Solve The Following Fraction Operations: 1) 2/3 + 5/8; 2) 7/12 - 3/8; 3) 11/16 - 5/8; 4) 6/35 + 3/10; 5) 8/15 + 3 4/9; 6) 4/15 + 7/12; 7) 11/48 - 5/36.

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Before diving into the specifics of adding and subtracting fractions, it's crucial to establish a firm understanding of what fractions represent and the terminology involved. A fraction is essentially a way to represent a part of a whole. It consists of two primary components: the numerator and the denominator. The numerator, which sits atop the fraction bar, indicates the number of parts we're considering. Conversely, the denominator, found beneath the fraction bar, signifies the total number of equal parts that make up the whole. For instance, in the fraction 2/3, the numerator (2) tells us we're dealing with two parts, while the denominator (3) reveals that the whole is divided into three equal parts. This fundamental understanding is crucial for grasping the concepts of fraction addition and subtraction.

Fractions come in various forms, each with its unique characteristics. The most common types include proper fractions, improper fractions, and mixed numbers. Proper fractions are those where the numerator is smaller than the denominator, such as 2/3 or 5/8. These fractions represent a value less than one whole. On the other hand, improper fractions have a numerator that is greater than or equal to the denominator, like 7/3 or 11/4. Improper fractions represent a value equal to or greater than one whole. Mixed numbers, as the name suggests, combine a whole number and a proper fraction, such as 3 4/9. It's essential to be comfortable converting between improper fractions and mixed numbers, as this skill is often required in fraction arithmetic. To convert an improper fraction to a mixed number, divide the numerator by the denominator. The quotient becomes the whole number part, the remainder becomes the new numerator, and the denominator stays the same. Conversely, to convert a mixed number to an improper fraction, multiply the whole number by the denominator, add the numerator, and place the result over the original denominator.

The simplest scenario in fraction arithmetic involves adding or subtracting fractions that share the same denominator. This is because the fractions are already expressed in terms of the same-sized parts of the whole. The core principle here is that we can directly add or subtract the numerators while keeping the denominator constant. Think of it like adding slices of the same-sized pie – if you have 2 slices and add 3 more slices, you simply have 5 slices in total.

For example, consider the addition problem 1/5 + 3/5. Both fractions have a denominator of 5, meaning they represent fifths of a whole. To add them, we add the numerators (1 + 3 = 4) and keep the denominator as 5, resulting in 4/5. The same logic applies to subtraction. If we have 7/8 and subtract 2/8, we subtract the numerators (7 - 2 = 5) while retaining the denominator of 8, yielding 5/8. It's crucial to remember that this method only works when the denominators are identical. If the denominators are different, we need to find a common denominator before we can proceed with addition or subtraction. This common denominator serves as a shared unit for the fractions, allowing us to combine them meaningfully.

When faced with adding or subtracting fractions with unlike denominators, the crucial first step is to identify the Least Common Denominator (LCD). The LCD is the smallest common multiple of the denominators of the given fractions. It serves as the unifying denominator that allows us to rewrite the fractions with a common base, enabling addition or subtraction. There are several methods to determine the LCD, and choosing the most efficient one depends on the specific numbers involved.

One common approach is the listing method. This involves listing the multiples of each denominator until a common multiple is found. For instance, to find the LCD of 1/4 and 1/6, we'd list multiples of 4 (4, 8, 12, 16, ...) and multiples of 6 (6, 12, 18, ...). The smallest number appearing in both lists is 12, making it the LCD. Another method, particularly useful for larger numbers, is prime factorization. This involves breaking down each denominator into its prime factors. For example, 12 can be factored as 2 x 2 x 3, and 18 can be factored as 2 x 3 x 3. To find the LCD, we take the highest power of each prime factor present in either factorization. In this case, we have 2^2 (from 12), 3^2 (from 18), and their product (2^2 x 3^2 = 36) is the LCD. Once the LCD is found, we rewrite each fraction with the LCD as the new denominator. This involves multiplying both the numerator and denominator of each fraction by a suitable factor that transforms the original denominator into the LCD. It's essential to remember that multiplying both the numerator and denominator by the same factor doesn't change the value of the fraction, only its representation.

Now, let's delve into the step-by-step process of adding and subtracting fractions with unlike denominators. This process builds upon the understanding of LCD and equivalent fractions, providing a clear methodology for solving these types of problems. The core idea is to transform the fractions into equivalent forms that share a common denominator, allowing for straightforward addition or subtraction of the numerators.

  • Step 1: Find the Least Common Denominator (LCD): As discussed earlier, the LCD is the smallest common multiple of the denominators. Employ the listing method, prime factorization, or any other preferred technique to determine the LCD accurately. This step is crucial as it sets the foundation for the subsequent steps.
  • Step 2: Rewrite the Fractions with the LCD: Once the LCD is identified, rewrite each fraction with the LCD as its new denominator. To do this, determine the factor by which you need to multiply the original denominator to obtain the LCD. Then, multiply both the numerator and the denominator of the fraction by that factor. This process ensures that you're creating an equivalent fraction with the desired denominator.
  • Step 3: Add or Subtract the Numerators: With the fractions now sharing a common denominator, you can proceed to add or subtract the numerators. Remember to keep the denominator the same. This step is analogous to adding or subtracting fractions with like denominators, as the fractions are now expressed in terms of the same-sized parts.
  • Step 4: Simplify the Result (if possible): After performing the addition or subtraction, the resulting fraction may be in its simplest form, or it may need further simplification. Look for a common factor between the numerator and the denominator. If a common factor exists, divide both the numerator and the denominator by that factor to reduce the fraction to its lowest terms. This step ensures that your answer is presented in its most concise and understandable form.

Let's apply these concepts to the specific examples provided:

1) 2/3 + 5/8

  • LCD: The LCD of 3 and 8 is 24.
  • Rewrite: 2/3 = (2 x 8) / (3 x 8) = 16/24; 5/8 = (5 x 3) / (8 x 3) = 15/24
  • Add: 16/24 + 15/24 = 31/24
  • Simplify: 31/24 is an improper fraction. Convert to a mixed number: 1 7/24

2) 7/12 - 3/8

  • LCD: The LCD of 12 and 8 is 24.
  • Rewrite: 7/12 = (7 x 2) / (12 x 2) = 14/24; 3/8 = (3 x 3) / (8 x 3) = 9/24
  • Subtract: 14/24 - 9/24 = 5/24
  • Simplify: 5/24 is in its simplest form.

3) 11/16 - 5/8

  • LCD: The LCD of 16 and 8 is 16.
  • Rewrite: 11/16 remains the same; 5/8 = (5 x 2) / (8 x 2) = 10/16
  • Subtract: 11/16 - 10/16 = 1/16
  • Simplify: 1/16 is in its simplest form.

4) 6/35 + 3/10

  • LCD: The LCD of 35 and 10 is 70.
  • Rewrite: 6/35 = (6 x 2) / (35 x 2) = 12/70; 3/10 = (3 x 7) / (10 x 7) = 21/70
  • Add: 12/70 + 21/70 = 33/70
  • Simplify: 33/70 is in its simplest form.

5) 8/15 + 3 4/9

  • Convert: Convert the mixed number to an improper fraction: 3 4/9 = (3 x 9 + 4) / 9 = 31/9
  • LCD: The LCD of 15 and 9 is 45.
  • Rewrite: 8/15 = (8 x 3) / (15 x 3) = 24/45; 31/9 = (31 x 5) / (9 x 5) = 155/45
  • Add: 24/45 + 155/45 = 179/45
  • Simplify: 179/45 is an improper fraction. Convert to a mixed number: 3 44/45

6) 4/15 + 7/12

  • LCD: The LCD of 15 and 12 is 60.
  • Rewrite: 4/15 = (4 x 4) / (15 x 4) = 16/60; 7/12 = (7 x 5) / (12 x 5) = 35/60
  • Add: 16/60 + 35/60 = 51/60
  • Simplify: Both 51 and 60 are divisible by 3: 51/60 = 17/20

7) 11/48 - 5/36

  • LCD: The LCD of 48 and 36 is 144.
  • Rewrite: 11/48 = (11 x 3) / (48 x 3) = 33/144; 5/36 = (5 x 4) / (36 x 4) = 20/144
  • Subtract: 33/144 - 20/144 = 13/144
  • Simplify: 13/144 is in its simplest form.

Adding and subtracting fractions is a fundamental skill in mathematics, with applications ranging from everyday life to advanced scientific calculations. By mastering the concepts of LCD, equivalent fractions, and the step-by-step procedures outlined in this guide, you can confidently tackle a wide range of fraction arithmetic problems. Remember to practice regularly and to break down complex problems into smaller, manageable steps. With dedication and a solid understanding of the underlying principles, you can achieve proficiency in fraction addition and subtraction.