Multiplication of Fractions

Oct 13, 2025 by miodeya.com 8 views

Introduction

In mathematics, multiplication of fractions and mixed numbers is a fundamental operation that requires a clear understanding of the concept. When multiplying fractions, we need to multiply the numerators and denominators separately and then simplify the resulting fraction. In the case of mixed numbers, we need to convert them to improper fractions before performing the multiplication. In this article, we will explore the multiplication of fractions and mixed numbers, and provide step-by-step solutions to the given problems.

Multiplication of Fractions

When multiplying fractions, we follow the rule that the numerator of the first fraction is multiplied by the numerator of the second fraction, and the denominator of the first fraction is multiplied by the denominator of the second fraction. The resulting fraction is then simplified by dividing both the numerator and denominator by their greatest common divisor (GCD).

Example (a): $\frac{3}{5} \times \frac{2}{9}$

To multiply these fractions, we multiply the numerators and denominators separately:

$\frac{3}{5} \times \frac{2}{9} = \frac{3 \times 2}{5 \times 9} = \frac{6}{45}$

Now, we simplify the resulting fraction by dividing both the numerator and denominator by their GCD, which is 3:

$\frac{6}{45} = \frac{6 ÷ 3}{45 ÷ 3} = \frac{2}{15}$

Therefore, the product of $\frac{3}{5}$ and $\frac{2}{9}$ is $\frac{2}{15}$ .

Example (b): $\frac{4}{7} \times \frac{9}{8}$

To multiply these fractions, we multiply the numerators and denominators separately:

$\frac{4}{7} \times \frac{9}{8} = \frac{4 \times 9}{7 \times 8} = \frac{36}{56}$

Now, we simplify the resulting fraction by dividing both the numerator and denominator by their GCD, which is 4:

$\frac{36}{56} = \frac{36 ÷ 4}{56 ÷ 4} = \frac{9}{14}$

Therefore, the product of $\frac{4}{7}$ and $\frac{9}{8}$ is $\frac{9}{14}$ .

Multiplication of Mixed Numbers

When multiplying mixed numbers, we need to convert them to improper fractions before performing the multiplication. To convert a mixed number to an improper fraction, we multiply the whole number part by the denominator and add the numerator, then write the result as a fraction with the original denominator.

Example (c): $2 \frac{3}{5} \times \frac{9}{8}$

To multiply this mixed number by the fraction, we first convert the mixed number to an improper fraction:

$2 \frac{3}{5} = \frac{(2 \times 5) + 3}{5} = \frac{10 + 3}{5} = \frac{13}{5}$

Now, we multiply the improper fraction by the fraction:

$\frac{13}{5} \times \frac{9}{8} = \frac{13 \times 9}{5 \times 8} = \frac{117}{40}$

Therefore, the product of $2 \frac{3}{5}$ and $\frac{9}{8}$ is $\frac{117}{40}$ .

Conclusion

In conclusion, multiplication of fractions and mixed numbers requires a clear understanding of the concept and the ability to simplify the resulting fraction. By following the rules of multiplication and simplification, we can find the product of fractions and mixed numbers with ease. In this article, we have explored the multiplication of fractions and mixed numbers, and provided step-by-step solutions to the given problems.

Key Takeaways

When multiplying fractions, we multiply the numerators and denominators separately and then simplify the resulting fraction.
When multiplying mixed numbers, we need to convert them to improper fractions before performing the multiplication.
To simplify a fraction, we divide both the numerator and denominator by their greatest common divisor (GCD).
To convert a mixed number to an improper fraction, we multiply the whole number part by the denominator and add the numerator, then write the result as a fraction with the original denominator.

Practice Problems

$\frac{2}{3} \times \frac{5}{9}$
$\frac{3}{4} \times \frac{7}{10}$
$3 \frac{2}{7} \times \frac{9}{12}$

Answer Key

$\frac{10}{27}$
$\frac{21}{40}$
$\frac{43}{28}$
Frequently Asked Questions (FAQs) on Multiplication of Fractions and Mixed Numbers =====================================================================================

Q: What is the rule for multiplying fractions?

A: When multiplying fractions, we multiply the numerators and denominators separately and then simplify the resulting fraction.

Q: How do I simplify a fraction after multiplying?

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

Q: What is the difference between multiplying fractions and multiplying mixed numbers?

A: When multiplying mixed numbers, we need to convert them to improper fractions before performing the multiplication.

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

A: To convert a mixed number to an improper fraction, we multiply the whole number part by the denominator and add the numerator, then write the result as a fraction with the original denominator.

Q: Can I multiply a fraction by a mixed number?

A: Yes, you can multiply a fraction by a mixed number. First, convert the mixed number to an improper fraction, then multiply the two fractions.

Q: What is the order of operations when multiplying fractions and mixed numbers?

A: The order of operations is:

Multiply the numerators and denominators separately.
Simplify the resulting fraction by dividing both the numerator and denominator by their GCD.
If the result is an improper fraction, convert it back to a mixed number.

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

A: Yes, you can use a calculator to multiply fractions and mixed numbers. However, it's always a good idea to check your answer by simplifying the fraction manually.

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

A: Some common mistakes to avoid include:

Not simplifying the fraction after multiplying.
Not converting a mixed number to an improper fraction before multiplying.
Not using the correct order of operations.

Q: How can I practice multiplying fractions and mixed numbers?

A: You can practice multiplying fractions and mixed numbers by working through examples and exercises, such as those found in this article. You can also try creating your own problems and solving them.

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

A: Multiplying fractions and mixed numbers has many real-world applications, such as:

Cooking and baking: When a recipe calls for a certain amount of a ingredient, you may need to multiply fractions or mixed numbers to get the correct amount.
Science and engineering: When working with measurements and conversions, you may need to multiply fractions or mixed numbers.
Finance: When calculating interest rates or investments, you may need to multiply fractions or mixed numbers.

Conclusion

In conclusion, multiplying fractions and mixed numbers is an essential skill that requires a clear understanding of the concept and the ability to simplify the resulting fraction. By following the rules of multiplication and simplification, and avoiding common mistakes, you can become proficient in multiplying fractions and mixed numbers.