Understanding the Difference Between Sum and Product of Fractions
Understanding the fundamental operations of fractions is a critical skill in mathematics. Both the sum and the product of fractions are essential in various fields, including engineering, physics, and finance. In this article, we will explore the difference between the sum and the product of the fractions 1/2 and 1/3. By following the steps meticulously, we will unravel the solution with clarity and detail.
The Sum of Fractions: 1/2 1/3
The sum of two fractions, in this case, 1/2 and 1/3, involves adding them together. However, to add these fractions, they must have a common denominator. The least common multiple (LCM) of the denominators 2 and 3 is 6. So, we express 1/2 and 1/3 as fractions with a common denominator:
1/2 3/6 and 1/3 2/6
Now, we add the fractions:
1/2 1/3 3/6 2/6 5/6
The Product of Fractions: 1/2 times; 1/3
The product of fractions involves multiplying them. When multiplying 1/2 and 1/3, we multiply their respective numerators and denominators:
1/2 times; 1/3 (1 times; 1) / (2 times; 3) 1/6
Calculating the Difference
To find the difference between the sum and the product of 1/2 and 1/3, we subtract the product from the sum:
Sum - Product 5/6 - 1/6 (5 - 1) / 6 4/6 2/3
Thus, the difference between the sum of 1/2 and 1/3 and the product of 1/2 and 1/3 is 2/3.
Alternative Methods
Let's verify this solution using alternative methods to ensure accuracy.
Method 1: Using Common Denominator
We know the LCM of 2 and 3 is 6. We can directly add and subtract the fractions:
1/2 1/3 3/6 2/6 5/6
1/2 × 1/3 1/6
5/6 - 1/6 4/6 2/3
Method 2: Direct Subtraction
By expressing the sum and product in a single equation, we can directly calculate the difference:
1/2 1/3 - 1/2 × 1/3 1/2 1/3 - 1/6
Convert 1/2 and 1/3 to the common denominator 6:
1/2 1/3 - 1/6 3/6 2/6 - 1/6 5/6 - 1/6 4/6 2/3
Conclusion
In conclusion, we have demonstrated that the difference between the sum of 1/2 and 1/3 and the product of 1/2 and 1/3 is 2/3. This calculation is a fundamental example of the application of basic arithmetic operations to fractions, highlighting the importance of understanding these concepts for more advanced mathematical and real-world problem-solving scenarios.