In the realm of mathematics, fractions play a crucial role in representing parts of a whole. Understanding how to add, subtract, multiply, and divide fractions is essential for solving a wide range of problems. Today, we will delve into the process of adding fractions, specifically focusing on the fractions 3 2/3 and 2 2/5. By the end of this post, you will have a clear understanding of how to add these fractions step by step.
Understanding Mixed Numbers
Before we dive into the addition process, it’s important to understand what mixed numbers are. A mixed number is a whole number and a proper fraction combined. For example, 3 2⁄3 is a mixed number where 3 is the whole number and 2⁄3 is the fractional part.
Converting Mixed Numbers to Improper Fractions
To add mixed numbers, it’s often easier to convert them into improper fractions. An improper fraction is a fraction where the numerator is greater than or equal to the denominator.
Let’s convert 3 2⁄3 and 2 2⁄5 into improper fractions.
For 3 2⁄3:
- Multiply the whole number by the denominator: 3 * 3 = 9
- Add the numerator to the result: 9 + 2 = 11
- The improper fraction is 11⁄3.
For 2 2⁄5:
- Multiply the whole number by the denominator: 2 * 5 = 10
- Add the numerator to the result: 10 + 2 = 12
- The improper fraction is 12⁄5.
Finding a Common Denominator
To add fractions, they must have a common denominator. The least common denominator (LCD) is the smallest number that both denominators can divide into without leaving a remainder.
For 11⁄3 and 12⁄5, the denominators are 3 and 5. The LCD of 3 and 5 is 15.
Converting to the Common Denominator
Now, we need to convert both fractions to have the denominator of 15.
For 11⁄3:
- Multiply both the numerator and the denominator by 5: (11 * 5) / (3 * 5) = 55⁄15
For 12⁄5:
- Multiply both the numerator and the denominator by 3: (12 * 3) / (5 * 3) = 36⁄15
Adding the Fractions
Now that both fractions have the same denominator, we can add them:
55⁄15 + 36⁄15 = 91⁄15
Converting Back to a Mixed Number
The result, 91⁄15, is an improper fraction. To convert it back to a mixed number:
- Divide the numerator by the denominator: 91 ÷ 15 = 6 with a remainder of 1
- The mixed number is 6 1⁄15.
Final Answer
Therefore, the sum of 3 2⁄3 and 2 2⁄5 is 6 1⁄15.
📝 Note: Always double-check your calculations to ensure accuracy, especially when converting between mixed numbers and improper fractions.
Adding fractions, especially mixed numbers, can seem daunting at first, but with practice, it becomes a straightforward process. By converting mixed numbers to improper fractions, finding a common denominator, and then adding the fractions, you can solve a wide range of problems involving fractions. This method ensures that you have a clear and accurate result every time.
In summary, we started with the mixed numbers 3 2⁄3 and 2 2⁄5, converted them to improper fractions, found a common denominator, added the fractions, and then converted the result back to a mixed number. This step-by-step process is essential for mastering fraction addition and can be applied to any set of fractions you encounter.
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