Write the product of the numerators over the product of the denominators

Oct 13, 2025 by miodeya.com 21 views

Introduction to Multiplying Fractions

Multiplying fractions is a fundamental concept in mathematics that helps us solve various problems in different fields, such as science, engineering, and finance. In this article, we will focus on multiplying two fractions, $\frac{11}{2}$ and $\frac{3}{7}$ , and simplify the result to its simplest form.

Understanding the Concept of Multiplying Fractions

To multiply fractions, we need to understand the concept of multiplying numbers and dividing numbers. When we multiply two numbers, we are essentially adding a number a certain number of times. For example, $3 \times 4$ means adding $3$ together $4$ times, which equals $12$ . On the other hand, when we divide a number by another number, we are essentially finding how many times the divisor fits into the dividend. For instance, $12 \div 3$ means finding how many times $3$ fits into $12$ , which equals $4$ .

Multiplying Fractions: A Step-by-Step Guide

To multiply fractions, we follow a simple step-by-step process:

Multiply the numerators: Multiply the numbers on top of the fractions, which are the numerators. In this case, we multiply $11$ and $3$ .
Multiply the denominators: Multiply the numbers on the bottom of the fractions, which are the denominators. In this case, we multiply $2$ and $7$ .
Write the product of the numerators over the product of the denominators: Write the result of multiplying the numerators over the result of multiplying the denominators.

Applying the Multiplication Rule to the Given Fractions

Now, let's apply the multiplication rule to the given fractions, $\frac{11}{2}$ and $\frac{3}{7}$ .

Step 1: Multiply the Numerators

To multiply the numerators, we simply multiply $11$ and $3$ .

$11 \times 3 = 33$

Step 2: Multiply the Denominators

To multiply the denominators, we simply multiply $2$ and $7$ .

$2 \times 7 = 14$

Step 3: Write the Product of the Numerators over the Product of the Denominators

Now, we write the product of the numerators over the product of the denominators.

$\frac{33}{14}$

Simplifying the Result

To simplify the result, we need to find the greatest common divisor (GCD) of the numerator and the denominator. The GCD is the largest number that divides both the numerator and the denominator without leaving a remainder.

Finding the Greatest Common Divisor (GCD)

To find the GCD of $33$ and $14$ , we can use the Euclidean algorithm or list the factors of each number.

Factors of $33$ : $1, 3, 11, 33$ Factors of $14$ : $1, 2, 7, 14$

The greatest common divisor of $33$ and $14$ is $1$ .

Simplifying the Fraction

Since the GCD of $33$ and $14$ is $1$ , the fraction $\frac{33}{14}$ is already in its simplest form.

Conclusion

In conclusion, multiplying fractions is a straightforward process that involves multiplying the numerators and denominators separately and then writing the product of the numerators over the product of the denominators. To simplify the result, we need to find the greatest common divisor of the numerator and the denominator and divide both numbers by the GCD. In this article, we multiplied the fractions $\frac{11}{2}$ and $\frac{3}{7}$ and simplified the result to its simplest form, which is $\frac{33}{14}$ .

Frequently Asked Questions

Q: What is the result of multiplying $\frac{11}{2}$ and $\frac{3}{7}$ ?

A: The result of multiplying $\frac{11}{2}$ and $\frac{3}{7}$ is $\frac{33}{14}$ .

Q: How do I simplify a fraction?

A: To simplify a fraction, you need to find the greatest common divisor (GCD) of the numerator and the denominator and divide both numbers by the GCD.

Q: What is the greatest common divisor (GCD) of $33$ and $14$ ?

A: The greatest common divisor of $33$ and $14$ is $1$ .

Final Answer

The final answer is: $\boxed{\frac{33}{14}}$

Introduction to Multiplying Fractions Q&A

Frequently Asked Questions

Q: What is the result of multiplying $\frac{11}{2}$ and $\frac{3}{7}$ ?

A: The result of multiplying $\frac{11}{2}$ and $\frac{3}{7}$ is $\frac{33}{14}$ .

Q: How do I multiply fractions?

A: To multiply fractions, you need to multiply the numerators and denominators separately and then write the product of the numerators over the product of the denominators.

Q: What is the difference between multiplying and dividing fractions?

A: When you multiply fractions, you are essentially adding a number a certain number of times. On the other hand, when you divide fractions, you are essentially finding how many times the divisor fits into the dividend.

Q: Can I simplify a fraction before multiplying it with another fraction?

A: Yes, you can simplify a fraction before multiplying it with another fraction. However, it is not always necessary to simplify the fractions before multiplying them.

Q: How do I find the greatest common divisor (GCD) of two numbers?

A: To find the GCD of two numbers, you can use the Euclidean algorithm or list the factors of each number.

Q: What is the greatest common divisor (GCD) of $33$ and $14$ ?

A: The greatest common divisor of $33$ and $14$ is $1$ .

Q: Can I multiply a fraction with a whole number?

A: Yes, you can multiply a fraction with a whole number. To do this, you need to multiply the numerator of the fraction by the whole number and keep the denominator the same.

Q: How do I multiply a fraction with a decimal number?

A: To multiply a fraction with a decimal number, you need to convert the decimal number to a fraction and then multiply the fractions.

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

A: Yes, you can multiply a fraction with a mixed number. To do this, you need to convert the mixed number to an improper fraction and then multiply the fractions.

Common Mistakes to Avoid

Mistake 1: Not Simplifying the Fractions

One common mistake when multiplying fractions is not simplifying the fractions before multiplying them. This can lead to incorrect results.

Mistake 2: Not Multiplying the Numerators and Denominators Separately

Another common mistake is not multiplying the numerators and denominators separately. This can also lead to incorrect results.

Mistake 3: Not Finding the Greatest Common Divisor (GCD)

A third common mistake is not finding the greatest common divisor (GCD) of the numerator and denominator. This can also lead to incorrect results.

Conclusion

In conclusion, multiplying fractions is a straightforward process that involves multiplying the numerators and denominators separately and then writing the product of the numerators over the product of the denominators. To simplify the result, we need to find the greatest common divisor of the numerator and the denominator and divide both numbers by the GCD. We hope this Q&A article has helped address common questions and doubts related to multiplying fractions.

Final Answer

The final answer is: $\boxed{\frac{33}{14}}$