Simplifying (2/3) / (1/3 + 2/3) + 1 A Step-by-Step Guide
In the realm of mathematics, simplifying complex fractions is a fundamental skill. This guide dives deep into the process of simplifying the complex fraction (2/3) / (1/3 + 2/3) + 1, providing a step-by-step approach that makes the process clear and understandable. Whether you're a student tackling homework or someone looking to brush up on your math skills, this guide offers a comprehensive explanation.
Understanding Complex Fractions
When dealing with complex fractions, the key is to remember that a fraction is simply a division operation. A complex fraction is a fraction where the numerator, the denominator, or both contain fractions themselves. Our focus expression, (2/3) / (1/3 + 2/3) + 1, exemplifies this perfectly. To simplify it, we must methodically address each part, ensuring accuracy and clarity in every step. The ultimate goal is to transform this complex expression into a simpler, more manageable form, which often means reducing it to a single fraction or a whole number. The ability to simplify complex fractions is crucial not only for academic success in mathematics but also for various real-world applications, such as in engineering, physics, and finance, where complex calculations involving fractions are commonplace. Therefore, mastering this skill is an investment in one's problem-solving capabilities across various domains.
Step 1 Combining Fractions in the Denominator
The first crucial step in simplifying the complex fraction (2/3) / (1/3 + 2/3) + 1 involves focusing on the denominator. Within the denominator, we find the expression 1/3 + 2/3. This is a straightforward addition of two fractions, but it's essential to perform it correctly to move forward in the simplification process. Both fractions share a common denominator, which is 3. This simplifies the addition significantly because we can directly add the numerators while keeping the denominator constant. Adding the numerators, 1 and 2, gives us 3. Therefore, 1/3 + 2/3 equals 3/3. This resulting fraction, 3/3, is particularly interesting because it represents a whole unit. Any fraction where the numerator and the denominator are the same simplifies to 1. Thus, 3/3 simplifies to 1. This simplification is a critical step because it transforms the complex denominator into a simple number, making the overall fraction easier to handle. The ability to recognize and simplify fractions like this is a foundational skill in mathematics, as it streamlines calculations and prevents unnecessary complications in more complex problems. This step highlights the importance of understanding basic fraction operations as a prerequisite for tackling more advanced mathematical concepts.
Step 2 Rewriting the Expression
Having simplified the denominator, the next strategic step is to rewrite the original expression, (2/3) / (1/3 + 2/3) + 1, incorporating the simplified denominator. We've established that 1/3 + 2/3 equals 1. Now, we substitute this value back into the original expression. This substitution transforms the complex fraction into a more manageable form. By replacing (1/3 + 2/3) with its simplified value of 1, the expression now reads (2/3) / 1 + 1. This rewritten expression is significantly less daunting than the original. It clearly shows a fraction, 2/3, being divided by 1, followed by the addition of 1. This step is crucial because it breaks down the complex fraction into simpler operations that are easier to understand and execute. Rewriting the expression after each simplification step is a good practice in mathematics, as it helps maintain clarity and reduces the chances of errors. It allows for a more organized and systematic approach to problem-solving, especially when dealing with complex mathematical expressions.
Step 3 Performing the Division
With the expression now rewritten as (2/3) / 1 + 1, the next critical step is to address the division operation. Dividing by 1 is a fundamental operation in mathematics, and it holds a unique property: any number divided by 1 remains unchanged. In our case, we are dividing the fraction 2/3 by 1. Applying the rule that any number divided by 1 is itself, we find that (2/3) / 1 simplifies to 2/3. This simplification is a straightforward yet crucial step in isolating the fractional part of the expression before moving on to the addition. Understanding this property of division by 1 is essential for simplifying various mathematical expressions and equations. It allows for quick and efficient calculations, especially in more complex problems where multiple operations are involved. After performing this division, the expression further simplifies, making it easier to handle the remaining operation. This step demonstrates the importance of knowing the basic rules of arithmetic and how they can be applied to simplify mathematical problems.
Step 4 Adding the Whole Number
Following the division, the expression now stands as 2/3 + 1. This step involves adding a fraction to a whole number, a common operation in mathematics that requires careful handling. To add a fraction and a whole number, we need to express the whole number as a fraction with the same denominator as the fraction we are adding. In this case, we have the fraction 2/3, so we need to express the whole number 1 as a fraction with a denominator of 3. The number 1 can be written as 3/3 because any number divided by itself equals 1. Now, we can rewrite the expression as 2/3 + 3/3. With both terms having the same denominator, we can simply add the numerators while keeping the denominator constant. Adding the numerators 2 and 3 gives us 5. Therefore, 2/3 + 3/3 equals 5/3. This resulting fraction, 5/3, is an improper fraction, meaning the numerator is greater than the denominator. This form is perfectly acceptable, but it can also be converted to a mixed number for better understanding in some contexts. This step highlights the importance of understanding fraction addition and the concept of equivalent fractions in simplifying mathematical expressions.
Step 5 Expressing the Result as a Mixed Number (Optional)
While the improper fraction 5/3 is a perfectly valid answer, it is often beneficial to express it as a mixed number. A mixed number consists of a whole number and a proper fraction, which can make the value of the fraction more intuitive. To convert the improper fraction 5/3 into a mixed number, we need to determine how many whole units are contained within the fraction. We do this by dividing the numerator (5) by the denominator (3). When we divide 5 by 3, we get 1 as the whole number quotient and 2 as the remainder. This tells us that 5/3 contains one whole unit and an additional fraction. The whole number part of the mixed number is the quotient, which is 1. The remainder, 2, becomes the numerator of the fractional part, and we keep the original denominator, which is 3. Therefore, the fractional part of the mixed number is 2/3. Combining the whole number part and the fractional part, we express 5/3 as the mixed number 1 2/3. This conversion provides a clearer understanding of the quantity represented by the fraction, as it explicitly shows the whole number component and the remaining fractional part. While expressing the result as a mixed number is optional, it often provides a more practical and intuitive understanding of the value, especially in real-world applications.
Final Answer
After meticulously following each step, we've successfully simplified the complex fraction (2/3) / (1/3 + 2/3) + 1. The initial steps involved simplifying the denominator, rewriting the expression, and performing the division. Finally, we added the whole number and, optionally, expressed the result as a mixed number. The final simplified form of the expression is 5/3, which can also be expressed as the mixed number 1 2/3. This result provides a clear and concise answer to the original complex fraction, demonstrating the power of methodical simplification in mathematics. The process we've undertaken highlights the importance of understanding basic arithmetic operations, fraction manipulation, and the properties of division and addition. By breaking down the complex fraction into smaller, manageable steps, we were able to arrive at the solution with confidence. This step-by-step approach is a valuable strategy for tackling any complex mathematical problem, ensuring accuracy and clarity in the solution. The final answer not only provides the numerical result but also showcases the effectiveness of a systematic approach in simplifying mathematical expressions.
