Find The Products In Simplest Form. 8 21 ⋅ 5 16 = □ \frac{8}{21} \cdot \frac{5}{16} = \square 21 8 ​ ⋅ 16 5 ​ = □ 12 25 ⋅ 15 16 = □ \frac{12}{25} \cdot \frac{15}{16} = \square 25 12 ​ ⋅ 16 15 ​ = □

Introduction

Multiplication of fractions is a fundamental concept in mathematics that involves multiplying two or more fractions together. In this article, we will explore the multiplication of fractions and provide step-by-step solutions to two problems: $\frac{8}{21} \cdot \frac{5}{16} = \square$ and $\frac{12}{25} \cdot \frac{15}{16} = \square$. We will also discuss the importance of simplifying products of fractions to their simplest form.

What is Multiplication of Fractions?

Multiplication of fractions is a process of multiplying two or more fractions together. It involves multiplying the numerators (the numbers on top) and the denominators (the numbers on the bottom) separately and then simplifying the resulting fraction to its simplest form.

Steps to Multiply Fractions

To multiply fractions, follow these steps:

1. Multiply the numerators together.
2. Multiply the denominators together.
3. Simplify the resulting fraction to its simplest form.

Example 1: Multiplying $\frac{8}{21} \cdot \frac{5}{16}$

To multiply $\frac{8}{21} \cdot \frac{5}{16}$, we will follow the steps outlined above.

Step 1: Multiply the Numerators

The numerators are 8 and 5. Multiply them together: $8 \cdot 5 = 40$.

Step 2: Multiply the Denominators

The denominators are 21 and 16. Multiply them together: $21 \cdot 16 = 336$.

Step 3: Simplify the Resulting Fraction

The resulting fraction is $\frac{40}{336}$. To simplify this fraction, we need to find the greatest common divisor (GCD) of 40 and 336.

Finding the Greatest Common Divisor (GCD)

To find the GCD of 40 and 336, we can use the Euclidean algorithm or list the factors of each number.

Factors of 40

The factors of 40 are: 1, 2, 4, 5, 8, 10, 20, 40.

Factors of 336

The factors of 336 are: 1, 2, 3, 4, 6, 7, 8, 12, 14, 16, 21, 24, 28, 42, 48, 56, 84, 112, 168, 336.

Greatest Common Divisor (GCD)

The greatest common divisor of 40 and 336 is 8.

Simplifying the Fraction

To simplify the fraction $\frac{40}{336}$, we will divide both the numerator and the denominator by the GCD, which is 8.

$\frac{40}{336} = \frac{40 \div 8}{336 \div 8} = \frac{5}{42}$

Example 2: Multiplying $\frac{12}{25} \cdot \frac{15}{16}$

To multiply $\frac{12}{25} \cdot \frac{15}{16}$, we will follow the steps outlined above.

Step 1: Multiply the Numerators

The numerators are 12 and 15. Multiply them together: $12 \cdot 15 = 180$.

Step 2: Multiply the Denominators

The denominators are 25 and 16. Multiply them together: $25 \cdot 16 = 400$.

Step 3: Simplify the Resulting Fraction

The resulting fraction is $\frac{180}{400}$. To simplify this fraction, we need to find the greatest common divisor (GCD) of 180 and 400.

Finding the Greatest Common Divisor (GCD)

To find the GCD of 180 and 400, we can use the Euclidean algorithm or list the factors of each number.

Factors of 180

The factors of 180 are: 1, 2, 3, 4, 5, 6, 9, 10, 12, 15, 18, 20, 30, 36, 45, 60, 90, 180.

Factors of 400

The factors of 400 are: 1, 2, 4, 5, 8, 10, 16, 20, 25, 40, 50, 80, 100, 200, 400.

Greatest Common Divisor (GCD)

The greatest common divisor of 180 and 400 is 20.

Simplifying the Fraction

To simplify the fraction $\frac{180}{400}$, we will divide both the numerator and the denominator by the GCD, which is 20.

$\frac{180}{400} = \frac{180 \div 20}{400 \div 20} = \frac{9}{20}$

Conclusion

In this article, we have explored the multiplication of fractions and provided step-by-step solutions to two problems: $\frac{8}{21} \cdot \frac{5}{16} = \square$ and $\frac{12}{25} \cdot \frac{15}{16} = \square$. We have also discussed the importance of simplifying products of fractions to their simplest form. By following the steps outlined above, you can multiply fractions and simplify the resulting fraction to its simplest form.

Introduction

Multiplication of fractions is a fundamental concept in mathematics that involves multiplying two or more fractions together. In this article, we will provide answers to frequently asked questions about multiplication of fractions, including how to multiply fractions, how to simplify fractions, and how to find the greatest common divisor (GCD).

Q&A

Q: What is the formula for multiplying fractions?

A: The formula for multiplying fractions is:

$\frac{a}{b} \cdot \frac{c}{d} = \frac{a \cdot c}{b \cdot d}$

Q: How do I multiply fractions with different denominators?

A: To multiply fractions with different denominators, you need to multiply the numerators together and the denominators together, and then simplify the resulting fraction.

Q: What is the greatest common divisor (GCD)?

A: The greatest common divisor (GCD) is the largest number that divides two or more numbers without leaving a remainder.

Q: How do I find the GCD of two numbers?

A: There are several ways to find the GCD of two numbers, including:

• Listing the factors of each number
• Using the Euclidean algorithm
• Using a calculator

Q: Why is it important to simplify fractions?

A: Simplifying fractions is important because it helps to:

• Reduce the complexity of the fraction
• Make it easier to compare fractions
• Make it easier to perform operations with fractions

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 then divide both numbers by the GCD.

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

A: Multiplying fractions involves multiplying the numerators together and the denominators together, while adding fractions involves adding the numerators together and keeping the same denominator.

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

A: Yes, you can multiply a fraction by a whole number by multiplying the numerator by the whole number and keeping the same denominator.

Q: Can I multiply a fraction by a decimal?

A: Yes, you can multiply a fraction by a decimal by converting the decimal to a fraction and then multiplying the fractions together.

Q: What is the order of operations for multiplying fractions?

A: The order of operations for multiplying fractions is:

1. Multiply the numerators together
2. Multiply the denominators together
3. Simplify the resulting fraction

Conclusion

In this article, we have provided answers to frequently asked questions about multiplication of fractions, including how to multiply fractions, how to simplify fractions, and how to find the greatest common divisor (GCD). By following the steps outlined above, you can multiply fractions and simplify the resulting fraction to its simplest form.

Additional Resources

• Multiplication of Fractions: A Step-by-Step Guide
• Simplifying Fractions: A Step-by-Step Guide
• Finding the Greatest Common Divisor (GCD): A Step-by-Step Guide

Frequently Asked Questions

• What is the formula for multiplying fractions?
• How do I multiply fractions with different denominators?
• What is the greatest common divisor (GCD)?
• How do I find the GCD of two numbers?
• Why is it important to simplify fractions?
• How do I simplify a fraction?
• What is the difference between multiplying fractions and adding fractions?
• Can I multiply a fraction by a whole number?
• Can I multiply a fraction by a decimal?
• What is the order of operations for multiplying fractions?

Related Articles

• Multiplication of Fractions: A Step-by-Step Guide
• Simplifying Fractions: A Step-by-Step Guide
• Finding the Greatest Common Divisor (GCD): A Step-by-Step Guide

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Note: The above article is a Q&A article that provides answers to frequently asked questions about multiplication of fractions. The article includes a list of questions and answers, as well as additional resources and related articles.