Calculate The Product Of 8/15 6/5 And 1/3

Calculating the Product of 8/15, 6/5, and 1/3: A Comprehensive Guide

This article will delve into the detailed process of calculating the product of the three fractions: 8/15, 6/5, and 1/3. We'll explore the fundamental principles of fraction multiplication, offer various methods for solving the problem, and discuss the application of these principles in more complex scenarios. This comprehensive guide aims to enhance your understanding of fraction arithmetic and equip you with the skills to tackle similar problems with confidence.

Understanding Fraction Multiplication

Before diving into the calculation, let's review the basic rules of multiplying fractions. Multiplying fractions is a straightforward process:

1. Multiply the Numerators: The numerators are the top numbers in each fraction. Multiply these numbers together to get the numerator of the product.

2. Multiply the Denominators: The denominators are the bottom numbers in each fraction. Multiply these numbers together to get the denominator of the product.

3. Simplify the Result (if possible): The resulting fraction should be simplified to its lowest terms. This means reducing the fraction to its simplest form by dividing both the numerator and the denominator by their greatest common divisor (GCD).

Calculating the Product: Step-by-Step

Let's apply these rules to calculate the product of 8/15, 6/5, and 1/3:

1. Set up the multiplication:

(8/15) * (6/5) * (1/3)

2. Multiply the numerators:

8 * 6 * 1 = 48

3. Multiply the denominators:

15 * 5 * 3 = 225

4. Form the resulting fraction:

48/225

5. Simplify the fraction:

To simplify 48/225, we need to find the greatest common divisor (GCD) of 48 and 225. The GCD of 48 and 225 is 3. Dividing both the numerator and the denominator by 3, we get:

48 ÷ 3 = 16 225 ÷ 3 = 75

Therefore, the simplified fraction is 16/75.

Alternative Methods for Calculation

While the above method is the most straightforward, there are alternative approaches that can simplify the calculation, especially when dealing with larger numbers or more fractions.

Method 1: Cancellation before Multiplication

This method involves simplifying the fractions before multiplying. We can cancel common factors between numerators and denominators. Observe the initial fractions:

(8/15) * (6/5) * (1/3)

Notice that:

• 6 and 15 share a common factor of 3 (6 = 23 and 15 = 53). We can cancel out the 3.
• 8 and 3 have no common factors.
• 2 and 5 have no common factors.

Let's rewrite the equation after canceling common factors:

(8/5) * (2/5) * (1/1)

Now, multiply the numerators: 8 * 2 * 1 = 16

And the denominators: 5 * 5 * 1 = 25

This gives us 16/25. Note: there was an error in the previous calculation which resulted in 16/75. This method reveals the correct answer to be 16/25

Method 2: Prime Factorization

Prime factorization is a powerful technique for simplifying fractions. It involves breaking down each number into its prime factors. Let's apply this to our problem:

• 8 = 2 * 2 * 2
• 15 = 3 * 5
• 6 = 2 * 3
• 5 = 5
• 3 = 3

Rewrite the fractions using their prime factors:

((222)/(35)) * ((23)/5) * (1/3)

Now, cancel out common factors in the numerators and denominators. You'll find that many factors cancel, leaving you with a much simpler fraction before you even perform the multiplication. After cancellation, you will be left with 16/25

Applications and Extensions

The principles of fraction multiplication extend far beyond this simple example. Understanding these principles is crucial for various mathematical and real-world applications, including:

• Geometry: Calculating areas and volumes often involves multiplying fractions representing lengths and widths.
• Probability: Determining the probability of multiple independent events occurring requires multiplying individual probabilities, which are often expressed as fractions.
• Cooking and Baking: Scaling recipes involves multiplying ingredient amounts by fractions.
• Engineering: Calculating ratios and proportions in engineering designs often requires working with fractions.

Advanced Concepts and Challenges

For those seeking more advanced challenges, consider these extensions:

• Multiplying more than three fractions: The principles remain the same; multiply all numerators and all denominators, then simplify.
• Mixed numbers: Convert mixed numbers (e.g., 1 1/2) into improper fractions before multiplying.
• Fractions with variables: Applying the same principles to algebraic expressions involving fractions.
• Solving equations with fractions: Using fraction multiplication to isolate variables in algebraic equations.

Conclusion

Calculating the product of 8/15, 6/5, and 1/3, though seemingly simple, offers a valuable opportunity to reinforce fundamental concepts in fraction arithmetic. Understanding the different methods – direct multiplication, cancellation, and prime factorization – provides a flexible and efficient approach to tackling such problems. Mastering these techniques opens the door to tackling more complex mathematical scenarios and real-world applications that require a solid understanding of fractions. Remember that consistent practice and a clear understanding of the underlying principles are key to success in fraction arithmetic. The correct answer, as shown by cancellation before multiplication and prime factorization, is 16/25.