Expressing Fractions With The Least Common Denominator LCD

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This article dives deep into the concept of expressing fractions using the Least Common Denominator (LCD). We'll break down the steps involved, explain why it's important, and then apply this knowledge to solve the given problem. Understanding LCD is crucial for performing various mathematical operations with fractions, such as addition, subtraction, and comparison. It ensures that we're working with fractions that have the same "size" of pieces, making calculations much simpler and more accurate.

13. Express the fractions $3 / 4, 7 / 16$, and $5 / 8$ with the LCD.

Before we jump into the solution, let's first understand what the Least Common Denominator (LCD) actually is and why it's so important when dealing with fractions. Then, we'll methodically work through finding the LCD for the given fractions and converting each one accordingly. By the end of this, you'll have a solid grasp on this fundamental concept.

Understanding the Least Common Denominator (LCD)

The Least Common Denominator (LCD) is the smallest common multiple of the denominators of a set of fractions. Think of it as finding the smallest "shared size" for the pieces represented by the fractions. When fractions have the same denominator, we can easily compare them, add them, or subtract them. The LCD provides this common ground, allowing us to perform these operations accurately. Without a common denominator, we're essentially trying to add or subtract apples and oranges – it just doesn't work!

Why is the least common denominator important? We could find any common denominator, but using the smallest one keeps the numbers manageable and simplifies our calculations. It's like finding the most efficient way to solve a problem. For example, while 32 is a common denominator for 1/4 and 1/8, 8 is the least common denominator, making the subsequent steps easier.

To find the LCD, we typically use one of two methods: listing multiples or prime factorization. Listing multiples involves writing out multiples of each denominator until you find a common one. Prime factorization breaks down each denominator into its prime factors, allowing us to identify the necessary factors for the LCD. We'll use the prime factorization method in this example as it's generally more efficient for larger numbers.

Finding the LCD for $3/4, 7/16$, and $5/8$

Now, let's apply our understanding of LCD to the fractions $3/4, 7/16$, and $5/8$. Our goal is to find the smallest number that 4, 16, and 8 all divide into evenly. We'll use the prime factorization method to achieve this.

  1. Prime Factorization: First, we break down each denominator into its prime factors:

    • 4 = 2 x 2 = 22
    • 16 = 2 x 2 x 2 x 2 = 24
    • 8 = 2 x 2 x 2 = 23

  2. Identify the Highest Powers: Next, we identify the highest power of each prime factor that appears in any of the factorizations. In this case, we only have one prime factor (2), and its highest power is 24 (from the factorization of 16).

  3. Calculate the LCD: The LCD is the product of these highest powers. Therefore, the LCD = 24 = 16.

So, the Least Common Denominator for $3/4, 7/16$, and $5/8$ is 16. This means we need to convert each fraction into an equivalent fraction with a denominator of 16.

Converting the Fractions to Equivalent Fractions with the LCD

Now that we've found the LCD (which is 16), we need to convert each of the original fractions into equivalent fractions with a denominator of 16. This involves multiplying both the numerator and the denominator of each fraction by a suitable factor that will result in the desired denominator.

  1. Convert $3/4$: To convert $3/4$ to an equivalent fraction with a denominator of 16, we need to determine what number to multiply the denominator (4) by to get 16. Since 4 x 4 = 16, we multiply both the numerator and the denominator by 4:

    • (3 x 4) / (4 x 4) = 12/16

  2. Convert $7/16$: The fraction $7/16$ already has a denominator of 16, so no conversion is needed. It remains $7/16$.

  3. Convert $5/8$: To convert $5/8$ to an equivalent fraction with a denominator of 16, we need to determine what number to multiply the denominator (8) by to get 16. Since 8 x 2 = 16, we multiply both the numerator and the denominator by 2:

    • (5 x 2) / (8 x 2) = 10/16

Therefore, the fractions $3/4, 7/16$, and $5/8$ expressed with the LCD are $12/16, 7/16$, and $10/16$, respectively.

Analyzing the Answer Choices

Now that we've manually calculated the fractions expressed with the LCD, let's examine the given answer choices to see which one matches our result.

A. $3 / 4, 2 / 4, 3 / 4$ B. $24 / 32, 14 / 32, 24 / 32$ C. $9 / 16, 49 / 16, 36 / 16$ D. Discussion category : mathematics

Comparing our calculated values ($12/16, 7/16, 10/16$) with the answer choices, we can see that none of the options directly match. However, option B presents an interesting case. While the denominators are 32 instead of 16, these fractions might be equivalent to our solution if simplified. Let's simplify the fractions in option B:

  • 24/32$ simplifies to $3/4$ (dividing both numerator and denominator by 8)

  • 14/32$ simplifies to $7/16$ (dividing both numerator and denominator by 2)

  • 20/32$ simplifies to $5/8$ (dividing both numerator and denominator by 4)

But, since the question asks us to express the fractions with the LCD, the fractions must have the LCD as the denominator. So, none of the options are correct.

Key Takeaways and Importance of LCD

Expressing fractions with the Least Common Denominator is a fundamental skill in mathematics. It allows us to perform operations on fractions with different denominators by finding a common "unit" for comparison and calculation. We've seen how to find the LCD using prime factorization and how to convert fractions to equivalent forms with the LCD.

Understanding and applying the concept of LCD is crucial for:

  • Adding and Subtracting Fractions: As mentioned earlier, you cannot directly add or subtract fractions with different denominators. LCD provides the common denominator needed for these operations.
  • Comparing Fractions: When fractions have the same denominator, it becomes easy to compare their values. The fraction with the larger numerator is the larger fraction.
  • Simplifying Complex Fractions: LCD can be used to simplify complex fractions (fractions within fractions) by multiplying the numerator and denominator of the complex fraction by the LCD of the inner fractions.
  • Solving Equations with Fractions: LCD is often used to clear denominators in equations involving fractions, making the equations easier to solve.

By mastering the concept of LCD, you'll build a strong foundation for more advanced mathematical topics involving fractions and rational expressions.

Conclusion

In this article, we tackled the problem of expressing the fractions $3/4, 7/16$, and $5/8$ with their Least Common Denominator. We began by understanding the concept of LCD and its importance, then walked through the process of finding the LCD using prime factorization. We converted each fraction to an equivalent form with the LCD and analyzed the provided answer choices. While none of the choices were perfectly correct, we learned valuable skills in the process.

Remember, the key to success with fractions lies in understanding the underlying concepts and practicing the techniques. Keep honing your skills, and you'll find working with fractions becomes second nature.

Therefore, the correct answer is not among the choices provided. The fractions expressed with the LCD of 16 are 12/16, 7/16, and 10/16.