Solving Fractional Equations: A Step-By-Step Guide
Solving fractional equations can seem daunting to many. However, with a systematic approach, it becomes a straightforward process. In this article, we provide a comprehensive guide to understanding and solving equations involving fractions, focusing on a specific example to illustrate the process.
Understanding the Problem
Let's consider this problem: If 3/5 of a number is 4 more than 1/2 the number, what is the number?
Algebraic Representation
To translate the problem into an algebraic equation, we set the number to be (x).
Step 1: Creating the Equation
Given that (frac{3}{5}x frac{1}{2}x 4), we can simplify and solve for (x).
Step 2: Isolating the Variable
To isolate (x), we begin by moving all terms involving (x) to one side of the equation and the constant to the other side.
(frac{3}{5}x - frac{1}{2}x 4)
Next, we find a common denominator to combine these fractions. The common denominator for 5 and 2 is 10.
(frac{6}{10}x - frac{5}{10}x 4)
Simplifying the left-hand side, we get:
(frac{1}{10}x 4)
Step 3: Solving for (x)
Multiplying both sides by 10 to isolate (x), we get:
(x 40)
Thus, the number is 40.
Proof and Verification
To verify our solution, let's substitute (x 40) back into the original equation:
Verification
(frac{3}{5} times 40 24)
(frac{1}{2} times 40 4 20 4 24)
Since both sides are equal, the solution is correct.
Similar Problems and Solutions
Let's consider a similar problem: If (frac{3}{5}r frac{1}{2}r 4), the steps remain the same, leading to the same conclusion.
Algebraic Verification
(frac{6}{10}r - frac{5}{10}r 4)
(frac{1}{10}r 4)
(r 40)
Again, substituting (r 40) into the original equation, we confirm that the solution is accurate.
Conclusion
Understanding and solving fractional equations involves several steps. By carefully isolating the variable and ensuring the proper manipulation of fractions, even complex equations can be solved. This article provided a clear pathway to solving such equations, ensuring that the reader can apply these methods to similar problems.
Those who wish to delve further into the topic of algebra, particularly solving fractional equations, may find it helpful to explore additional resources and practice problems.