Solving 5/2 × -7/11 + 5/2 × 3/11 + 1/11 × 5/2 With Distributive Property
In this article, we will delve into the solution of the mathematical expression 5/2 × -7/11 + 5/2 × 3/11 + 1/11 × 5/2 using the distributive property. This property is a fundamental concept in algebra and arithmetic, allowing us to simplify complex expressions by factoring out common terms. By understanding and applying the distributive property, we can make calculations easier and more efficient. Let's break down the problem step by step to see how this works.
Understanding the Distributive Property
The distributive property is a cornerstone of arithmetic and algebra, allowing us to simplify expressions involving multiplication and addition or subtraction. At its core, the distributive property states that multiplying a single term by a sum or difference inside parentheses is the same as multiplying the term by each part of the sum or difference individually and then adding or subtracting the results. Mathematically, this can be expressed as:
- a × (b + c) = a × b + a × c
- a × (b - c) = a × b - a × c
Where 'a', 'b', and 'c' represent any numbers. This property is incredibly versatile and is used extensively in simplifying algebraic expressions, solving equations, and performing mental calculations efficiently. The beauty of the distributive property lies in its ability to break down complex problems into smaller, more manageable parts. For instance, consider the expression 3 × (10 + 2). Instead of first adding 10 and 2 and then multiplying by 3, we can distribute the 3 across both terms: 3 × 10 + 3 × 2. This gives us 30 + 6, which equals 36. This approach is particularly useful when dealing with larger numbers or algebraic variables.
In the context of our problem, 5/2 × -7/11 + 5/2 × 3/11 + 1/11 × 5/2, we can identify a common term that can be factored out, making the calculation significantly simpler. This is where the distributive property shines, allowing us to transform the expression into a more manageable form. By recognizing the common factor and applying the distributive property, we can avoid tedious calculations and arrive at the solution more efficiently. The key is to identify the common term and apply the property in the correct manner, which we will demonstrate in the subsequent sections.
Identifying Common Factors
Before we can apply the distributive property, it's crucial to identify common factors within the expression 5/2 × -7/11 + 5/2 × 3/11 + 1/11 × 5/2. A common factor is a number or expression that is a factor of two or more terms in an expression. In this case, we are looking for a term that appears in each part of the sum. Upon closer inspection, we can see that the fraction 5/2 is present in the first two terms, 5/2 × -7/11 and 5/2 × 3/11. However, it might not be immediately obvious that 5/2 is also a factor in the third term, 1/11 × 5/2. To make it clearer, we can rewrite the third term using the commutative property of multiplication, which states that the order of multiplication does not change the result (a × b = b × a). Thus, 1/11 × 5/2 is the same as 5/2 × 1/11.
Now, our expression looks like this: 5/2 × -7/11 + 5/2 × 3/11 + 5/2 × 1/11. It's much easier to see that 5/2 is indeed a common factor in all three terms. Identifying common factors is a fundamental step in simplifying expressions, not only with the distributive property but also in various other algebraic manipulations. By recognizing common factors, we can reduce the complexity of the expression and make it easier to handle. This step sets the stage for applying the distributive property, allowing us to factor out the common term and simplify the expression significantly. In the next section, we will apply the distributive property to this expression using the common factor we've identified.
Applying the Distributive Property
Now that we've identified the common factor of 5/2 in the expression 5/2 × -7/11 + 5/2 × 3/11 + 5/2 × 1/11, we can proceed to apply the distributive property. The distributive property, as discussed earlier, allows us to factor out a common term from an expression. In this case, we will factor out 5/2 from each term. Recall that the distributive property states that a × b + a × c = a × (b + c). Applying this in reverse, we can rewrite our expression by factoring out 5/2:
5/2 × -7/11 + 5/2 × 3/11 + 5/2 × 1/11 = 5/2 × (-7/11 + 3/11 + 1/11)
By factoring out 5/2, we have simplified the expression into a product of 5/2 and the sum of the fractions -7/11, 3/11, and 1/11. This significantly reduces the complexity of the calculation. Instead of performing three separate multiplications, we now only need to perform one multiplication after simplifying the sum inside the parentheses. This is the power of the distributive property – it transforms a series of multiplications and additions into a simpler, more manageable form. The next step is to simplify the expression inside the parentheses, which involves adding the fractions. This is a straightforward process since all the fractions have the same denominator. Once we've simplified the expression inside the parentheses, we can then multiply the result by 5/2 to obtain the final answer. Applying the distributive property effectively streamlines the calculation process, making it easier to arrive at the correct solution. Let's proceed with simplifying the sum of the fractions in the next section.
Simplifying the Expression Inside Parentheses
After applying the distributive property, our expression now looks like this: 5/2 × (-7/11 + 3/11 + 1/11). The next step is to simplify the expression inside the parentheses. This involves adding the fractions -7/11, 3/11, and 1/11. Since all the fractions have the same denominator (11), we can simply add their numerators. The sum of the numerators is -7 + 3 + 1. Let's calculate this sum:
-7 + 3 + 1 = -4 + 1 = -3
So, the sum of the fractions inside the parentheses is -3/11. Now, we can replace the expression inside the parentheses with its simplified form:
5/2 × (-3/11)
Simplifying expressions within parentheses is a crucial step in solving mathematical problems. It helps to reduce the complexity of the equation and makes it easier to perform further calculations. In this case, by adding the fractions, we have transformed the sum of three fractions into a single fraction, which simplifies the final multiplication. This process of simplifying complex expressions into more manageable forms is a fundamental skill in mathematics. With the expression inside the parentheses now simplified to -3/11, we are ready for the final step: multiplying 5/2 by -3/11. This will give us the final answer to the original problem. In the next section, we will perform this multiplication and present the solution.
Final Calculation and Solution
Having simplified the expression to 5/2 × (-3/11), we are now ready to perform the final calculation and arrive at the solution. To multiply two fractions, we multiply their numerators together and their denominators together. In this case, we will multiply the numerators 5 and -3, and then multiply the denominators 2 and 11:
(5 × -3) / (2 × 11)
Multiplying the numerators gives us:
5 × -3 = -15
Multiplying the denominators gives us:
2 × 11 = 22
So, the result of the multiplication is -15/22. Therefore, the solution to the original expression 5/2 × -7/11 + 5/2 × 3/11 + 1/11 × 5/2 is -15/22. This final step demonstrates how the distributive property, combined with basic arithmetic operations, allows us to solve complex expressions systematically. By identifying common factors, applying the distributive property, simplifying expressions within parentheses, and performing the final multiplication, we have successfully navigated the problem to its solution. The result, -15/22, represents the simplified form of the original expression. This exercise highlights the importance of understanding and applying mathematical properties to solve problems efficiently and accurately. In conclusion, the distributive property is a powerful tool that simplifies calculations and makes complex problems more approachable.
Conclusion
In conclusion, we have successfully solved the expression 5/2 × -7/11 + 5/2 × 3/11 + 1/11 × 5/2 by effectively using the distributive property. This property allowed us to simplify the expression by factoring out the common term, 5/2, and then performing a single multiplication after simplifying the sum of the fractions. The steps we followed included identifying the common factor, applying the distributive property, simplifying the expression inside the parentheses, and finally, performing the multiplication to arrive at the solution, which is -15/22. This process highlights the power and efficiency of the distributive property in simplifying complex mathematical expressions. By understanding and applying this property, we can tackle similar problems with greater ease and accuracy. The distributive property is not just a mathematical rule; it is a tool that enhances our problem-solving capabilities and makes mathematical calculations more manageable. Through this example, we have demonstrated how breaking down a problem into smaller, more manageable steps can lead to a clear and concise solution. The ability to recognize common factors and apply the appropriate properties is a fundamental skill in mathematics, and mastering these skills will undoubtedly benefit anyone facing mathematical challenges. We hope this detailed explanation has provided a clear understanding of how to solve such expressions using the distributive property, empowering you to apply this knowledge in various mathematical contexts.
