Calculate The Following Expression: 5 9 × 9 8 \frac{5}{9} \times \frac{9}{8} 9 5 ​ × 8 9 ​

Introduction

In mathematics, fractions are a fundamental concept used to represent a part of a whole. When dealing with complex fractions, it's essential to simplify them to make calculations easier and more manageable. In this article, we will focus on simplifying the expression $\frac{5}{9} \times \frac{9}{8}$ using step-by-step instructions.

Understanding the Expression

The given expression is a product of two fractions: $\frac{5}{9}$ and $\frac{9}{8}$. To simplify this expression, we need to understand the concept of multiplying fractions. When multiplying fractions, we multiply the numerators (the numbers on top) and multiply the denominators (the numbers on the bottom).

Step 1: Multiply the Numerators

To multiply the numerators, we simply multiply the numbers on top: $5 \times 9 = 45$.

Step 2: Multiply the Denominators

To multiply the denominators, we simply multiply the numbers on the bottom: $9 \times 8 = 72$.

Step 3: Write the Product as a Fraction

Now that we have multiplied the numerators and denominators, we can write the product as a fraction: $\frac{45}{72}$.

Simplifying the Fraction

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

Finding the GCD

To find the GCD of 45 and 72, we can use the Euclidean algorithm or list the factors of each number. The factors of 45 are 1, 3, 5, 9, 15, and 45. The factors of 72 are 1, 2, 3, 4, 6, 8, 9, 12, 18, 24, 36, and 72. The greatest common divisor of 45 and 72 is 9.

Simplifying the Fraction

Now that we have found the GCD, we can simplify the fraction by dividing both the numerator and denominator by the GCD: $\frac{45 \div 9}{72 \div 9} = \frac{5}{8}$.

Conclusion

In conclusion, simplifying complex fractions requires a step-by-step approach. By multiplying the numerators and denominators, finding the GCD, and simplifying the fraction, we can arrive at the final answer. In this article, we simplified the expression $\frac{5}{9} \times \frac{9}{8}$ to $\frac{5}{8}$.

Real-World Applications

Simplifying complex fractions has numerous real-world applications in mathematics, science, and engineering. For example, in physics, simplifying complex fractions can help us calculate the trajectory of an object under the influence of gravity. In engineering, simplifying complex fractions can help us design more efficient systems and structures.

Common Mistakes to Avoid

When simplifying complex fractions, there are several common mistakes to avoid. These include:

• Not finding the GCD: Failing to find the GCD can result in an unsimplified fraction.
• Not dividing both the numerator and denominator by the GCD: Failing to divide both the numerator and denominator by the GCD can result in an unsimplified fraction.
• Not checking for common factors: Failing to check for common factors can result in an unsimplified fraction.

Tips and Tricks

When simplifying complex fractions, here are some tips and tricks to keep in mind:

• Use the Euclidean algorithm: The Euclidean algorithm is a fast and efficient way to find the GCD of two numbers.
• List the factors: Listing the factors of each number can help you find the GCD.
• Check for common factors: Checking for common factors can help you simplify the fraction.

Conclusion

$$$$

Introduction

In our previous article, we discussed the step-by-step process of simplifying complex fractions. In this article, we will answer some frequently asked questions (FAQs) about simplifying complex fractions.

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, you need to follow these steps:

1. Multiply the numerators and denominators of the fractions.
2. Find the greatest common divisor (GCD) of the numerator and denominator.
3. Divide both the numerator and denominator by the GCD.

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

A: The GCD is the largest number that divides both the numerator and denominator without leaving a remainder.

Q: How do I find the GCD?

A: There are several ways to find the GCD, including:

• Using the Euclidean algorithm
• Listing the factors of each number
• Using a calculator

Q: What is the difference between a simplified fraction and an unsimplified fraction?

A: A simplified fraction is a fraction that has been reduced to its simplest form, while an unsimplified fraction is a fraction that has not been reduced to its simplest form.

Q: Why is it important to simplify complex fractions?

A: Simplifying complex fractions is important because it makes calculations easier and more manageable. It also helps to avoid errors and ensures that the final answer is accurate.

Q: Can I simplify a complex fraction with a variable?

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

Q: What are some common mistakes to avoid when simplifying complex fractions?

A: Some common mistakes to avoid when simplifying complex fractions include:

• Not finding the GCD
• Not dividing both the numerator and denominator by the GCD
• Not checking for common factors

Q: How do I check for common factors?

A: To check for common factors, you need to list the factors of each number and look for any common factors.

Q: What are some real-world applications of simplifying complex fractions?

A: Simplifying complex fractions has numerous real-world applications in mathematics, science, and engineering. For example, in physics, simplifying complex fractions can help us calculate the trajectory of an object under the influence of gravity. In engineering, simplifying complex fractions can help us design more efficient systems and structures.

Q: Can I use a calculator to simplify complex fractions?

A: Yes, you can use a calculator to simplify complex fractions. However, it's always a good idea to double-check your work to ensure that the final answer is accurate.

Conclusion

In conclusion, simplifying complex fractions requires a step-by-step approach. By following the steps outlined in this article, you can simplify complex fractions and arrive at the final answer. Remember to avoid common mistakes and use algebraic techniques to simplify fractions with variables.