Computing the Given Fraction

Oct 13, 2025 by miodeya.com 130 views

Introduction

In mathematics, factorials and fractions are fundamental concepts that are used extensively in various mathematical operations. A factorial of a number is the product of all positive integers less than or equal to that number. On the other hand, a fraction is a way of expressing a part of a whole as a ratio of two numbers. In this article, we will focus on computing the value of a given fraction involving a factorial.

Understanding Factorials

A factorial of a number n, denoted by n!, is the product of all positive integers less than or equal to n. For example, the factorial of 5 (denoted by 5!) is equal to 5 × 4 × 3 × 2 × 1 = 120. Factorials are used extensively in mathematics, particularly in combinatorics and algebra.

Computing the Given Fraction

The given fraction is $\frac{71}{3!}$ . To compute this fraction, we need to first calculate the value of 3!. As mentioned earlier, 3! is equal to 3 × 2 × 1 = 6. Now, we can substitute this value into the fraction:

$\frac{71}{3!} = \frac{71}{6}$

To simplify this fraction, we can divide the numerator (71) by the denominator (6). This gives us:

$\frac{71}{6} = 11.83$

Evaluating the Options

Now that we have computed the value of the given fraction, we can evaluate the options provided:

a. 1680 b. 840 c. 21 d. 4

Comparing the computed value (11.83) with the options, we can see that none of the options match the computed value. However, we can try to find the closest option by rounding the computed value to the nearest whole number.

Rounding the Computed Value

Rounding the computed value (11.83) to the nearest whole number gives us 12. However, none of the options match this value either. We can try to round the computed value to the nearest multiple of 10, which gives us 10 or 20. However, none of the options match these values either.

Conclusion

In conclusion, the computed value of the given fraction ( $\frac{71}{3!}$ ) is 11.83, which does not match any of the options provided. Therefore, the correct answer is not among the options provided.

Final Answer

The final answer is not among the options provided. However, if we had to choose an option based on the computed value, we would choose option A (1680) as the closest option.

Discussion

This problem requires a good understanding of factorials and fractions. The student needs to be able to compute the value of a factorial and then use it to simplify a fraction. The student also needs to be able to evaluate the options provided and choose the closest one based on the computed value.

Additional Resources

For more information on factorials and fractions, please refer to the following resources:

FAQs

Q: What is a factorial? A: A factorial of a number n is the product of all positive integers less than or equal to n.
Q: What is a fraction? A: A fraction is a way of expressing a part of a whole as a ratio of two numbers.
Q: How do I compute the value of a factorial? A: To compute the value of a factorial, you need to multiply all positive integers less than or equal to the given number.
Q: How do I simplify a fraction? A: To simplify a fraction, you need to divide the numerator by the denominator.
Factorial and Fraction Q&A =============================

Introduction

In our previous article, we discussed computing the value of a given fraction involving a factorial. In this article, we will provide a Q&A section to help students understand factorials and fractions better.

Q&A

Q: What is a factorial?

A: A factorial of a number n is the product of all positive integers less than or equal to n. For example, the factorial of 5 (denoted by 5!) is equal to 5 × 4 × 3 × 2 × 1 = 120.

Q: What is a fraction?

A: A fraction is a way of expressing a part of a whole as a ratio of two numbers. For example, the fraction 1/2 represents one half of a whole.

Q: How do I compute the value of a factorial?

A: To compute the value of a factorial, you need to multiply all positive integers less than or equal to the given number. For example, to compute the value of 5!, you need to multiply 5 × 4 × 3 × 2 × 1 = 120.

Q: How do I simplify a fraction?

A: To simplify a fraction, you need to divide the numerator by the denominator. For example, to simplify the fraction 12/4, you need to divide 12 by 4, which gives you 3.

Q: What is the difference between a factorial and a fraction?

A: A factorial is a product of all positive integers less than or equal to a given number, while a fraction is a way of expressing a part of a whole as a ratio of two numbers.

Q: Can I use a calculator to compute the value of a factorial?

A: Yes, you can use a calculator to compute the value of a factorial. However, it's always a good idea to understand the concept behind the calculation.

Q: Can I simplify a fraction with a negative number?

A: Yes, you can simplify a fraction with a negative number. For example, to simplify the fraction -12/4, you need to divide -12 by 4, which gives you -3.

Q: Can I use a fraction to represent a decimal number?

A: Yes, you can use a fraction to represent a decimal number. For example, the decimal number 0.5 can be represented as the fraction 1/2.

Q: Can I use a factorial to represent a decimal number?

A: No, you cannot use a factorial to represent a decimal number. Factorials are used to represent products of integers, while decimal numbers are used to represent fractions with a denominator of 10 or a power of 10.

Q: Can I use a fraction to represent a percentage?

A: Yes, you can use a fraction to represent a percentage. For example, the percentage 25% can be represented as the fraction 1/4.

Q: Can I use a factorial to represent a percentage?

A: No, you cannot use a factorial to represent a percentage. Factorials are used to represent products of integers, while percentages are used to represent fractions with a denominator of 100.

Conclusion

In conclusion, factorials and fractions are fundamental concepts in mathematics that are used extensively in various mathematical operations. Understanding the concept of factorials and fractions is essential to solving problems involving these concepts. We hope that this Q&A section has helped you understand factorials and fractions better.

Additional Resources

For more information on factorials and fractions, please refer to the following resources:

FAQs

Q: What is a factorial? A: A factorial of a number n is the product of all positive integers less than or equal to n.
Q: What is a fraction? A: A fraction is a way of expressing a part of a whole as a ratio of two numbers.
Q: How do I compute the value of a factorial? A: To compute the value of a factorial, you need to multiply all positive integers less than or equal to the given number.
Q: How do I simplify a fraction? A: To simplify a fraction, you need to divide the numerator by the denominator.