Equivalent Expression To 1/3 A Step-by-Step Solution
In the realm of mathematics, identifying equivalent expressions is a fundamental skill. It allows us to simplify complex equations, solve problems more efficiently, and gain a deeper understanding of mathematical relationships. This article delves into the process of determining which expression is equivalent to $ rac{1}{3}$, providing a step-by-step analysis of several options. We will explore the underlying principles of algebraic manipulation, fraction simplification, and equivalent expression identification. Whether you are a student grappling with algebra or a seasoned math enthusiast, this guide will equip you with the knowledge and techniques to confidently tackle such problems.
Before diving into the specific options, it's crucial to grasp the concept of equivalent expressions. In essence, equivalent expressions are mathematical statements that, despite appearing different, hold the same value for all possible values of the variable. For example, $2x + 4$ and $2(x + 2)$ are equivalent expressions because they yield the same result regardless of the value assigned to $x$. Identifying equivalent expressions often involves simplifying, factoring, or manipulating algebraic expressions to reveal their underlying equivalence. This process requires a solid understanding of mathematical operations, including addition, subtraction, multiplication, division, and the order of operations. It also necessitates familiarity with algebraic properties such as the distributive property, commutative property, and associative property. In the context of fractions, equivalent expressions can be generated by multiplying or dividing both the numerator and denominator by the same non-zero value. This principle is particularly relevant when dealing with rational expressions, where the numerator and denominator are polynomials. By mastering the techniques of identifying equivalent expressions, you can enhance your problem-solving skills and gain a more profound appreciation for the elegance and consistency of mathematics.
Now, let's turn our attention to the specific expressions presented in the problem. We are tasked with determining which of the following expressions is equivalent to $ rac{1}{3}$:
Each of these expressions involves rational functions, which are fractions where the numerator and denominator are polynomials. To determine equivalence, we need to simplify each expression and compare the result to $\frac{1}{3}$. This simplification process may involve combining fractions with common denominators, factoring polynomials, and canceling common factors. It's crucial to pay close attention to the order of operations and to ensure that all algebraic manipulations are performed correctly. A common mistake is to incorrectly apply the distributive property or to fail to identify common factors. By carefully analyzing each expression and systematically simplifying it, we can confidently identify the one that is equivalent to $\frac{1}{3}$. This process not only provides the solution to the problem but also reinforces our understanding of algebraic manipulation and fraction simplification.
Expression 1: $\frac{3 d}{3 d+1}+\frac{1}{3 d+1}$
The first expression we need to examine is $ rac{3 d}{3 d+1}+\frac{1}{3 d+1}$. To determine if this expression is equivalent to $ rac{1}{3}$, we must simplify it. Notice that both fractions have a common denominator of $3d + 1$. This allows us to combine the numerators directly:
Now, we observe that the numerator and denominator are identical. As long as $3d + 1$ is not equal to zero (which would make the fraction undefined), we can simplify this expression further:
Therefore, the first expression simplifies to 1, which is not equal to $ rac{1}{3}$. This eliminates the first option as a potential solution. This process highlights the importance of simplifying expressions to their most basic form before making comparisons. By combining fractions with common denominators and recognizing identical numerators and denominators, we were able to quickly determine that this expression is not equivalent to the target value.
Expression 2: $\frac{d+5}{3 d+3}-\frac{4}{3 d+3}$
Let's analyze the second expression: $\frac{d+5}{3 d+3}-\frac{4}{3 d+3}$. Similar to the previous expression, we have two fractions with a common denominator, which is $3d + 3$. This allows us to combine the numerators by subtracting the second numerator from the first:
Now, we can factor out a 3 from the denominator:
We notice that the factor $(d + 1)$ appears in both the numerator and the denominator. As long as $d + 1$ is not equal to zero, we can cancel this common factor:
Thus, the second expression simplifies to $\frac{1}{3}$, which is the target value. This indicates that the second expression is indeed equivalent to $\frac{1}{3}$. This example demonstrates the power of factoring and canceling common factors in simplifying rational expressions. By recognizing the common factor of $(d + 1)$, we were able to reduce the expression to its simplest form and confirm its equivalence to the target value.
Expression 3: $\frac{2}{3 d}-\frac{1}{3 d}$
Now, let's examine the third expression: $\frac{2}{3 d}-\frac{1}{3 d}$. Again, we have two fractions with a common denominator, $3d$. We can combine the numerators by subtracting the second numerator from the first:
The simplified expression is $\frac{1}{3d}$. This expression is equivalent to $ rac{1}{3}$ only when $d = 1$. However, for other values of $d$, the expression will have a different value. Therefore, this expression is not generally equivalent to $\frac{1}{3}$. This analysis highlights the importance of considering the variable's role in determining equivalence. While the expression may be equal to $ rac{1}{3}$ for a specific value of $d$, it is not equivalent in the general sense, as it does not hold true for all values of $d$.
Expression 4: $\frac{d}{d+3}+\frac{1}{d+3}$
Finally, we analyze the fourth expression: $\frac{d}{d+3}+\frac{1}{d+3}$. The fractions have a common denominator of $d + 3$, so we can combine the numerators:
The simplified expression is $\frac{d + 1}{d + 3}$. This expression cannot be simplified further. It is not equivalent to $\frac{1}{3}$ for all values of $d$. To see this, we can try substituting a few values for $d$. For example, if $d = 0$, the expression becomes $\frac{1}{3}$, but if $d = 1$, the expression becomes $ rac{2}{4} = \frac{1}{2}$, which is not equal to $\frac{1}{3}$. Therefore, this expression is not generally equivalent to $\frac{1}{3}$. This example reinforces the concept that an expression must hold true for all values of the variable to be considered equivalent. By substituting different values for $d$, we can quickly verify whether an expression is a potential solution.
In conclusion, after carefully analyzing all four expressions, we have determined that the expression $ rac{d+5}{3 d+3}-\frac{4}{3 d+3}$ is the only one that is equivalent to $ rac{1}{3}$. This was achieved by simplifying each expression and comparing the result to the target value. The process involved combining fractions with common denominators, factoring polynomials, and canceling common factors. This exercise demonstrates the importance of mastering algebraic manipulation techniques to solve mathematical problems effectively. By understanding the principles of equivalent expressions and practicing simplification strategies, you can confidently tackle a wide range of mathematical challenges. Remember to always pay close attention to the order of operations, to consider the variable's role, and to verify your results to ensure accuracy. With consistent practice and a solid understanding of the underlying concepts, you can excel in identifying and manipulating equivalent expressions in mathematics.
The expression equivalent to $\frac{1}{3}$ is:
