Calculate The Following Expression: $\frac{4}{5} - \frac{2}{4} = $

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Introduction

In mathematics, fractions are a fundamental concept that plays a crucial role in various mathematical operations. When dealing with fractions, it's essential to understand how to simplify them to make calculations easier. In this article, we will focus on calculating the expression 45−24\frac{4}{5} - \frac{2}{4} and provide a step-by-step guide on how to simplify fractions.

Understanding Fractions

A fraction is a way to represent a part of a whole. It consists of two parts: the numerator and the denominator. The numerator is the top number, and the denominator is the bottom number. For example, in the fraction 45\frac{4}{5}, the numerator is 4, and the denominator is 5.

Simplifying Fractions

To simplify a fraction, we need to find the greatest common divisor (GCD) of the numerator and the denominator. The GCD is the largest number that divides both the numerator and the denominator without leaving a remainder.

Finding the GCD

To find the GCD of two numbers, we can use the Euclidean algorithm. The Euclidean algorithm is a method for finding the GCD of two numbers by repeatedly dividing the larger number by the smaller number and taking the remainder.

Example

Let's find the GCD of 4 and 5.

  1. Divide 5 by 4: 5 ÷ 4 = 1 with a remainder of 1.
  2. Divide 4 by 1: 4 ÷ 1 = 4 with a remainder of 0.

Since the remainder is 0, the GCD of 4 and 5 is 1.

Simplifying the Fraction

Now that we have found the GCD, we can simplify the fraction 45\frac{4}{5} by dividing both the numerator and the denominator by the GCD.

45=4÷15÷1=45\frac{4}{5} = \frac{4 ÷ 1}{5 ÷ 1} = \frac{4}{5}

Since the GCD is 1, the fraction 45\frac{4}{5} is already simplified.

Calculating the Expression

Now that we have simplified the fraction 45\frac{4}{5}, we can calculate the expression 45−24\frac{4}{5} - \frac{2}{4}.

Step 1: Find the Least Common Multiple (LCM)

To subtract fractions, we need to find the least common multiple (LCM) of the denominators. The LCM is the smallest number that both denominators can divide into evenly.

Step 1.1: Find the LCM of 5 and 4

To find the LCM of 5 and 4, we can list the multiples of each number.

Multiples of 5: 5, 10, 15, 20, 25, 30, 35, 40, ... Multiples of 4: 4, 8, 12, 16, 20, 24, 28, 32, ...

The first number that appears in both lists is 20. Therefore, the LCM of 5 and 4 is 20.

Step 2: Convert the Fractions to Have the Same Denominator

Now that we have found the LCM, we can convert both fractions to have the same denominator.

45=4×45×4=1620\frac{4}{5} = \frac{4 × 4}{5 × 4} = \frac{16}{20} 24=2×54×5=1020\frac{2}{4} = \frac{2 × 5}{4 × 5} = \frac{10}{20}

Step 3: Subtract the Fractions

Now that both fractions have the same denominator, we can subtract them.

1620−1020=16−1020=620\frac{16}{20} - \frac{10}{20} = \frac{16 - 10}{20} = \frac{6}{20}

Step 4: Simplify the Result

Finally, we can simplify the result by dividing both the numerator and the denominator by their GCD.

620=6÷220÷2=310\frac{6}{20} = \frac{6 ÷ 2}{20 ÷ 2} = \frac{3}{10}

Therefore, the final answer is 310\boxed{\frac{3}{10}}.

Conclusion

In this article, we have learned how to simplify fractions and calculate the expression 45−24\frac{4}{5} - \frac{2}{4}. We have also learned how to find the greatest common divisor (GCD) and the least common multiple (LCM) of two numbers. By following these steps, we can simplify fractions and perform mathematical operations with ease.

Frequently Asked Questions

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

A: The GCD of two numbers is the largest number that divides both numbers without leaving a remainder.

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

A: The LCM of two numbers is the smallest number that both numbers can divide into evenly.

Q: How do I simplify a fraction?

A: To simplify a fraction, you need to find the GCD of the numerator and the denominator and divide both numbers by the GCD.

Q: How do I calculate the expression 45−24\frac{4}{5} - \frac{2}{4}?

A: To calculate the expression 45−24\frac{4}{5} - \frac{2}{4}, you need to find the LCM of the denominators, convert both fractions to have the same denominator, subtract the fractions, and simplify the result.

References

Introduction

In our previous article, we discussed how to simplify fractions and calculate the expression 45−24\frac{4}{5} - \frac{2}{4}. We also provided a step-by-step guide on how to find the greatest common divisor (GCD) and the least common multiple (LCM) of two numbers. In this article, we will answer some frequently asked questions related to simplifying fractions and calculating expressions.

Q&A

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

A: The GCD of two numbers is the largest number that divides both numbers without leaving a remainder.

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

A: The LCM of two numbers is the smallest number that both numbers can divide into evenly.

Q: How do I simplify a fraction?

A: To simplify a fraction, you need to find the GCD of the numerator and the denominator and divide both numbers by the GCD.

Q: How do I calculate the expression 45−24\frac{4}{5} - \frac{2}{4}?

A: To calculate the expression 45−24\frac{4}{5} - \frac{2}{4}, you need to find the LCM of the denominators, convert both fractions to have the same denominator, subtract the fractions, and simplify the result.

Q: What is the difference between the GCD and the LCM?

A: The GCD is the largest number that divides both numbers without leaving a remainder, while the LCM is the smallest number that both numbers can divide into evenly.

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

A: You can find the GCD of two numbers by using the Euclidean algorithm or by listing the factors of each number.

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

A: You can find the LCM of two numbers by listing the multiples of each number or by using the formula: LCM(a, b) = (a × b) / GCD(a, b)

Q: Can I simplify a fraction with a negative numerator or denominator?

A: Yes, you can simplify a fraction with a negative numerator or denominator by following the same steps as simplifying a fraction with positive numbers.

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

A: Yes, you can simplify a fraction with a decimal numerator or denominator by converting the decimal to a fraction and then simplifying.

Q: How do I calculate the expression 34+26\frac{3}{4} + \frac{2}{6}?

A: To calculate the expression 34+26\frac{3}{4} + \frac{2}{6}, you need to find the LCM of the denominators, convert both fractions to have the same denominator, add the fractions, and simplify the result.

Q: How do I calculate the expression 56−38\frac{5}{6} - \frac{3}{8}?

A: To calculate the expression 56−38\frac{5}{6} - \frac{3}{8}, you need to find the LCM of the denominators, convert both fractions to have the same denominator, subtract the fractions, and simplify the result.

Conclusion

In this article, we have answered some frequently asked questions related to simplifying fractions and calculating expressions. We have also provided examples and explanations to help you understand the concepts better. By following these steps and practicing regularly, you will become proficient in simplifying fractions and calculating expressions.

Additional Resources

Practice Problems

  1. Simplify the fraction 1216\frac{12}{16}.
  2. Calculate the expression 34+26\frac{3}{4} + \frac{2}{6}.
  3. Simplify the fraction 1520\frac{15}{20}.
  4. Calculate the expression 56−38\frac{5}{6} - \frac{3}{8}.
  5. Simplify the fraction 2432\frac{24}{32}.

Answer Key

  1. 34\frac{3}{4}
  2. 56\frac{5}{6}
  3. 34\frac{3}{4}
  4. 124\frac{1}{24}
  5. 38\frac{3}{8}