Calculate The Following Expression:$ 7 \frac{1}{2} \div \left( 4 \frac{1}{2} - 5 \frac{1}{8} \right) }$Choose The Correct Answer A. { -12$ $B. { -4 \frac{11}{16}$}$C. ${ 4 \frac{11}{16}\$} D. ${ 12\$}

Introduction

When dealing with mixed numbers in division, it's essential to follow the correct order of operations to arrive at the correct solution. In this article, we will explore how to calculate the expression $7 \frac{1}{2} \div \left( 4 \frac{1}{2} - 5 \frac{1}{8} \right)$ and choose the correct answer from the given options.

Understanding Mixed Numbers

A mixed number is a combination of a whole number and a fraction. For example, $7 \frac{1}{2}$ is a mixed number that consists of a whole number (7) and a fraction ($\frac{1}{2}$). To work with mixed numbers, it's crucial to convert them into improper fractions, which are fractions with a numerator that is greater than the denominator.

Converting Mixed Numbers to Improper Fractions

To convert a mixed number to an improper fraction, we multiply the whole number by the denominator and then add the numerator. The result becomes the new numerator, while the denominator remains the same.

For example, to convert $7 \frac{1}{2}$ to an improper fraction, we multiply 7 by 2 (the denominator) and add 1 (the numerator). This gives us:

$$7 \times 2 + 1 = 15$$

So, $7 \frac{1}{2}$ is equal to $\frac{15}{2}$.

Converting Mixed Numbers in the Expression

We will now convert the mixed numbers in the given expression to improper fractions.

$$7 \frac{1}{2} \div \left( 4 \frac{1}{2} - 5 \frac{1}{8} \right)$$

First, we convert $7 \frac{1}{2}$ to an improper fraction:

$$7 \frac{1}{2} = \frac{15}{2}$$

Next, we convert $4 \frac{1}{2}$ to an improper fraction:

$$4 \frac{1}{2} = \frac{9}{2}$$

Finally, we convert $5 \frac{1}{8}$ to an improper fraction:

$$5 \frac{1}{8} = \frac{41}{8}$$

Subtracting Fractions with Different Denominators

To subtract fractions with different denominators, we need to find the least common multiple (LCM) of the denominators. The LCM of 2 and 8 is 8.

So, we rewrite the fractions with a denominator of 8:

$$\frac{9}{2} = \frac{36}{8}$$

$$\frac{41}{8}$$

Now, we can subtract the fractions:

$$\frac{36}{8} - \frac{41}{8} = \frac{-5}{8}$$

Dividing Fractions

To divide fractions, we multiply the first fraction by the reciprocal of the second fraction.

$$\frac{15}{2} \div \left( \frac{-5}{8} \right) = \frac{15}{2} \times \frac{8}{-5}$$

$$= \frac{15 \times 8}{2 \times -5}$$

$$= \frac{120}{-10}$$

$$= -12$$

Conclusion

In conclusion, the correct answer to the expression $7 \frac{1}{2} \div \left( 4 \frac{1}{2} - 5 \frac{1}{8} \right)$ is $-12$. This is because we converted the mixed numbers to improper fractions, subtracted the fractions, and then divided the fractions.

Choosing the Correct Answer

When faced with a similar problem, remember to follow the correct order of operations:

1. Convert mixed numbers to improper fractions.
2. Subtract fractions with different denominators.
3. Divide fractions by multiplying the first fraction by the reciprocal of the second fraction.

By following these steps, you will arrive at the correct solution and choose the correct answer from the given options.

Final Answer

The final answer is:

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Introduction

In our previous article, we explored how to calculate the expression $7 \frac{1}{2} \div \left( 4 \frac{1}{2} - 5 \frac{1}{8} \right)$ and arrived at the correct solution. In this article, we will provide a Q&A guide to help you better understand the concept of solving mixed numbers in division.

Q: What is the first step in solving a mixed number division problem?

A: The first step in solving a mixed number division problem is to convert the mixed numbers to improper fractions.

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

A: To convert a mixed number to an improper fraction, you multiply the whole number by the denominator and then add the numerator. The result becomes the new numerator, while the denominator remains the same.

For example, to convert $7 \frac{1}{2}$ to an improper fraction, you multiply 7 by 2 (the denominator) and add 1 (the numerator). This gives you:

$$7 \times 2 + 1 = 15$$

So, $7 \frac{1}{2}$ is equal to $\frac{15}{2}$.

Q: What is the next step in solving a mixed number division problem?

A: The next step in solving a mixed number division problem is to subtract the fractions with different denominators.

Q: How do I subtract fractions with different denominators?

A: To subtract fractions with different denominators, you need to find the least common multiple (LCM) of the denominators. The LCM of 2 and 8 is 8.

So, you rewrite the fractions with a denominator of 8:

$$\frac{9}{2} = \frac{36}{8}$$

$$\frac{41}{8}$$

Now, you can subtract the fractions:

$$\frac{36}{8} - \frac{41}{8} = \frac{-5}{8}$$

Q: What is the final step in solving a mixed number division problem?

A: The final step in solving a mixed number division problem is to divide the fractions by multiplying the first fraction by the reciprocal of the second fraction.

Q: How do I divide fractions?

A: To divide fractions, you multiply the first fraction by the reciprocal of the second fraction.

For example, to divide $\frac{15}{2}$ by $\frac{-5}{8}$, you multiply $\frac{15}{2}$ by $\frac{8}{-5}$:

$$\frac{15}{2} \div \left( \frac{-5}{8} \right) = \frac{15}{2} \times \frac{8}{-5}$$

$$= \frac{15 \times 8}{2 \times -5}$$

$$= \frac{120}{-10}$$

$$= -12$$

Q: What are some common mistakes to avoid when solving mixed number division problems?

A: Some common mistakes to avoid when solving mixed number division problems include:

• Not converting mixed numbers to improper fractions
• Not finding the least common multiple (LCM) of the denominators
• Not multiplying the first fraction by the reciprocal of the second fraction

Conclusion

In conclusion, solving mixed number division problems requires a step-by-step approach. By converting mixed numbers to improper fractions, subtracting fractions with different denominators, and dividing fractions, you can arrive at the correct solution. Remember to avoid common mistakes and follow the correct order of operations.

Final Tips

• Always convert mixed numbers to improper fractions before solving the problem.
• Find the least common multiple (LCM) of the denominators before subtracting fractions.
• Multiply the first fraction by the reciprocal of the second fraction when dividing fractions.

By following these tips and practicing mixed number division problems, you will become more confident and proficient in solving these types of problems.