Problem 14 - Elementary and Intermediate Algebra

Question

Solve each of these number problems. See Example \(1 .\) If the same number is subtracted from both the numerator and the denominator of \(\frac{11}{13},\) the result is \(\frac{3}{4} .\) Find the number.

Step-by-Step Solution

Answer

The number is 5.

Step 1: Set Up the Equation

Let's designate the number we need to find as \( x \). If \( x \) is subtracted from both the numerator and the denominator of \( \frac{11}{13} \), the result is \( \frac{3}{4} \). This gives us the equation: \[ \frac{11 - x}{13 - x} = \frac{3}{4}. \]

Step 2: Cross Multiply

To eliminate the fractions, use cross multiplication on the equation. This means multiplying the denominator on the left by the numerator on the right and vice versa: \[ 4(11 - x) = 3(13 - x). \]

Step 3: Distribute and Simplify

Expand both sides by distributing the multiplication: \[ 44 - 4x = 39 - 3x. \]

Step 4: Rearrange the Equation

To isolate \( x \), move all terms involving \( x \) to one side and constant terms to the other side: \[ 44 - 39 = 4x - 3x. \] Simplifying gives: \[ 5 = x. \]

Step 5: Verify the Solution

Substitute \( x = 5 \) back into the original equation to check if it holds true: \[ \frac{11 - 5}{13 - 5} = \frac{6}{8} = \frac{3}{4}. \] Both sides are equal, confirming \( x = 5 \) as the correct solution.

Key Concepts

Rational Equations

Rational equations are equations that involve fractions, specifically those with polynomials in the numerator and denominator. In this type of equation, the goal is often to find the value of a variable that satisfies the equation.
Key characteristics of rational equations include:

• The presence of a fraction with a variable, in either the numerator, the denominator, or both.
• Approaches for finding the solution often involve eliminating the fraction to simplify solving it more straightforwardly.

One typical method for solving rational equations is to "clear the fraction." This often involves multiplication or cross multiplication, which we'll detail in the next section. Rational equations require careful attention to restrictions, particularly those that make the denominator zero, as those are undefined values in mathematics.
Rational equations appear in various real-life situations, like calculating ratios, rates, or any scenario involving proportional relationships. When tackling them, focus on systematic steps to rearrange and simplify the equation.

Cross Multiplication

Cross multiplication is a technique used to solve rational equations that equate two fractions. In essence, it helps to simplify the equation by eliminating the fractions altogether.
Here's how cross multiplication works:

• Consider two fractions set equal, like \( \frac{a}{b} = \frac{c}{d} \).
• Multiply the numerator of the first fraction (\(a\)) by the denominator of the second fraction (\(d\)).
• Then, multiply the numerator of the second fraction (\(c\)) by the denominator of the first fraction (\(b\)).
• Set the two products equal, yielding the equation \( a \times d = b \times c \).

With cross multiplication, the fractions are replaced by a simpler equation without fractions, making it easier to solve for the variable. In the original exercise, this method transformed \( \frac{11 - x}{13 - x} = \frac{3}{4} \) into the equation \( 4(11 - x) = 3(13 - x) \), averting the intricacies of handling fractional terms later in the process.

Numerator and Denominator

In math, fractions are made up of two main components: the numerator and the denominator. Understanding these is crucial when working with rational equations.
The **numerator** is the top part of the fraction. It indicates the number of parts we have. For example, in \( \frac{11- x}{13-x} \), \(11-x\) is the numerator, which represents how the value changes when a number is subtracted.

• The numerator explains what is being counted or subtracted.
• In rational equations, changing the numerator can modify the entire value of the fraction.

On the other hand, the **denominator** is the bottom part of the fraction, showing the total number of equal parts in the whole. In \( \frac{11-x}{13-x} \), the denominator is \(13-x\), affected as well by subtracting the same number.

• The denominator helps determine the size of each part.
• As with rational equations, any changes here will influence the fraction's value drastically.

Together, they form the fraction, and any operations executed on them (like subtracting a number from both) alter the fraction's overall value, crucial in solving rational equations effectively.