Which Products Are Greater Than $2 \frac{5}{6}$?A. $\frac{1}{8} \times 2 \frac{5}{6}$ B. \$\frac{6}{5} \times 2 \frac{5}{6}$[/tex\] C. $\frac{5}{6} \times 2 \frac{5}{6}$ D. $2 \frac{5}{6} \times

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Which Products are Greater than $2 \frac{5}{6}$?

Understanding the Problem

To determine which products are greater than $2 \frac{5}{6}$, we need to evaluate each option by multiplying the given fractions and mixed numbers. This will help us compare the results and identify the products that exceed $2 \frac{5}{6}$.

Option A: $\frac{1}{8} \times 2 \frac{5}{6}$

To evaluate this option, we need to convert the mixed number $2 \frac{5}{6}$ to an improper fraction. We can do this by multiplying the whole number part (2) by the denominator (6) and then adding the numerator (5).

256=(2×6)+56=1762 \frac{5}{6} = \frac{(2 \times 6) + 5}{6} = \frac{17}{6}

Now, we can multiply this improper fraction by $\frac{1}{8}$.

18×176=1×178×6=1748\frac{1}{8} \times \frac{17}{6} = \frac{1 \times 17}{8 \times 6} = \frac{17}{48}

Since $\frac{17}{48}$ is less than $\frac{17}{6}$, option A is not greater than $2 \frac{5}{6}$.

Option B: $\frac{6}{5} \times 2 \frac{5}{6}$

To evaluate this option, we need to convert the mixed number $2 \frac{5}{6}$ to an improper fraction, as we did in option A.

256=1762 \frac{5}{6} = \frac{17}{6}

Now, we can multiply this improper fraction by $\frac{6}{5}$.

65×176=6×175×6=10230=5115\frac{6}{5} \times \frac{17}{6} = \frac{6 \times 17}{5 \times 6} = \frac{102}{30} = \frac{51}{15}

Since $\frac{51}{15}$ is greater than $\frac{17}{6}$, option B is greater than $2 \frac{5}{6}$.

Option C: $\frac{5}{6} \times 2 \frac{5}{6}$

To evaluate this option, we need to convert the mixed number $2 \frac{5}{6}$ to an improper fraction, as we did in option A.

256=1762 \frac{5}{6} = \frac{17}{6}

Now, we can multiply this improper fraction by $\frac{5}{6}$.

56×176=5×176×6=8536\frac{5}{6} \times \frac{17}{6} = \frac{5 \times 17}{6 \times 6} = \frac{85}{36}

Since $\frac{85}{36}$ is greater than $\frac{17}{6}$, option C is greater than $2 \frac{5}{6}$.

Option D: $2 \frac{5}{6} \times \frac{5}{6}$

To evaluate this option, we need to convert the mixed number $2 \frac{5}{6}$ to an improper fraction, as we did in option A.

256=1762 \frac{5}{6} = \frac{17}{6}

Now, we can multiply this improper fraction by $\frac{5}{6}$.

176×56=17×56×6=8536\frac{17}{6} \times \frac{5}{6} = \frac{17 \times 5}{6 \times 6} = \frac{85}{36}

Since $\frac{85}{36}$ is greater than $\frac{17}{6}$, option D is greater than $2 \frac{5}{6}$.

Conclusion

Based on our calculations, options B, C, and D are greater than $2 \frac{5}{6}$. Therefore, the correct answers are:

  • Option B: $\frac{6}{5} \times 2 \frac{5}{6}$
  • Option C: $\frac{5}{6} \times 2 \frac{5}{6}$
  • Option D: $2 \frac{5}{6} \times \frac{5}{6}$

These options result in products that exceed $2 \frac{5}{6}$, making them the correct choices.
Frequently Asked Questions (FAQs) about Multiplying Mixed Numbers and Fractions

Q: What is the first step in multiplying mixed numbers and fractions?

A: The first step is to convert the mixed number to an improper fraction. This involves multiplying the whole number part by the denominator and then adding the numerator.

Q: How do I multiply a mixed number by a fraction?

A: To multiply a mixed number by a fraction, follow these steps:

  1. Convert the mixed number to an improper fraction.
  2. Multiply the improper fraction by the fraction.
  3. Simplify the result, if possible.

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

A: Multiplying mixed numbers involves converting the mixed number to an improper fraction before multiplying, whereas multiplying fractions involves multiplying two or more fractions together.

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

A: Yes, you can multiply a mixed number by a whole number. To do this, convert the mixed number to an improper fraction and then multiply it by the whole number.

Q: How do I simplify the result of multiplying mixed numbers and fractions?

A: To simplify the result, look for common factors between the numerator and denominator. Cancel out any common factors to simplify the fraction.

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

A: Some common mistakes to avoid include:

  • Failing to convert the mixed number to an improper fraction
  • Multiplying the whole number part by the numerator instead of the denominator
  • Failing to simplify the result
  • Not canceling out common factors between the numerator and denominator

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

A: Yes, you can use a calculator to multiply mixed numbers and fractions. However, it's still important to understand the steps involved in multiplying mixed numbers and fractions to ensure accuracy.

Q: How do I multiply multiple mixed numbers and fractions together?

A: To multiply multiple mixed numbers and fractions together, follow these steps:

  1. Convert each mixed number to an improper fraction.
  2. Multiply the improper fractions together.
  3. Simplify the result, if possible.

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

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

  • Cooking and recipe scaling
  • Building and construction
  • Finance and accounting
  • Science and engineering

By understanding how to multiply mixed numbers and fractions, you can apply these skills to a variety of real-world situations.