Calculate The Following Expression:${ \frac{2}{3} - \frac{1}{2} \times \frac{3}{4} }$

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Introduction

Fractions are an essential part of mathematics, and understanding how to simplify complex fractions is crucial for solving various mathematical problems. In this article, we will focus on calculating the expression 23−12×34\frac{2}{3} - \frac{1}{2} \times \frac{3}{4}, which involves subtracting and multiplying fractions. We will break down the problem into manageable steps and provide a clear explanation of each step.

Understanding the Problem

The given expression is a combination of subtraction and multiplication of fractions. To simplify this expression, we need to follow the order of operations (PEMDAS):

  1. Parentheses: None in this case
  2. Exponents: None in this case
  3. Multiplication and Division: Evaluate from left to right
  4. Addition and Subtraction: Evaluate from left to right

Step 1: Multiply the Fractions

The first step is to multiply the fractions 12\frac{1}{2} and 34\frac{3}{4}. To multiply fractions, we multiply the numerators and denominators separately:

12×34=1×32×4=38\frac{1}{2} \times \frac{3}{4} = \frac{1 \times 3}{2 \times 4} = \frac{3}{8}

Step 2: Subtract the Result from the First Fraction

Now that we have the result of the multiplication, we can subtract it from the first fraction 23\frac{2}{3}:

23−38\frac{2}{3} - \frac{3}{8}

To subtract fractions, we need to have a common denominator. The least common multiple (LCM) of 3 and 8 is 24. We can rewrite the fractions with the common denominator:

23=2×83×8=1624\frac{2}{3} = \frac{2 \times 8}{3 \times 8} = \frac{16}{24}

38=3×38×3=924\frac{3}{8} = \frac{3 \times 3}{8 \times 3} = \frac{9}{24}

Now we can subtract the fractions:

1624−924=16−924=724\frac{16}{24} - \frac{9}{24} = \frac{16 - 9}{24} = \frac{7}{24}

Conclusion

In this article, we simplified the complex fraction expression 23−12×34\frac{2}{3} - \frac{1}{2} \times \frac{3}{4} by following the order of operations and using the rules for subtracting and multiplying fractions. We broke down the problem into manageable steps and provided a clear explanation of each step. By understanding how to simplify complex fractions, we can solve various mathematical problems with confidence.

Tips and Tricks

  • When multiplying fractions, multiply the numerators and denominators separately.
  • When subtracting fractions, find the least common multiple (LCM) of the denominators and rewrite the fractions with the common denominator.
  • Always follow the order of operations (PEMDAS) when simplifying complex expressions.

Real-World Applications

Simplifying complex fractions is an essential skill in various fields, including:

  • Finance: Understanding how to simplify complex fractions can help you calculate interest rates, investment returns, and other financial metrics.
  • Science: Simplifying complex fractions is crucial in scientific calculations, such as calculating the area of a circle or the volume of a sphere.
  • Engineering: Understanding how to simplify complex fractions can help you design and optimize systems, such as electrical circuits or mechanical systems.

Introduction

In our previous article, we explored the concept of simplifying complex fractions and provided a step-by-step guide on how to calculate the expression 23−12×34\frac{2}{3} - \frac{1}{2} \times \frac{3}{4}. In this article, we will address some of the most frequently asked questions about 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 the order of operations (PEMDAS):

  1. Parentheses: None in this case
  2. Exponents: None in this case
  3. Multiplication and Division: Evaluate from left to right
  4. Addition and Subtraction: Evaluate from left to right

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

A: The least common multiple (LCM) is the smallest multiple that two or more numbers have in common.

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

A: To find the LCM of two numbers, list the multiples of each number and find the smallest multiple that appears in both lists.

Q: Can I simplify a complex fraction by canceling out common factors?

A: Yes, you can simplify a complex fraction by canceling out common factors in the numerator and denominator.

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

A: A fraction is a ratio of two numbers, while a decimal is a way of representing a fraction as a series of digits.

Q: Can I convert a fraction to a decimal?

A: Yes, you can convert a fraction to a decimal by dividing the numerator by the denominator.

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

A: To simplify a complex fraction with negative numbers, follow the same steps as before, but be careful with the signs.

Q: Can I simplify a complex fraction with variables?

A: Yes, you can simplify a complex fraction with variables by following the same steps as before, but be careful with the variables.

Common Mistakes

  • Forgetting to follow the order of operations (PEMDAS)
  • Not finding the least common multiple (LCM) of the denominators
  • Canceling out common factors incorrectly
  • Not being careful with negative numbers and variables

Tips and Tricks

  • Always follow the order of operations (PEMDAS)
  • Find the least common multiple (LCM) of the denominators
  • Be careful with negative numbers and variables
  • Practice, practice, practice!

Real-World Applications

Simplifying complex fractions is an essential skill in various fields, including:

  • Finance: Understanding how to simplify complex fractions can help you calculate interest rates, investment returns, and other financial metrics.
  • Science: Simplifying complex fractions is crucial in scientific calculations, such as calculating the area of a circle or the volume of a sphere.
  • Engineering: Understanding how to simplify complex fractions can help you design and optimize systems, such as electrical circuits or mechanical systems.

By mastering the art of simplifying complex fractions, you can solve a wide range of mathematical problems with confidence and accuracy.