Problem 12 - Calculus Early Transcendentals: Pearson New International Edition

Question

simplify as much as possible. Be sure to remove all parentheses and reduce all fractions. $$ \frac{\frac{1}{2}-\frac{3}{4}+\frac{7}{8}}{\frac{1}{2}+\frac{3}{4}-\frac{7}{8}} $$

Step-by-Step Solution

Answer

The simplified expression is \( \frac{5}{3} \).

Step 1: Simplifying the Numerator

Start by simplifying the numerator \( \frac{1}{2} - \frac{3}{4} + \frac{7}{8} \). To do this, find a common denominator. The least common multiple of 2, 4, and 8 is 8. Convert each fraction: \( \frac{1}{2} = \frac{4}{8} \), \( \frac{3}{4} = \frac{6}{8} \), and \( \frac{7}{8} \) remains the same. Now rewrite the expression as \( \frac{4}{8} - \frac{6}{8} + \frac{7}{8} \). Simplifying this gives \( \frac{4 - 6 + 7}{8} = \frac{5}{8} \).

Step 2: Simplifying the Denominator

Next, simplify the denominator \( \frac{1}{2} + \frac{3}{4} - \frac{7}{8} \) in the expression. Again, use 8 as the common denominator. Convert each fraction: \( \frac{1}{2} = \frac{4}{8} \), \( \frac{3}{4} = \frac{6}{8} \), and \( \frac{7}{8} \). Rewrite the expression as \( \frac{4}{8} + \frac{6}{8} - \frac{7}{8} \). Simplifying this gives \( \frac{4 + 6 - 7}{8} = \frac{3}{8} \).

Step 3: Simplifying the Overall Fraction

Now that we have the simplified numerator and denominator, the overall expression becomes \( \frac{\frac{5}{8}}{\frac{3}{8}} \). To simplify this, multiply by the reciprocal of the denominator: \( \frac{5}{8} \times \frac{8}{3} = \frac{5}{1} \times \frac{1}{3} = \frac{5}{3} \). The 8's (numerator and denominator) cancel each other out, simplifying to \( \frac{5}{3} \).

Step 4: Final Simplification Check

We have simplified the fraction to \( \frac{5}{3} \). Since 5 and 3 are coprime (have no common factors other than 1), \( \frac{5}{3} \) is in its simplest form.

Key Concepts

Numerator and Denominator

In any fraction, the concept of the numerator and denominator is crucial for understanding fraction simplification. The numerator is the top number that denotes how many parts we are focusing on. On the other hand, the denominator is the bottom number, which shows the total number of equal parts. For example, in the fraction \( \frac{1}{2} \), 1 is the numerator, while 2 is the denominator.

When simplifying fractions, like in the original exercise, the aim is often to make the numerator and denominator as small as possible, while keeping the fraction equivalent. This helps in comparing and calculating fractions more easily. Simplifying fractions involves dividing both the numerator and denominator by their greatest common divisor (GCD), if possible.

• The numerator is about what part of the division we are interested in.
• The denominator tells us how many parts the whole is divided into.
• Simplification aims to reduce both elements by using division.

Just like in the given problem, this understanding helps navigate through the steps of finding common denominators and simplifying expressions.

Common Denominator

Finding a common denominator is a central step when adding or subtracting fractions. It is crucial because it allows the fractions to be converted into equivalent forms with matching denominators. This process is essential for calculating differences accurately.

A common denominator is needed when fractions have different denominators. The goal is to find a common value that each denominator can divide into without leaving a remainder. In the original exercise, we needed to compare fractions \( \frac{1}{2} \), \( \frac{3}{4} \), and \( \frac{7}{8} \). The least common multiple (LCM) of 2, 4, and 8 is found to be 8. Therefore, all fractions were rewritten with 8 as the denominator, facilitating straightforward operations.

• Transforming fractions to have a common denominator makes operations simpler.
• The LCM is the smallest number that all denominators can divide into.
• Adapting fractions to a common denominator aids in their accurate addition/subtraction.

Once transformed, operations can be smoothly executed, yielding simplified results without errors.

Reciprocal

The concept of a reciprocal is invaluable in fraction division and simplification. A reciprocal of a fraction is created by flipping the fraction, inverting its numerator and denominator. For instance, the reciprocal of \( \frac{3}{8} \) is \( \frac{8}{3} \).

In the original exercise, once we achieved \( \frac{\frac{5}{8}}{\frac{3}{8}} \), we needed to simplify this complex fraction. This was achieved by multiplying the numerator \( \frac{5}{8} \) by the reciprocal of the denominator \( \frac{3}{8} \), which is \( \frac{8}{3} \). Multiplication with the reciprocal is comparable to dividing by the original fraction.

• Reciprocals present an intuitive way to handle fraction division.
• Flipping the numerator and denominator creates a reciprocal.
• Multiplying by a reciprocal acts similarly to dividing by the original fraction.

This operation allowed the cancellation of the common terms (8s in this case), further simplifying the fraction to its simplest form. Understanding and using reciprocals effectively reduces the complexity in dealing with fractions.