Solving 1/r + 1/t = 1/c For R A Step-by-Step Guide
Introduction
In the realm of mathematics, solving equations for a specific variable is a fundamental skill. This article delves into the process of isolating r in the equation 1/r + 1/t = 1/c. This type of equation frequently appears in various contexts, such as problems involving rates, work, and parallel resistances in electrical circuits. Understanding how to manipulate and solve such equations is crucial for students and professionals alike. We will break down the steps involved in solving for r, ensuring a clear and comprehensive understanding of the underlying principles. Let’s embark on this mathematical journey and master the art of isolating variables.
Understanding the Equation
Before diving into the solution, it's essential to understand the equation 1/r + 1/t = 1/c. This equation represents a relationship between three variables: r, t, and c. Each term in the equation is a reciprocal, meaning it's the inverse of the variable. Equations of this form often arise in scenarios where combined rates or proportions are involved. For instance, in physics, this equation can model the total resistance of two resistors connected in parallel. In work-rate problems, it might represent the combined time it takes two individuals working together to complete a task. The ability to manipulate and solve this equation has far-reaching applications in various fields, making it a valuable skill to acquire.
Step-by-Step Solution
To solve for r, we need to isolate it on one side of the equation. Here's a step-by-step breakdown of the process:
1. Isolate the Term with r
The first step is to isolate the term containing r, which is 1/r. To do this, we subtract 1/t from both sides of the equation:
1/r + 1/t - 1/t = 1/c - 1/t
This simplifies to:
1/r = 1/c - 1/t
2. Find a Common Denominator
Now, we need to combine the fractions on the right side of the equation. To do this, we find a common denominator for 1/c and 1/t. The common denominator is the least common multiple (LCM) of c and t, which is ct. We rewrite the fractions with the common denominator:
1/r = (t/ct) - (c/ct)
3. Combine the Fractions
Now that the fractions have a common denominator, we can combine them:
1/r = (t - c) / ct
4. Take the Reciprocal
To solve for r, we need to get it out of the denominator. We can do this by taking the reciprocal of both sides of the equation. The reciprocal of 1/r is r, and the reciprocal of (t - c) / ct is ct / (t - c):
r = ct / (t - c)
5. Final Solution
Therefore, the solution for r is:
r = ct / (t - c)
This is the final expression for r in terms of c and t. This step-by-step solution ensures clarity and accuracy in solving the equation.
Common Mistakes to Avoid
When solving equations like this, there are several common mistakes that students often make. Being aware of these pitfalls can help you avoid them and ensure accurate solutions. One common mistake is failing to find a common denominator before combining fractions. This can lead to incorrect simplification and ultimately an incorrect answer. Another mistake is forgetting to take the reciprocal of both sides after isolating 1/r. It's crucial to remember that you're solving for r, not 1/r. A third common mistake is incorrectly distributing negative signs when combining fractions or simplifying expressions. Pay close attention to the signs of each term to avoid errors. By being mindful of these common mistakes, you can improve your accuracy and confidence in solving similar equations.
Practical Applications
The equation 1/r + 1/t = 1/c and its solution for r have practical applications in various fields. In physics, this equation is used to calculate the equivalent resistance of two resistors connected in parallel. Here, r would represent the equivalent resistance, t and c would represent the resistances of the individual resistors. In engineering, particularly in electrical circuit analysis, understanding how to manipulate this equation is crucial for designing and analyzing circuits. In mathematics, this equation can arise in problems involving rates of work. For example, if two people are working together on a task, r might represent the combined time it takes them to complete the task, while t and c represent the individual times. These practical applications highlight the importance of mastering this type of equation and its solution.
Example Problems
To solidify your understanding, let's work through a couple of example problems.
Example 1
Solve for r when c = 6 and t = 3.
Using the formula r = ct / (t - c), we substitute the given values:
r = (6 * 3) / (3 - 6)
r = 18 / (-3)
r = -6
Example 2
Solve for r when c = 10 and t = 5.
Again, using the formula r = ct / (t - c), we substitute the values:
r = (10 * 5) / (5 - 10)
r = 50 / (-5)
r = -10
These examples demonstrate how to apply the formula to find r given specific values for c and t. Working through examples like these is an effective way to reinforce your understanding and build problem-solving skills.
Conclusion
In conclusion, solving for r in the equation 1/r + 1/t = 1/c involves a series of algebraic manipulations, including isolating the term with r, finding a common denominator, combining fractions, and taking the reciprocal. The final solution, r = ct / (t - c), provides a clear expression for r in terms of c and t. This skill is not only valuable in mathematics but also has practical applications in physics, engineering, and other fields. By understanding the step-by-step process and avoiding common mistakes, you can confidently solve similar equations and apply them to real-world problems. Mastering these fundamental algebraic techniques will undoubtedly enhance your problem-solving abilities and open doors to more advanced mathematical concepts. Remember to practice regularly and apply these techniques in various contexts to solidify your understanding.
