Solving \frac{70}{5} + \frac{85}{5} - \frac{15}{5} A Step By Step Guide
Navigating the world of fractions can sometimes feel like traversing a complex maze, especially when faced with expressions involving addition and subtraction. However, with a solid understanding of the fundamental principles and a step-by-step approach, even seemingly daunting problems can be tackled with confidence. In this comprehensive guide, we will delve into the intricacies of solving the expression \frac{70}{5}+\frac{85}{5}-\frac{15}{5}, breaking down each step and illuminating the underlying concepts. Our journey will not only equip you with the skills to solve this specific problem but also empower you to confidently handle a wide range of fraction arithmetic challenges.
Understanding the Basics: Fractions and Common Denominators
Before we dive into the specific problem at hand, let's first revisit the fundamental concepts of fractions and common denominators. A fraction, in its essence, represents a part of a whole. It consists of two key components: the numerator, which signifies the number of parts we have, and the denominator, which indicates the total number of equal parts that make up the whole. For instance, in the fraction \frac{70}{5}, 70 is the numerator, representing the number of parts we possess, while 5 is the denominator, indicating that the whole is divided into 5 equal parts.
The concept of a common denominator is pivotal when it comes to adding or subtracting fractions. A common denominator is a shared denominator among two or more fractions, allowing us to perform arithmetic operations on their numerators. Think of it like comparing apples to apples – we need a common unit of measurement before we can add or subtract them meaningfully. In the context of fractions, the common denominator provides this shared unit, enabling us to combine or differentiate the fractional parts accurately.
When fractions already share a common denominator, the process of addition and subtraction becomes remarkably straightforward. We simply add or subtract the numerators while keeping the denominator constant. This is precisely the scenario we encounter in our problem, \frac{70}{5}+\frac{85}{5}-\frac{15}{5}, where all three fractions share the common denominator of 5. This simplifies the problem significantly, allowing us to focus on the numerators and perform the arithmetic operations directly.
Step-by-Step Solution: Solving \frac{70}{5}+\frac{85}{5}-\frac{15}{5}
Now that we have a firm grasp of the foundational concepts, let's embark on the step-by-step solution of our problem, \frac{70}{5}+\frac{85}{5}-\frac{15}{5}. The beauty of this expression lies in the fact that all fractions share the same denominator, 5, making the arithmetic operations significantly easier.
Step 1: Combine the Numerators
Since the denominators are identical, we can directly combine the numerators according to the operations indicated. In this case, we have addition followed by subtraction. Therefore, we can rewrite the expression as:
(70 + 85 - 15) / 5
This step effectively consolidates the fractional parts into a single numerator, simplifying the expression and paving the way for the next step.
Step 2: Perform the Arithmetic Operations
Now, let's perform the arithmetic operations in the numerator. Following the order of operations (addition and subtraction from left to right), we first add 70 and 85, which gives us 155. Then, we subtract 15 from 155, resulting in 140. So, our expression now becomes:
140 / 5
This step reduces the expression to a single fraction, making it easier to simplify further.
Step 3: Simplify the Fraction
The final step involves simplifying the fraction \frac{140}{5}. To do this, we need to find the greatest common divisor (GCD) of the numerator and the denominator and divide both by it. In this case, the GCD of 140 and 5 is 5. Dividing both the numerator and the denominator by 5, we get:
(140 ÷ 5) / (5 ÷ 5) = 28 / 1 = 28
Therefore, the simplified answer to the expression \frac{70}{5}+\frac{85}{5}-\frac{15}{5} is 28. This completes our step-by-step solution, demonstrating how to effectively handle fraction arithmetic with common denominators.
Alternative Approach: Simplifying Before Combining
While we solved the problem by combining the numerators first and then simplifying, there's another equally valid approach: simplifying each fraction individually before performing the addition and subtraction. This method can sometimes make the calculations easier, especially when dealing with larger numbers. Let's explore this alternative approach to solve \frac{70}{5}+\frac{85}{5}-\frac{15}{5}.
Step 1: Simplify Each Fraction
We begin by simplifying each fraction individually. This involves dividing the numerator and denominator of each fraction by their greatest common divisor (GCD).
- \frac70}{5}**{1} = 14.
- \frac85}{5}**{1} = 17.
- \frac15}{5}**{1} = 3.
By simplifying each fraction individually, we have transformed the original expression into:
14 + 17 - 3
This step eliminates the fractions altogether, making the subsequent arithmetic operations significantly simpler.
Step 2: Perform the Arithmetic Operations
Now, we simply perform the addition and subtraction operations in the order they appear. Adding 14 and 17, we get 31. Subtracting 3 from 31, we arrive at 28.
14 + 17 - 3 = 31 - 3 = 28
This alternative approach yields the same result, 28, as our previous method. It highlights the flexibility in solving mathematical problems and demonstrates that different paths can lead to the same correct answer. Choosing the most efficient approach often depends on the specific problem and individual preferences.
Real-World Applications: Why Fraction Arithmetic Matters
While solving mathematical expressions like \frac{70}{5}+\frac{85}{5}-\frac{15}{5} might seem like an abstract exercise, the underlying principles of fraction arithmetic are deeply interwoven into our daily lives. From measuring ingredients in a recipe to calculating distances on a map, fractions play a crucial role in various practical scenarios. Understanding how to add, subtract, multiply, and divide fractions empowers us to make informed decisions and solve real-world problems effectively.
Consider, for instance, a scenario where you're baking a cake and need to adjust the recipe to make a larger or smaller batch. The recipe might call for \frac{3}{4} cup of flour, but you only want to make half the recipe. To determine the new amount of flour needed, you would need to multiply the fraction by \frac{1}{2}. Similarly, if you're planning a road trip and need to calculate the distance you can travel on a certain amount of fuel, you might encounter fractions representing fuel consumption rates or distances between cities. In these situations, a solid understanding of fraction arithmetic is indispensable.
Moreover, fractions are fundamental in various professional fields, including engineering, finance, and construction. Engineers use fractions to calculate stresses and strains in structures, financial analysts rely on fractions to determine investment returns, and construction workers employ fractions to measure materials and ensure precise dimensions. Mastering fraction arithmetic, therefore, not only enhances our problem-solving abilities in everyday situations but also opens doors to a wide range of career opportunities.
Conclusion: Mastering Fraction Arithmetic for Mathematical Proficiency
In this comprehensive guide, we have meticulously dissected the expression \frac{70}{5}+\frac{85}{5}-\frac{15}{5}, demonstrating two distinct approaches to arrive at the solution. We explored the foundational concepts of fractions and common denominators, emphasizing their importance in performing arithmetic operations. We then presented a step-by-step solution, combining the numerators first and simplifying the resulting fraction. Subsequently, we delved into an alternative approach, simplifying each fraction individually before combining them.
Both methods yielded the same answer, 28, underscoring the flexibility and versatility of mathematical problem-solving. Furthermore, we highlighted the real-world applications of fraction arithmetic, showcasing its relevance in everyday scenarios and various professional fields. By understanding fractions and mastering the techniques for adding, subtracting, multiplying, and dividing them, we equip ourselves with essential tools for navigating the mathematical landscape and tackling practical challenges effectively.
Fraction arithmetic is not merely a collection of rules and procedures; it's a fundamental building block for mathematical proficiency. By embracing the concepts and practicing the techniques, we can unlock a deeper understanding of mathematics and its power to shape our world.
