we would invert the fraction and multiply by 3, resulting in $ \frac{3}{2} $.
Q5: Can I divide a whole number by a fraction?
A5: Yes, you can divide a whole number by a fraction. To do this, you need to invert the fraction and multiply by the whole number. For example, if we have the whole number 4 and we want to divide it by the fraction $ \frac{1}{2} $, we would invert the fraction and multiply by 4, resulting in 8.
Q6: What is the result of dividing a fraction by itself?
A6: The result of dividing a fraction by itself is 1. For example, if we have the fraction $ \frac{1}{2} $ and we divide it by itself, the result would be 1.
Q7: Can I divide a negative fraction by a positive fraction?
A7: Yes, you can divide a negative fraction by a positive fraction. To do this, you need to invert the second fraction and multiply. For example, if we have the negative fraction $ -\frac{1}{2} $ and we want to divide it by the positive fraction $ \frac{2}{3} $, we would invert the second fraction and multiply, resulting in $ -\frac{3}{4} $.
Q8: Can I divide a positive fraction by a negative fraction?
A8: Yes, you can divide a positive fraction by a negative fraction. To do this, you need to invert the second fraction and multiply. For example, if we have the positive fraction $ \frac{1}{2} $ and we want to divide it by the negative fraction $ -\frac{2}{3} $, we would invert the second fraction and multiply, resulting in $ -\frac{3}{4} $.
Q9: Can I divide a mixed number by a fraction?
A9: Yes, you can divide a mixed number by a fraction. To do this, you need to convert the mixed number to an improper fraction and then divide. For example, if we have the mixed number 2 1/2 and we want to divide it by the fraction $ \frac{1}{3} $, we would convert the mixed number to an improper fraction and then divide.
Q10: Can I divide a fraction by a decimal?
A10: Yes, you can divide a fraction by a decimal. To do this, you need to convert the decimal to a fraction and then divide. For example, if we have the fraction $ \frac{1}{2} $ and we want to divide it by the decimal 0.5, we would convert the decimal to a fraction and then divide.
Conclusion
Division of fractions can be a bit tricky to understand, but with practice and patience, you can master the concept. By following the rules for division of fractions and addressing common questions and doubts, you can become more confident and accurate in your calculations. Remember to invert the second fraction and multiply, and to multiply the numerators and denominators together. With this knowledge, you will be able to tackle division of fractions with ease and accuracy.