What's The Least Common Denominator Of 3/4 4/5 And 2/3

Finding the Least Common Denominator (LCD) of 3/4, 4/5, and 2/3

This article will comprehensively guide you through the process of determining the least common denominator (LCD) of the fractions 3/4, 4/5, and 2/3. We'll explore different methods, delve into the underlying mathematical concepts, and provide practical examples to solidify your understanding. Finding the LCD is a crucial skill in adding and subtracting fractions, a fundamental concept in arithmetic and algebra. Understanding this process will empower you to tackle more complex mathematical problems with confidence.

Understanding Least Common Denominator (LCD)

Before we dive into calculating the LCD of 3/4, 4/5, and 2/3, let's establish a solid foundation. The denominator of a fraction represents the total number of equal parts into which a whole is divided. The numerator represents how many of those parts are being considered.

The least common denominator (LCD) is the smallest multiple that is common to all the denominators in a set of fractions. It's the smallest number that all the denominators can divide into evenly. Finding the LCD is essential when adding or subtracting fractions because it allows us to express the fractions with a common denominator, making the addition or subtraction straightforward. Without a common denominator, directly adding or subtracting numerators would be incorrect.

Method 1: Listing Multiples

One straightforward method to find the LCD is to list the multiples of each denominator until you find the smallest common multiple. Let's apply this method to our fractions: 3/4, 4/5, and 2/3.

Finding Multiples of Each Denominator:

• Multiples of 4: 4, 8, 12, 16, 20, 24, 28, 32, 36, 40, 44, 48, 52, 56, 60...
• Multiples of 5: 5, 10, 15, 20, 25, 30, 35, 40, 45, 50, 55, 60...
• Multiples of 3: 3, 6, 9, 12, 15, 18, 21, 24, 27, 30, 33, 36, 39, 42, 45, 48, 51, 54, 57, 60...

By examining the lists, we can see that the smallest number that appears in all three lists is 60. Therefore, the least common denominator (LCD) of 3/4, 4/5, and 2/3 is 60.

Converting Fractions to the LCD

Now that we've found the LCD, we can convert each fraction to an equivalent fraction with a denominator of 60:

• 3/4: To get a denominator of 60, we multiply both the numerator and denominator by 15 (60/4 = 15). This gives us (3 * 15) / (4 * 15) = 45/60.
• 4/5: To get a denominator of 60, we multiply both the numerator and denominator by 12 (60/5 = 12). This gives us (4 * 12) / (5 * 12) = 48/60.
• 2/3: To get a denominator of 60, we multiply both the numerator and denominator by 20 (60/3 = 20). This gives us (2 * 20) / (3 * 20) = 40/60.

Now all three fractions have the same denominator, making them ready for addition or subtraction.

Method 2: Prime Factorization

A more efficient method, especially for larger numbers, is using prime factorization. This method involves breaking down each denominator into its prime factors. The prime factors are the prime numbers that multiply together to give the original number. A prime number is a whole number greater than 1 that has only two divisors: 1 and itself.

Prime Factorization of Denominators:

• 4: 2 x 2 = 2²
• 5: 5 (5 is a prime number)
• 3: 3 (3 is a prime number)

Finding the LCD using Prime Factors:

1. Identify all prime factors: We have 2, 3, and 5.
2. Take the highest power of each prime factor: The highest power of 2 is 2², the highest power of 3 is 3¹, and the highest power of 5 is 5¹.
3. Multiply the highest powers together: 2² x 3 x 5 = 4 x 3 x 5 = 60

Therefore, the LCD of 3/4, 4/5, and 2/3 is 60, confirming the result from Method 1. This method is particularly useful when dealing with larger denominators or a greater number of fractions.

Method 3: Using the Least Common Multiple (LCM)

The least common denominator (LCD) is directly related to the least common multiple (LCM). The LCM of a set of numbers is the smallest number that is a multiple of all the numbers in the set. Since the denominators are integers, finding the LCM is equivalent to finding the LCD. There are various methods to calculate the LCM, such as the prime factorization method described above.

Finding the LCM (and thus the LCD)

Using prime factorization (as demonstrated above), we found that the prime factorization of 4, 5, and 3 are 2², 5, and 3, respectively. The LCM is found by multiplying the highest powers of each prime factor present: 2² * 3 * 5 = 60. Therefore, the LCD of 3/4, 4/5, and 2/3 is 60.

Applying the LCD in Calculations

Now that we've determined the LCD of 3/4, 4/5, and 2/3 to be 60, let's see how it's used in calculations. Let's say we want to add these fractions:

3/4 + 4/5 + 2/3

1. Convert to equivalent fractions with the LCD: As calculated earlier, this becomes: 45/60 + 48/60 + 40/60

2. Add the numerators: 45 + 48 + 40 = 133

3. Keep the common denominator: The result is 133/60.

4. Simplify (if possible): This fraction can be simplified to a mixed number: 2 13/60.

Conclusion

Finding the least common denominator (LCD) is a fundamental skill in mathematics, crucial for accurately performing operations with fractions. This article has explored three different methods for finding the LCD: listing multiples, prime factorization, and utilizing the least common multiple (LCM). While listing multiples is straightforward for smaller numbers, prime factorization provides a more efficient approach for larger numbers or a greater number of fractions. Understanding these methods equips you with the necessary tools to confidently tackle fraction-based problems and advance your mathematical skills. Remember, mastering the concept of LCD is essential not only for arithmetic but also for more advanced mathematical concepts in algebra and beyond. Practice these methods to build proficiency and confidence in your mathematical abilities. The ability to find the LCD efficiently is a cornerstone of mathematical fluency.