Problem 45 - Prealgebra

Question

The denominator of a fraction is 4 more than the numerator. If both the numerator and denominator are increased by \(1,\) the resulting fraction equals \(\frac{1}{2} .\) Find the original fraction.

Step-by-Step Solution

Answer

The original fraction is \( \frac{3}{7} \).

Step 1: Define Variables

Let the numerator of the original fraction be \( x \). Then the denominator is \( x + 4 \), since it is 4 more than the numerator. Therefore, the original fraction can be expressed as \( \frac{x}{x+4} \).

Step 2: Translate Conditions into Equation

According to the problem, when both the numerator and denominator are increased by 1, the fraction becomes \( \frac{1}{2} \). This results in the equation: \[\frac{x + 1}{x + 4 + 1} = \frac{1}{2}.\]Simplifying the denominator gives: \[\frac{x + 1}{x + 5} = \frac{1}{2}.\]

Step 3: Solve the Equation

Cross-multiply the equation from Step 2 to get rid of the fractions:\[2(x + 1) = 1(x + 5).\]Simplify this equation:\[2x + 2 = x + 5.\]

Step 4: Isolate the Variable

Subtract \( x \) from both sides to solve for \( x \):\[2x - x = 5 - 2,\]which simplifies to:\[x = 3.\]

Step 5: Determine the Original Fraction

Substitute \( x = 3 \) back into the original fraction to find the numerator and denominator:\[\text{Numerator} = x = 3,\]\[\text{Denominator} = x + 4 = 7.\]Thus, the original fraction is \( \frac{3}{7} \).

Key Concepts

Understanding the Numerator

In any fraction, the numerator is the top number that signifies how many parts of a whole we are considering. It sits above the fraction bar. For instance, in the fraction \( \frac{3}{7} \), the number \( 3 \) is the numerator. It's important to note that the numerator can be any integer, positive or negative, and it dictates the portion of the denominator that is being represented.
When attempting to solve fraction problems like the one given, the numerator plays a crucial role in forming the correct equation for solving the problem. Here, in the exercise, the numerator is represented by \( x \), which is our unknown value that we need to find. Knowing the numerator and its relationship to the denominator allows us to establish a connection that aids in solving the equation involving fractions.

Decoding the Denominator

The denominator is the number below the fraction line, representing the total number of equal parts the whole is divided into. In the fraction \( \frac{3}{7} \), the number \( 7 \) is the denominator. It sets the context for the fraction by showing what fraction of the total is being discussed.

• The denominator is key in determining the scale of the fraction. For example, \( \frac{1}{2} \) signals that we are dealing with halves, while \( \frac{1}{8} \) involves eighths.
• The denominator in the exercise is more complex, described as being 4 more than the numerator.

This relationship is crucial as it allows us to translate conditions into an equation as seen when both the numerator and denominator are altered by 1. Understanding this relationship guides us in forming the equation \( \frac{x+1}{x+5} = \frac{1}{2} \), from which we solve for \( x \).

Steps to Equation Solving

Solving equations involving fractions requires a systematic approach, often starting with translating a word problem into equations. This task typically involves a few key steps:- **Define the Variables:** Identify what each variable represents. In our problem, \( x \) represents the numerator, helping us form a fraction \( \frac{x}{x+4} \).
- **Set Up the Equation:** Once the problem conditions are clear, set up equations based on the relationships between components of the fraction. Here, it means adding 1 to both numerator and denominator leading to \( \frac{x+1}{x+5} = \frac{1}{2} \).
- **Cross-Multiplication:** A helpful technique here is cross-multiplying to eliminate the fractions, thus simplifying the equation into a linear form, like \( 2(x + 1) = 1(x + 5) \).
- **Isolate and Solve the Variable:** Finally, manipulate the equation to isolate the variable (\( x \)) so that we can solve for it. This involves simplifying and rearranging terms until you solve for \( x = 3 \).
Each step builds on the last and requires careful execution to solve the equations involving both the numerator and denominator effectively. Understanding these steps lays a strong foundation for tackling any fraction-based equation problems in algebra.