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 provides a detailed explanation of how to calculate the product of the fractions 8/15, 6/5, and 1/3. We'll explore various methods, delve into the underlying mathematical concepts, and offer practical applications to solidify your understanding. By the end, you'll not only know the answer but also possess a strong grasp of fraction multiplication.

Understanding Fraction Multiplication

Before diving into the specific calculation, let's review the fundamental principles of multiplying fractions. The process is surprisingly straightforward:

• Multiply the numerators: The numerators are the top numbers in the fractions. Simply multiply them together.
• Multiply the denominators: The denominators are the bottom numbers. Multiply these together as well.
• Simplify the resulting fraction: Reduce the final fraction to its simplest form by finding the greatest common divisor (GCD) of the numerator and denominator and dividing both by it.

This process can be summarized with the formula:

(a/b) * (c/d) * (e/f) = (a * c * e) / (b * d * f)

Where 'a', 'c', and 'e' are the numerators, and 'b', 'd', and 'f' are the denominators.

Step-by-Step Calculation: 8/15 * 6/5 * 1/3

Now, let's apply this knowledge to calculate the product of 8/15, 6/5, and 1/3:

1. Multiply the Numerators:

8 * 6 * 1 = 48

2. Multiply the Denominators:

15 * 5 * 3 = 225

3. Form the Resulting Fraction:

This gives us the fraction 48/225.

4. Simplify the Fraction:

To simplify 48/225, we need to find the greatest common divisor (GCD) of 48 and 225. The GCD 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: Simplifying Before Multiplication

While the above method is perfectly valid, we can often simplify the calculation by canceling common factors before multiplying. This can make the numbers smaller and easier to work with. Let's illustrate this approach:

1. Identify Common Factors:

Look for common factors between any numerator and any denominator. In our example:

• 8 and 15 share no common factors.
• 6 and 15 share a common factor of 3. (6 = 2 * 3 and 15 = 3 * 5)
• 6 and 5 share no common factors.
• 6 and 3 share a common factor of 3.
• 5 and 15 share a common factor of 5.

2. Cancel Common Factors:

We can cancel out the common factors:

• Divide 6 by 3 and 15 by 3: This leaves us with 2/5 and 1/5.
• Alternatively, divide 6 by 3 and 3 by 3: this leaves 2 and 1.

The expression now becomes:

(8/5) * (2/5) * (1/1) or (8/15) * (2/1) * (1/1)

3. Multiply the Simplified Numerators and Denominators:

Now, multiply the simplified numerators and denominators:

8 * 2 * 1 = 16 5 * 5 * 1 = 25

or

8 * 2 * 1 = 16 15 * 1 * 1 = 15

This gives us the fractions 16/25 or 16/15.

4. Checking for further simplification

Since no common factor is shared between the numerator and denominator in 16/25, this is the most simplified form. However, we need to clarify which approach was correct. The correct approach is by dividing 6 and 15 by 3 first and then 6 and 3 by 3, which yields the solution of 16/75 as calculated in the original step-by-step method.

Practical Applications of Fraction Multiplication

Understanding fraction multiplication isn't just an academic exercise; it has numerous real-world applications. Consider these examples:

• Cooking and Baking: Scaling recipes up or down requires multiplying fractions. For example, if a recipe calls for 1/2 cup of flour and you want to make a double batch, you need to calculate 2 * (1/2).

• Construction and Engineering: Calculating material quantities often involves fraction multiplication. Determining the amount of wood needed for a project, for instance, might require multiplying fractional lengths and widths.

• Finance: Calculating interest or portions of investments often involves multiplying fractions.

Troubleshooting Common Mistakes

Here are some common mistakes to avoid when multiplying fractions:

• Forgetting to simplify: Always simplify the resulting fraction to its lowest terms. Leaving the fraction unsimplified is considered incomplete.

• Incorrect cancellation: Make sure you're only canceling common factors between numerators and denominators, not between two numerators or two denominators.

• Arithmetic errors: Double-check your multiplication and division.

Conclusion

Calculating the product of 8/15, 6/5, and 1/3, resulting in 16/75, is a straightforward process that involves multiplying numerators, multiplying denominators, and simplifying the resulting fraction. Mastering fraction multiplication provides a fundamental skill applicable to diverse fields. By understanding the underlying principles and practicing regularly, you can confidently tackle more complex fraction problems. Remember to always check your work and consider simplifying before multiplying to make the calculation more efficient.