Finding The Missing Rational Number Sum -5 And One Number Is -13/6

In the realm of mathematics, particularly when dealing with rational numbers, we often encounter problems that require us to find a missing piece of the puzzle. One such problem involves finding a rational number when we know the sum of two rational numbers and one of the numbers themselves. This article delves into a step-by-step approach to solving this type of problem, providing a clear and concise explanation that will help you master this essential mathematical concept.

Understanding Rational Numbers

Before we dive into the problem, let's first define what rational numbers are. Rational numbers are numbers that can be expressed as a fraction ${\frac{p}{q}}$, where p and q are integers, and q is not equal to zero. Examples of rational numbers include integers (e.g., -3, 0, 5), fractions (e.g., ${\frac{1}{2}}$, ${\frac{-3}{4}}$), and terminating or repeating decimals (e.g., 0.25, 0.333...). Understanding this fundamental concept is crucial for tackling problems involving the sum of rational numbers.

The Problem: Sum of Two Rational Numbers

Now, let's consider the specific problem at hand: The sum of two rational numbers is -5. If one of the numbers is ${\frac{-13}{6}}$, we need to find the other number. This problem requires us to utilize our understanding of rational numbers and algebraic manipulation to arrive at the solution. The key here is to translate the word problem into a mathematical equation and then solve for the unknown variable. This process involves identifying the knowns and unknowns, setting up the equation, and then using algebraic techniques to isolate the unknown variable.

Setting up the Equation

To begin, let's represent the unknown rational number as x. According to the problem, the sum of x and ${\frac{-13}{6}}$ is -5. We can express this as an equation:

x + ${\frac{-13}{6}}$ = -5

This equation is the foundation for solving the problem. It clearly states the relationship between the unknown rational number (x), the given rational number (${\frac{-13}{6}}$), and their sum (-5). The next step involves using algebraic manipulation to isolate x and find its value. This will involve adding the additive inverse of ${\frac{-13}{6}}$ to both sides of the equation, which will effectively cancel out the ${\frac{-13}{6}}$ term on the left side and leave x isolated.

Solving for the Unknown

To solve for x, we need to isolate it on one side of the equation. We can do this by adding the additive inverse of ${\frac{-13}{6}}$, which is ${\frac{13}{6}}$, to both sides of the equation:

x + ${\frac{-13}{6}}$ + ${\frac{13}{6}}$ = -5 + ${\frac{13}{6}}$

This step is crucial because it maintains the equality of the equation while moving us closer to isolating x. Adding ${\frac{13}{6}}$ to both sides cancels out the ${\frac{-13}{6}}$ term on the left side, leaving x by itself. On the right side, we now have to add -5 and ${\frac{13}{6}}$, which requires finding a common denominator.

Now, we need to add -5 and ${\frac{13}{6}}$. To do this, we need to express -5 as a fraction with a denominator of 6. We can do this by multiplying -5 by ${\frac{6}{6}}$:

-5 = ${\frac{-5}{1}}$ * ${\frac{6}{6}}$ = ${\frac{-30}{6}}$

Now we can rewrite the equation as:

x = ${\frac{-30}{6}}$ + ${\frac{13}{6}}$

Adding the fractions, we get:

x = ${\frac{-30 + 13}{6}}$ = ${\frac{-17}{6}}$

Therefore, the other rational number is ${\frac{-17}{6}}$. This is the final answer to the problem. We have successfully found the missing rational number by setting up an equation and using algebraic manipulation to solve for the unknown variable. This process demonstrates the importance of understanding rational numbers and their properties, as well as the ability to apply algebraic techniques to solve mathematical problems.

Verification

To ensure our answer is correct, we can verify it by adding the two rational numbers together and checking if the sum is indeed -5:

${\frac{-13}{6}}$ + ${\frac{-17}{6}}$ = ${\frac{-13 - 17}{6}}$ = ${\frac{-30}{6}}$ = -5

Since the sum is -5, our answer is correct. This verification step is an essential part of problem-solving in mathematics. It allows us to catch any potential errors and ensure that our solution is accurate. By verifying our answer, we can have confidence in our understanding of the problem and the methods we used to solve it.

Conclusion

In this article, we have explored how to find a rational number when the sum of two rational numbers and one of the numbers is given. We learned that the key to solving this type of problem is to set up an equation and use algebraic manipulation to isolate the unknown variable. We also emphasized the importance of understanding rational numbers and their properties, as well as the need to verify our answers to ensure accuracy. By following these steps, you can confidently tackle similar problems and strengthen your understanding of rational numbers and algebra. The ability to solve problems involving rational numbers is a fundamental skill in mathematics, and mastering this skill will open doors to more advanced mathematical concepts and applications.

Find the other rational number if the sum of two rational numbers is -5, and one of the numbers is -13/6.

Finding the Missing Rational Number Sum -5 and One Number is -13/6