What's The Product Of 3 2/3 And 14 2/5

What's the Product of 3 2/3 and 14 2/5? A Deep Dive into Mixed Number Multiplication

This article will thoroughly explore the process of multiplying mixed numbers, using the example of 3 2/3 and 14 2/5. We'll cover the fundamental steps, provide alternative methods, discuss the importance of simplification, and delve into practical applications. This comprehensive guide aims to equip you with a solid understanding of this mathematical concept, improving your numeracy skills and enhancing your problem-solving abilities.

Understanding Mixed Numbers

Before diving into the multiplication, let's solidify our understanding of mixed numbers. A mixed number combines a whole number and a fraction. For instance, 3 2/3 represents three whole units and two-thirds of another unit. Similarly, 14 2/5 represents fourteen whole units and two-fifths of another unit. Understanding this representation is crucial for accurately performing calculations.

Method 1: Converting to Improper Fractions

The most common and generally recommended method for multiplying mixed numbers involves converting them into improper fractions. Improper fractions have a numerator larger than or equal to their denominator.

Step 1: Convert 3 2/3 to an improper fraction:

To do this, we multiply the whole number (3) by the denominator (3), add the numerator (2), and keep the same denominator (3). This gives us:

(3 * 3) + 2 = 11

Therefore, 3 2/3 becomes 11/3.

Step 2: Convert 14 2/5 to an improper fraction:

Following the same process, we multiply the whole number (14) by the denominator (5), add the numerator (2), and keep the same denominator (5):

(14 * 5) + 2 = 72

Therefore, 14 2/5 becomes 72/5.

Step 3: Multiply the improper fractions:

Now that we have both mixed numbers as improper fractions, we can multiply them:

(11/3) * (72/5) = (11 * 72) / (3 * 5) = 792 / 15

Step 4: Simplify the resulting fraction:

The fraction 792/15 can be simplified by finding the greatest common divisor (GCD) of the numerator and denominator. The GCD of 792 and 15 is 3. Dividing both the numerator and the denominator by 3 gives us:

792 / 3 = 264 15 / 3 = 5

Therefore, the simplified fraction is 264/5.

Step 5: Convert back to a mixed number (optional):

While 264/5 is a perfectly acceptable answer, we can convert it back to a mixed number for easier interpretation. To do this, we divide the numerator (264) by the denominator (5):

264 ÷ 5 = 52 with a remainder of 4

This means that 264/5 is equal to 52 4/5.

Therefore, the product of 3 2/3 and 14 2/5 is 52 4/5.

Method 2: Distributive Property (Less Efficient, but Illustrative)

While less efficient for this specific problem, understanding the distributive property can be valuable for more complex scenarios. This method involves expanding the multiplication using the distributive property.

We can rewrite 3 2/3 as 3 + 2/3 and 14 2/5 as 14 + 2/5. Then, we multiply using the distributive property:

(3 + 2/3) * (14 + 2/5) = 3(14 + 2/5) + 2/3(14 + 2/5)

This expands to:

(3 * 14) + (3 * 2/5) + (2/3 * 14) + (2/3 * 2/5) = 42 + 6/5 + 28/3 + 4/15

Now, we need to find a common denominator for all the fractions, which is 15. Converting and adding:

42 + 18/15 + 140/15 + 4/15 = 42 + 162/15

Simplifying the fraction:

162/15 = 54/5 = 10 4/5

Finally, add the whole numbers:

42 + 10 4/5 = 52 4/5

As you can see, this method is more complex and prone to errors compared to the improper fraction method. However, it demonstrates the underlying principles of distributive multiplication.

Importance of Simplification

Simplifying fractions is crucial for obtaining the most concise and easily understandable answer. A simplified fraction represents the same value as the unsimplified fraction but in its most reduced form. In our example, simplifying 792/15 to 264/5, and further to 52 4/5, makes the answer much clearer and easier to use in further calculations or real-world applications.

Practical Applications

Multiplying mixed numbers has numerous practical applications in various fields:

• Cooking and Baking: Scaling recipes up or down often involves multiplying mixed number quantities (e.g., 1 1/2 cups of flour multiplied by 2).
• Construction and Engineering: Calculating dimensions and material quantities frequently requires multiplying mixed numbers (e.g., finding the area of a rectangular space with dimensions 12 1/2 feet by 8 2/3 feet).
• Sewing and Tailoring: Determining fabric requirements or adjusting pattern sizes involves calculations with mixed numbers.
• Finance and Budgeting: Calculating percentage increases or decreases, interest, or portions of a budget might involve multiplying mixed numbers.

Conclusion: Mastering Mixed Number Multiplication

Mastering the multiplication of mixed numbers is a fundamental skill with broad applicability. The conversion to improper fractions method provides a straightforward and efficient approach. While alternative methods exist, understanding the core concept and applying the correct technique is crucial for accurate and efficient problem-solving. Remember to always simplify your answers to obtain the most concise and readily understandable results. By understanding the process and practicing regularly, you'll confidently handle mixed number multiplication in any context. This skill will undoubtedly enhance your mathematical abilities and prove invaluable in various aspects of life. The product of 3 2/3 and 14 2/5, after careful calculation and simplification, is definitively 52 4/5. Now you have a comprehensive understanding of how to arrive at this answer and the underlying principles involved.