Calculate The Following Expression:$\[ \frac{1}{2} - \frac{1}{4} - \frac{4}{64} \\]A. \[$\frac{3}{16}\$\] B. \[$\frac{14}{64}\$\] C. \[$\frac{1}{8}\$\] D. \[$\frac{1}{128}\$\]

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Introduction

Fractions are an essential part of mathematics, and simplifying complex fractions is a crucial skill to master. In this article, we will focus on calculating the expression 12βˆ’14βˆ’464\frac{1}{2} - \frac{1}{4} - \frac{4}{64} and explore the different options available. We will break down the problem step by step, using simple language and clear explanations to ensure that readers understand the concept.

Understanding the Expression

The given expression is 12βˆ’14βˆ’464\frac{1}{2} - \frac{1}{4} - \frac{4}{64}. To simplify this expression, we need to find a common denominator for all three fractions. The least common multiple (LCM) of 2, 4, and 64 is 64.

Finding the Common Denominator

To find the common denominator, we need to express each fraction with a denominator of 64.

  • 12=1Γ—322Γ—32=3264\frac{1}{2} = \frac{1 \times 32}{2 \times 32} = \frac{32}{64}
  • 14=1Γ—164Γ—16=1664\frac{1}{4} = \frac{1 \times 16}{4 \times 16} = \frac{16}{64}
  • 464\frac{4}{64} is already in the correct form.

Subtracting the Fractions

Now that we have a common denominator, we can subtract the fractions.

3264βˆ’1664βˆ’464=32βˆ’16βˆ’464=1264\frac{32}{64} - \frac{16}{64} - \frac{4}{64} = \frac{32 - 16 - 4}{64} = \frac{12}{64}

Simplifying the Result

The fraction 1264\frac{12}{64} can be simplified by dividing both the numerator and the denominator by their greatest common divisor (GCD), which is 4.

1264=12Γ·464Γ·4=316\frac{12}{64} = \frac{12 \div 4}{64 \div 4} = \frac{3}{16}

Conclusion

In conclusion, the correct answer to the expression 12βˆ’14βˆ’464\frac{1}{2} - \frac{1}{4} - \frac{4}{64} is 316\frac{3}{16}. This result can be obtained by finding a common denominator, subtracting the fractions, and simplifying the result.

Comparison with Other Options

Let's compare our result with the other options available.

  • Option A: 316\frac{3}{16} - This is the correct answer.
  • Option B: 1464\frac{14}{64} - This is not the correct answer, as it does not match our result.
  • Option C: 18\frac{1}{8} - This is not the correct answer, as it does not match our result.
  • Option D: 1128\frac{1}{128} - This is not the correct answer, as it does not match our result.

Tips and Tricks

When simplifying complex fractions, it's essential to follow these tips and tricks:

  • Find the least common multiple (LCM): The LCM of the denominators is the common denominator.
  • Express each fraction with the common denominator: This ensures that all fractions have the same denominator.
  • Subtract the fractions: Once all fractions have the same denominator, you can subtract them.
  • Simplify the result: Divide both the numerator and the denominator by their greatest common divisor (GCD) to simplify the result.

By following these tips and tricks, you can simplify complex fractions with ease and accuracy.

Real-World Applications

Simplifying complex fractions has numerous real-world applications, including:

  • Finance: When calculating interest rates, investments, or loans, complex fractions are often involved.
  • Science: In physics, chemistry, and biology, complex fractions are used to represent rates, ratios, and proportions.
  • Engineering: In civil engineering, mechanical engineering, and electrical engineering, complex fractions are used to design and analyze systems.

Introduction

In our previous article, we explored the concept of simplifying complex fractions and calculated the expression 12βˆ’14βˆ’464\frac{1}{2} - \frac{1}{4} - \frac{4}{64}. In this article, we will answer some frequently asked questions (FAQs) related to simplifying complex fractions.

Q&A

Q: What is a complex fraction?

A: A complex fraction is a fraction that contains one or more fractions in its numerator or denominator.

Q: How do I simplify a complex fraction?

A: To simplify a complex fraction, follow these steps:

  1. Find the least common multiple (LCM) of the denominators.
  2. Express each fraction with the common denominator.
  3. Subtract the fractions.
  4. Simplify the result by dividing both the numerator and the denominator by their greatest common divisor (GCD).

Q: What is the least common multiple (LCM)?

A: The least common multiple (LCM) of a set of numbers is the smallest number that is a multiple of each of the numbers in the set.

Q: How do I find the LCM of two or more numbers?

A: To find the LCM of two or more numbers, follow these steps:

  1. List the multiples of each number.
  2. Identify the smallest number that appears in the list of multiples.
  3. The LCM is the smallest number that appears in the list of multiples.

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

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

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

A: To find the GCD of two or more numbers, follow these steps:

  1. List the factors of each number.
  2. Identify the largest number that appears in the list of factors.
  3. The GCD is the largest number that appears in the list of factors.

Q: Can I simplify a complex fraction with a variable in the numerator or denominator?

A: Yes, you can simplify a complex fraction with a variable in the numerator or denominator. However, you need to follow the same steps as before, and you may need to use algebraic techniques to simplify the result.

Q: How do I simplify a complex fraction with a negative sign?

A: To simplify a complex fraction with a negative sign, follow these steps:

  1. Change the sign of the numerator and the denominator.
  2. Simplify the result using the steps outlined above.

Q: Can I simplify a complex fraction with a decimal in the numerator or denominator?

A: Yes, you can simplify a complex fraction with a decimal in the numerator or denominator. However, you need to follow the same steps as before, and you may need to use decimal arithmetic to simplify the result.

Tips and Tricks

When simplifying complex fractions, it's essential to follow these tips and tricks:

  • Use a common denominator: Express each fraction with a common denominator to make it easier to subtract the fractions.
  • Simplify the result: Divide both the numerator and the denominator by their greatest common divisor (GCD) to simplify the result.
  • Check your work: Verify that your result is correct by plugging it back into the original expression.

Real-World Applications

Simplifying complex fractions has numerous real-world applications, including:

  • Finance: When calculating interest rates, investments, or loans, complex fractions are often involved.
  • Science: In physics, chemistry, and biology, complex fractions are used to represent rates, ratios, and proportions.
  • Engineering: In civil engineering, mechanical engineering, and electrical engineering, complex fractions are used to design and analyze systems.

In conclusion, simplifying complex fractions is a crucial skill that has numerous real-world applications. By following the steps outlined in this article and using the tips and tricks provided, you can simplify complex fractions with ease and accuracy.