Which Shows How The Distributive Property Can Be Used To Evaluate 7 × 8 4 5 7 \times 8 \frac{4}{5} 7 × 8 5 4 ​ ?A. 56 + 28 5 = 56 + 5 3 5 = 61 3 5 56 + \frac{28}{5} = 56 + 5 \frac{3}{5} = 61 \frac{3}{5} 56 + 5 28 ​ = 56 + 5 5 3 ​ = 61 5 3 ​ B. 56 × 28 5 = 1568 5 = 313 3 5 56 \times \frac{28}{5} = \frac{1568}{5} = 313 \frac{3}{5} 56 × 5 28 ​ = 5 1568 ​ = 313 5 3 ​ C.

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Introduction

The distributive property is a fundamental concept in mathematics that allows us to simplify complex expressions by breaking them down into more manageable parts. In this article, we will explore how the distributive property can be used to evaluate the expression $7 \times 8 \frac{4}{5}$.

Understanding the Distributive Property

The distributive property states that for any real numbers $a$, $b$, and $c$, the following equation holds:

$a(b + c) = ab + ac$

This property can be applied to both addition and multiplication, and it is a powerful tool for simplifying complex expressions.

Evaluating $7 \times 8 \frac{4}{5}$

To evaluate the expression $7 \times 8 \frac{4}{5}$, we can use the distributive property to break it down into simpler parts. We can start by converting the mixed number $8 \frac{4}{5}$ to an improper fraction:

$8 \frac{4}{5} = \frac{40}{5} + \frac{4}{5} = \frac{44}{5}$

Now, we can use the distributive property to evaluate the expression:

$7 \times 8 \frac{4}{5} = 7 \times \frac{44}{5}$

Using the distributive property, we can rewrite this expression as:

$7 \times \frac{44}{5} = \frac{7 \times 44}{5}$

Now, we can simplify the numerator by multiplying $7$ and $44$:

$\frac{7 \times 44}{5} = \frac{308}{5}$

Finally, we can simplify the fraction by dividing the numerator and denominator by their greatest common divisor, which is $1$:

$\frac{308}{5} = 61 \frac{3}{5}$

Alternative Solution

Some students may choose to evaluate the expression $7 \times 8 \frac{4}{5}$ by converting the mixed number to an improper fraction and then multiplying the numerators and denominators:

$7 \times 8 \frac{4}{5} = 7 \times \frac{40}{5} + \frac{4}{5}$

Using the distributive property, we can rewrite this expression as:

$7 \times \frac{40}{5} + \frac{4}{5} = \frac{7 \times 40}{5} + \frac{4}{5}$

Now, we can simplify the numerator by multiplying $7$ and $40$:

$\frac{7 \times 40}{5} + \frac{4}{5} = \frac{280}{5} + \frac{4}{5}$

Finally, we can simplify the fraction by dividing the numerator and denominator by their greatest common divisor, which is $5$:

$\frac{280}{5} + \frac{4}{5} = 56 + \frac{4}{5}$

Now, we can simplify the fraction by converting it to a mixed number:

$56 + \frac{4}{5} = 56 + 0 \frac{4}{5} = 56 + 0 \frac{4}{5} = 56 + 5 \frac{3}{5} = 61 \frac{3}{5}$

Conclusion

In this article, we have explored how the distributive property can be used to evaluate the expression $7 \times 8 \frac{4}{5}$. We have shown that by breaking down the expression into simpler parts and using the distributive property, we can simplify the expression and arrive at the correct answer. We have also provided an alternative solution that demonstrates the importance of converting mixed numbers to improper fractions and multiplying the numerators and denominators.

Practice Problems

1. Evaluate the expression $5 \times 9 \frac{2}{3}$ using the distributive property.
2. Evaluate the expression $3 \times 7 \frac{1}{2}$ using the distributive property.
3. Evaluate the expression $2 \times 6 \frac{3}{4}$ using the distributive property.

Answer Key

1. $45 \frac{1}{3}$
2. $21 \frac{1}{4}$
3. $13 \frac{1}{8}$
   Frequently Asked Questions: Mastering the Distributive Property ================================================================

Q: What is the distributive property?

A: The distributive property is a fundamental concept in mathematics that allows us to simplify complex expressions by breaking them down into more manageable parts. It states that for any real numbers $a$, $b$, and $c$, the following equation holds:

$a(b + c) = ab + ac$

Q: How do I apply the distributive property to evaluate an expression?

A: To apply the distributive property, you need to follow these steps:

1. Break down the expression into simpler parts.
2. Use the distributive property to rewrite the expression.
3. Simplify the expression by multiplying the numerators and denominators.

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, such as $8 \frac{4}{5}$. An improper fraction is a fraction where the numerator is greater than the denominator, such as $\frac{44}{5}$.

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

A: To convert a mixed number to an improper fraction, you need to follow these steps:

1. Multiply the whole number by the denominator.
2. Add the numerator to the product.
3. Write the result as an improper 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. For example, the GCD of $308$ and $5$ is $1$.

Q: How do I simplify a fraction by dividing the numerator and denominator by their GCD?

A: To simplify a fraction by dividing the numerator and denominator by their GCD, you need to follow these steps:

1. Find the GCD of the numerator and denominator.
2. Divide the numerator and denominator by the GCD.
3. Write the result as a simplified fraction.

Q: What are some common mistakes to avoid when applying the distributive property?

A: Some common mistakes to avoid when applying the distributive property include:

• Not breaking down the expression into simpler parts.
• Not using the distributive property to rewrite the expression.
• Not simplifying the expression by multiplying the numerators and denominators.
• Not converting mixed numbers to improper fractions.

Q: How can I practice applying the distributive property?

A: You can practice applying the distributive property by working through practice problems, such as:

• Evaluating expressions with mixed numbers and improper fractions.
• Simplifying fractions by dividing the numerator and denominator by their GCD.
• Applying the distributive property to real-world problems.

Q: What are some real-world applications of the distributive property?

A: The distributive property has many real-world applications, such as:

• Simplifying complex expressions in algebra and calculus.
• Evaluating expressions with mixed numbers and improper fractions in finance and accounting.
• Applying the distributive property to real-world problems in science and engineering.

Conclusion

In this article, we have answered some frequently asked questions about the distributive property and provided tips and tricks for mastering this fundamental concept in mathematics. We hope that this article has been helpful in clarifying any doubts you may have had about the distributive property and has provided you with the confidence to apply it to a wide range of problems.