Solving Fraction Problems A Step-by-Step Guide

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This article dives deep into solving a classic fraction problem, providing a detailed, step-by-step explanation that's easy to follow. If you've ever struggled with word problems involving fractions, this guide is for you. We'll break down the problem, translate the words into mathematical equations, and walk you through the algebraic steps to arrive at the solution. Let's embark on this mathematical journey together!

Understanding the Problem Statement

Let's first carefully analyze the problem statement. The question presents us with a fraction puzzle, rich in mathematical relationships and conditions. Our mission is to decode these relationships and unveil the fraction's true identity. The problem presents a scenario where we are given clues about a fraction and need to determine its value. The core of the problem lies in understanding the relationships between the numerator and the denominator, and how these relationships change when we manipulate the fraction. The wording might seem a bit intricate at first, but by breaking it down piece by piece, we can transform it into manageable mathematical expressions. So, let's dissect the problem statement, identify the key pieces of information, and set the stage for our solution.

The problem states:

  1. The denominator of a fraction exceeds the numerator by 2. This tells us there's a direct relationship between the top (numerator) and bottom (denominator) of the fraction. The denominator is larger than the numerator, specifically by a value of 2. This is our first crucial piece of information, and we'll translate it into an algebraic equation shortly.
  2. If 5 be added to the numerator, the fraction increases by unity. This is the second key piece of information. It describes what happens when we change the numerator – adding 5 to it. The result of this change is that the entire fraction's value goes up by 1. This gives us another equation to work with, linking the original fraction to its modified form.
  3. The fraction is: (a) 3/5 (b) 1/3 (c) 7/9 (d) None of these. This part presents us with multiple-choice options. While we could try plugging each option back into the original problem to see if it fits, our goal is to solve the problem algebraically. This gives us a systematic and reliable way to find the answer. Also, note the crucial option (d) “None of these”, which alerts us to the possibility that the correct answer might not be explicitly listed, adding a layer of analytical depth to our problem-solving approach.

Our objective is to find the original fraction that satisfies both conditions. We'll use algebra to represent the unknowns (numerator and denominator) and create equations based on the given information. By solving these equations, we can pinpoint the fraction's exact value.

Translating Words into Equations

The beauty of mathematics lies in its ability to transform real-world scenarios into abstract equations. In this section, we'll translate the word problem's statements into algebraic expressions. This is a crucial step because it allows us to manipulate the problem using the precise rules of mathematics. We'll represent the unknown quantities with variables and use the given relationships to form equations. This transformation from words to equations is the bridge that allows us to apply our mathematical tools and find the solution.

Let's denote:

  • The numerator of the fraction as x.
  • The denominator of the fraction as y.

Now, we can translate the first statement into an equation:

"The denominator of a fraction exceeds the numerator by 2" can be written as:

  • y = x + 2

This equation captures the fundamental relationship between the numerator and denominator. It tells us that the denominator (y) is always 2 more than the numerator (x). This is our first equation, and it forms the foundation for our solution.

Next, let's translate the second statement. The original fraction can be represented as x/y. When we add 5 to the numerator, the new fraction becomes (x + 5)/y. The problem states that this new fraction is equal to the original fraction plus 1. So, we can write this as:

  • (x + 5) / y = x / y + 1

This is our second equation. It captures the change in the fraction's value when we modify the numerator. It links the original fraction (x/y) to its modified form ((x + 5)/y) and the increase of 1. This equation is slightly more complex than the first, but it provides the second piece of the puzzle we need to solve for our unknowns.

We now have a system of two equations with two unknowns (x and y):

  1. y = x + 2
  2. (x + 5) / y = x / y + 1

This system of equations represents the core of our problem. By solving this system, we will find the values of x and y, which will give us the numerator and denominator of the fraction. The next step is to employ algebraic techniques to solve these equations and unveil the solution.

Solving the System of Equations

With our problem neatly translated into a system of two equations, we're ready to employ the power of algebra to find the values of x and y. There are several methods we could use to solve this system, such as substitution or elimination. Here, we'll use the substitution method, as it's particularly well-suited to this problem. Our goal is to manipulate the equations in a way that isolates the variables and allows us to determine their numerical values. This section is where our algebraic skills take center stage, as we carefully navigate the steps to arrive at the solution.

Let's start with our system of equations:

  1. y = x + 2
  2. (x + 5) / y = x / y + 1

The first equation is already solved for y, which makes it ideal for substitution. We can substitute the expression for y from the first equation into the second equation. This will eliminate y from the second equation, leaving us with an equation in terms of x only.

Substituting y = x + 2 into the second equation, we get:

  • (x + 5) / (x + 2) = x / (x + 2) + 1

Now, we have a single equation with one unknown, x. To solve for x, we need to get rid of the fractions. We can do this by multiplying both sides of the equation by the common denominator, which is (x + 2). This is a crucial step in simplifying the equation and making it easier to solve.

Multiplying both sides by (x + 2), we get:

  • (x + 5) = x + (x + 2)

Now we have a linear equation, which is much easier to solve. Let's simplify the equation by combining like terms:

  • x + 5 = x + x + 2
  • x + 5 = 2x + 2

To isolate x, we can subtract x from both sides:

  • 5 = x + 2

Then, subtract 2 from both sides:

  • x = 3

We've successfully found the value of x, which is the numerator of our fraction! Now that we know x, we can use the first equation (y = x + 2) to find the value of y.

Substituting x = 3 into y = x + 2, we get:

  • y = 3 + 2
  • y = 5

So, y = 5, which is the denominator of our fraction. We've now found both the numerator (x) and the denominator (y). The next step is to put these values together to form the fraction and verify our solution.

Constructing the Fraction and Verifying the Solution

With the values of x and y determined, we're in the final stretch of solving our fraction puzzle. We found that x = 3 (the numerator) and y = 5 (the denominator). Now, we'll assemble these pieces to form the fraction and then rigorously verify if our solution satisfies the original problem's conditions. This verification step is crucial because it ensures that our algebraic manipulations have led us to the correct answer. It's a final check to confirm that our solution aligns perfectly with the problem's initial statements.

Therefore, the fraction is x/y = 3/5. This is our candidate solution. But before we declare victory, we must verify that this fraction satisfies the conditions given in the problem statement. This verification process is not just a formality; it's a critical step in ensuring the accuracy of our solution.

Let's revisit the original conditions:

  1. The denominator of a fraction exceeds the numerator by 2.

    • In our fraction, 3/5, the denominator (5) indeed exceeds the numerator (3) by 2. So, this condition is satisfied.

  2. If 5 be added to the numerator, the fraction increases by unity.

    • If we add 5 to the numerator of 3/5, we get (3 + 5) / 5 = 8/5.
    • Now, we need to check if 8/5 is equal to the original fraction (3/5) plus 1.
    • 3/5 + 1 = 3/5 + 5/5 = 8/5
    • So, the second condition is also satisfied.

Since our fraction, 3/5, satisfies both conditions, we can confidently conclude that it is the correct solution to the problem. This verification step has reinforced our confidence in the accuracy of our algebraic manipulations and the validity of our answer.

Conclusion: The Fraction Revealed

In this comprehensive exploration, we successfully solved a classic fraction word problem. We began by carefully understanding the problem statement, then translated the words into a system of algebraic equations. We skillfully employed the substitution method to solve for the unknowns, revealing the numerator and denominator of the fraction. Finally, we meticulously verified our solution against the original conditions, ensuring its accuracy.

The fraction is 3/5, which corresponds to option (a) in the original problem statement.

This journey through the problem-solving process highlights the power of algebra in unraveling mathematical puzzles. By breaking down complex problems into smaller, manageable steps, we can conquer even the most challenging mathematical hurdles. The key takeaways from this exercise are the importance of careful translation, systematic equation solving, and thorough verification. These skills are invaluable not just in mathematics, but in problem-solving across various domains.

This problem serves as a fantastic example of how mathematical concepts can be applied to real-world scenarios. Fractions are a fundamental part of mathematics, and understanding how to manipulate them is crucial for a wide range of applications. The ability to translate word problems into equations is a critical skill in mathematics and beyond. This process involves identifying the key information, representing unknowns with variables, and expressing relationships as equations. By mastering this skill, you can unlock the power of mathematics to solve a vast array of problems.

By mastering these techniques, you'll be well-equipped to tackle similar fraction problems and more complex mathematical challenges. Keep practicing, and you'll continue to sharpen your problem-solving skills!