Multiplying Fractions How To Simplify 7/4 Times 6/5

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Introduction: Grasping the Fundamentals of Fraction Multiplication

In the realm of mathematics, multiplying fractions might initially seem daunting, but it's a fundamental operation that becomes remarkably straightforward once you grasp the core concepts. Fractions, representing parts of a whole, play a crucial role in various mathematical fields and everyday scenarios. Understanding how to multiply them efficiently is an essential skill for students, professionals, and anyone seeking to enhance their mathematical proficiency. This comprehensive guide will delve into the step-by-step process of multiplying fractions, with a particular focus on simplifying the results to their simplest form. We'll tackle the example of $\frac{7}{4} \times \frac{6}{5}$, breaking down each step to ensure clarity and comprehension. So, let's embark on this journey to master the art of fraction multiplication!

What are Fractions? A Quick Recap

Before we dive into the multiplication process, let's briefly revisit the concept of fractions. A fraction represents a part of a whole and is expressed in the form $\frac{a}{b}$, where 'a' is the numerator (the number of parts we have) and 'b' is the denominator (the total number of parts the whole is divided into). For instance, in the fraction $\frac{1}{2}$, 1 is the numerator, and 2 is the denominator, indicating that we have one part out of a total of two parts.

The Golden Rule of Multiplying Fractions

The fundamental rule for multiplying fractions is delightfully simple: multiply the numerators together to get the new numerator, and multiply the denominators together to get the new denominator. Mathematically, this can be expressed as:

$$\frac{a}{b} \times \frac{c}{d} = \frac{a \times c}{b \times d}$$

This seemingly simple rule forms the cornerstone of fraction multiplication and provides a clear path to solving a wide array of problems. Let's now apply this rule to our example.

Step-by-Step Guide: Multiplying $\frac{7}{4} \times \frac{6}{5}$

Step 1: Multiply the Numerators

Our first step in multiplying fractions is to multiply the numerators of the two fractions. In our case, we have $\frac{7}{4} \times \frac{6}{5}$. The numerators are 7 and 6. Multiplying them gives us:

$$7 \times 6 = 42$$

This result, 42, will be the numerator of our product fraction.

Step 2: Multiply the Denominators

Next, we move on to the denominators. Again, looking at $\frac{7}{4} \times \frac{6}{5}$, the denominators are 4 and 5. Multiplying these together, we get:

$$4 \times 5 = 20$$

This result, 20, will be the denominator of our product fraction.

Step 3: Form the Product Fraction

Now that we have both the new numerator (42) and the new denominator (20), we can form the product fraction:

$$\frac{42}{20}$$

This fraction represents the result of multiplying $\frac{7}{4}$ and $\frac{6}{5}$. However, our task isn't complete yet. We need to simplify this fraction to its simplest form.

Simplifying Fractions: The Art of Reduction

Simplifying a fraction means reducing it to its lowest terms. This involves finding the greatest common factor (GCF) of the numerator and the denominator and then dividing both by this factor. The GCF is the largest number that divides both the numerator and the denominator without leaving a remainder. Simplifying fractions is crucial for expressing answers in their most concise and understandable form. It also helps in comparing fractions more easily and performing further calculations with them.

Step 4: Finding the Greatest Common Factor (GCF)

To simplify $\frac{42}{20}$, we need to find the GCF of 42 and 20. There are several methods to find the GCF, including listing the factors of each number or using the prime factorization method. Let's use the listing method for this example.

Factors of 42: 1, 2, 3, 6, 7, 14, 21, 42

Factors of 20: 1, 2, 4, 5, 10, 20

The common factors of 42 and 20 are 1 and 2. The greatest among these is 2. Therefore, the GCF of 42 and 20 is 2.

Step 5: Dividing by the GCF

Now that we've identified the GCF as 2, we divide both the numerator and the denominator of $\frac{42}{20}$ by 2:

$$\frac{42 \div 2}{20 \div 2} = \frac{21}{10}$$

The resulting fraction, $\frac{21}{10}$, is the simplified form of $\frac{42}{20}$. This means that $\frac{21}{10}$ represents the same value as $\frac{42}{20}$, but it is expressed in its lowest terms.

Converting Improper Fractions to Mixed Numbers (Optional)

The fraction $\frac{21}{10}$ is an improper fraction because the numerator (21) is greater than the denominator (10). While $\frac{21}{10}$ is a perfectly valid answer, it is often preferable to express improper fractions as mixed numbers, especially in everyday contexts. A mixed number consists of a whole number and a proper fraction (where the numerator is less than the denominator).

Step 6: Converting to a Mixed Number

To convert $\frac{21}{10}$ to a mixed number, we divide the numerator (21) by the denominator (10):

$$21 \div 10 = 2 \text{ with a remainder of } 1$$

The quotient, 2, becomes the whole number part of the mixed number. The remainder, 1, becomes the numerator of the fractional part, and the denominator remains the same (10). Thus, the mixed number equivalent of $\frac{21}{10}$ is:

$$2 \frac{1}{10}$$

This means that $\frac{21}{10}$ is equal to 2 whole units and $\frac{1}{10}$ of another unit.

The Final Answer: $\frac{21}{10}$ or $2 \frac{1}{10}$

Therefore, the product of $\frac{7}{4} \times \frac{6}{5}$, expressed as a fraction in its simplest form, is $\frac{21}{10}$. Alternatively, if you prefer to express it as a mixed number, the answer is $2 \frac{1}{10}$. Both forms are mathematically correct and represent the same value.

Key Takeaways: Mastering Fraction Multiplication

Let's recap the essential steps involved in multiplying fractions and simplifying the result:

1. Multiply the numerators: Multiply the top numbers of the fractions together.
2. Multiply the denominators: Multiply the bottom numbers of the fractions together.
3. Form the product fraction: Write the new numerator over the new denominator.
4. Simplify the fraction: Find the greatest common factor (GCF) of the numerator and denominator and divide both by the GCF.
5. (Optional) Convert to a mixed number: If the fraction is improper (numerator > denominator), divide the numerator by the denominator to express it as a mixed number.

By following these steps diligently, you can confidently multiply fractions and express the answers in their simplest form. Practice is key to mastering this skill, so try working through various examples to solidify your understanding.

Real-World Applications: Why Fraction Multiplication Matters

Understanding fraction multiplication isn't just about acing math tests; it has practical applications in numerous real-world scenarios. Here are a few examples:

• Cooking and Baking: Recipes often involve fractions when measuring ingredients. For instance, if you need to double a recipe that calls for $\frac{3}{4}$ cup of flour, you'll need to multiply $\frac{3}{4}$ by 2.
• Construction and Carpentry: Builders and carpenters frequently work with fractions when measuring materials, cutting wood, or calculating dimensions.
• Finance: Fractions are used in various financial calculations, such as determining interest rates, calculating discounts, and dividing investments.
• Science and Engineering: Many scientific and engineering calculations involve fractions, such as determining ratios, calculating proportions, and working with measurements.

These are just a few examples, but they illustrate how fraction multiplication is a valuable skill that transcends the classroom and finds its place in various aspects of our lives.

Conclusion: Embracing the Power of Fractions

Multiplying fractions is a fundamental mathematical operation that unlocks a world of possibilities. By understanding the simple rule of multiplying numerators and denominators and mastering the art of simplification, you can confidently tackle a wide range of problems. Remember, practice makes perfect, so keep honing your skills and exploring the fascinating world of fractions. Whether you're baking a cake, building a house, or analyzing data, the ability to multiply fractions effectively will serve you well. So, embrace the power of fractions and watch your mathematical prowess soar!