Finding The Missing Rational Number A Step By Step Guide

Hey guys! Ever found yourself staring at a math problem that seems like a jumbled mess of fractions and negative signs? Don't worry, we've all been there. Let's break down one of these problems together and see how easy it can be to solve.

Understanding the Basics of Rational Numbers

Before diving into the problem, let's quickly recap what rational numbers are. Simply put, a rational number is any number that can be expressed as a fraction ${ \frac{p}{q} }$, where p and q are integers, and q is not zero. Think of it as a way to represent parts of a whole or ratios between quantities. This includes integers (like 2, -5, or 0), fractions (like ${ \frac{1}{2} }$, ${ \frac{-3}{4} }$, or ${ \frac{5}{7} }$), and even terminating or repeating decimals (like 0.5 or 0.333...). Rational numbers are the building blocks of many mathematical concepts, so getting comfy with them is super important.

Adding and Subtracting Rational Numbers

Now, when it comes to adding or subtracting rational numbers, there's one golden rule: you need a common denominator. Imagine trying to add apples and oranges – it doesn't quite work, right? You need a common unit, like “fruits.” Similarly, fractions need a common denominator to be added or subtracted meaningfully. This common denominator allows us to express the fractions in terms of the same “wholes,” making the addition or subtraction straightforward. So, if you're faced with adding ${ \frac{1}{3} }$ and ${ \frac{1}{4} }$, you'd first find the least common multiple (LCM) of 3 and 4, which is 12. Then, you'd convert both fractions to have a denominator of 12: ${ \frac{1}{3} }$ becomes ${ \frac{4}{12} }$ and ${ \frac{1}{4} }$ becomes ${ \frac{3}{12} }$. Now you can easily add them: ${ \frac{4}{12} + \frac{3}{12} = \frac{7}{12} }$. This fundamental concept is crucial for tackling more complex problems involving rational numbers.

Problem Setup

Let's think about how we frame these problems. The core idea here is that we're dealing with an equation where the sum of two numbers is given, and we know one of those numbers. It's like saying, “I have two boxes of chocolates, and together they have 50 chocolates. If one box has 20 chocolates, how many are in the other box?” We'd naturally subtract 20 from 50 to find the answer. The same logic applies to rational numbers. We're essentially solving for a missing addend. To put it in mathematical terms, if we have two rational numbers, let’s call them x and y, and their sum is z, we can write this as x + y = z. If we know y and z, our mission is to find x. This involves rearranging the equation to isolate x on one side, which leads us to the subtraction we mentioned earlier. Understanding this basic setup is key because it provides a clear roadmap for solving the problem. The problem gives us the sum, ${ -\frac{43}{52} }$, and one of the numbers, ${ \frac{5}{52} }$. Our goal is to find the other number. Let's call the unknown number x. So, the equation we need to solve is:

${ x + \frac{5}{52} = -\frac{43}{52} }$

Solving for the Unknown Rational Number

Okay, let's get down to business and solve this thing! We've got our equation set up, and now it's all about isolating that unknown variable, x. Remember, the key to solving equations is to keep both sides balanced. Whatever operation we perform on one side, we have to do the same on the other side. In our case, we want to get x all by itself on the left side of the equation. Currently, we have ${ x + \frac{5}{52} = -\frac{43}{52} }$. The ${ \frac{5}{52} }$ is the pesky term that's keeping x from being alone. So, how do we get rid of it? We use the magic of inverse operations!

Using Inverse Operations

The inverse operation of addition is subtraction. So, to undo the addition of ${ \frac{5}{52} }$, we're going to subtract ${ \frac{5}{52} }$ from both sides of the equation. This keeps the equation balanced and helps us isolate x. Here's what that looks like:

${ x + \frac{5}{52} - \frac{5}{52} = -\frac{43}{52} - \frac{5}{52} }$

On the left side, the ${ + \frac{5}{52} }$ and the ${ - \frac{5}{52} }$ cancel each other out, leaving us with just x. Awesome! Now, let's focus on the right side of the equation. We have ${ -\frac{43}{52} - \frac{5}{52} }$. Since both fractions have the same denominator (52), we can directly subtract the numerators. It's like saying, “I owe 43 dollars, and then I owe another 5 dollars. How much do I owe in total?”

Performing the Subtraction

To subtract these fractions, we simply subtract the numerators:

${ -43 - 5 = -48 }$

So, the right side of the equation becomes ${ -\frac{48}{52} }$. Now our equation looks like this:

${ x = -\frac{48}{52} }$

We're almost there! We've found the value of x, but it's always a good idea to simplify our answer if possible. This means reducing the fraction to its lowest terms.

Simplifying the Result

Alright, we've landed on ${ x = -\frac{48}{52} }$, which is a solid answer, but let's make it even sleeker by simplifying the fraction. Simplifying fractions is like giving them a makeover – we want to make them look their best by reducing them to their simplest form. This means finding the greatest common divisor (GCD) of the numerator and the denominator and then dividing both by that GCD. So, what exactly is the greatest common divisor? Well, it's the largest number that divides evenly into both the numerator and the denominator. In simpler terms, it's the biggest number that can go into both 48 and 52 without leaving a remainder.

Finding the Greatest Common Divisor (GCD)

There are a couple of ways we can find the GCD. One way is to list out the factors (the numbers that divide evenly) of both numbers and see which one is the largest they have in common. Another way, which can be particularly handy for larger numbers, is to use the Euclidean algorithm. But for 48 and 52, let's stick to listing the factors:

• Factors of 48: 1, 2, 3, 4, 6, 8, 12, 16, 24, 48
• Factors of 52: 1, 2, 4, 13, 26, 52

Looking at these lists, we can see that the greatest common factor is 4. That's our GCD! Now that we've found the GCD, we're ready to simplify the fraction.

Dividing by the GCD

To simplify ${ -\frac{48}{52} }$, we divide both the numerator and the denominator by the GCD, which is 4:

${ \frac{-48 \div 4}{52 \div 4} = \frac{-12}{13} }$

So, the simplified fraction is ${ -\frac{12}{13} }$. This is the same value as ${ -\frac{48}{52} }$, but it's expressed in its simplest form. Think of it like this: ${ -\frac{48}{52} }$ is like saying “48 out of 52 slices are gone,” while ${ -\frac{12}{13} }$ is like saying “12 out of 13 slices are gone.” They represent the same amount, just in different terms. And with that, we've simplified our result and found the missing rational number!

Final Answer

So, after all that awesome math-ing, we've arrived at our final answer! The other number is ${ -\frac{12}{13} }$. We took a problem that looked a little intimidating at first glance and broke it down into manageable steps. We remembered the basics of rational numbers, set up our equation, used inverse operations to solve for the unknown, and then simplified our answer to its sleekest form. Give yourself a pat on the back – you’ve nailed it!

To recap, we started with the equation:

${ x + \frac{5}{52} = -\frac{43}{52} }$

We subtracted ${ \frac{5}{52} }$ from both sides:

${ x = -\frac{43}{52} - \frac{5}{52} }$

We performed the subtraction:

${ x = -\frac{48}{52} }$

And finally, we simplified the fraction:

${ x = -\frac{12}{13} }$

So, the other number is indeed ${ \bf{-\frac{12}{13}} }$. And there you have it, folks! Math problems like these might seem tricky at first, but with a bit of understanding of the fundamentals and a step-by-step approach, you can conquer them all. Keep practicing, and you'll become a rational number whiz in no time!