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: Simplify the Numerator

Start by simplifying the numerator: \( \frac{1}{2} - \frac{3}{4} + \frac{7}{8} \). To do this, find a common denominator for these fractions, which is 8. Express each fraction with this common denominator to get: \( \frac{4}{8} - \frac{6}{8} + \frac{7}{8} \). Then, perform the addition and subtraction: \( \frac{4}{8} - \frac{6}{8} + \frac{7}{8} = \frac{5}{8} \).

Step 2: Simplify the Denominator

Now, simplify the denominator: \( \frac{1}{2} + \frac{3}{4} - \frac{7}{8} \). Again, convert each term to have a common denominator of 8: \( \frac{4}{8} + \frac{6}{8} - \frac{7}{8} \). Perform the operations: \( \frac{4}{8} + \frac{6}{8} - \frac{7}{8} = \frac{3}{8} \).

Step 3: Form the New Fraction

Replace the original fraction with the simplified numerator and denominator obtained from the previous steps: \( \frac{\frac{5}{8}}{\frac{3}{8}} \).

Step 4: Divide the Fractions

To simplify \( \frac{\frac{5}{8}}{\frac{3}{8}} \), divide the fractions by multiplying by the reciprocal of the denominator: \( \frac{5}{8} \times \frac{8}{3} = \frac{5 \times 8}{8 \times 3} = \frac{5}{3} \).

Step 5: Simplified Expression

The simplified form of the given expression is \( \frac{5}{3} \).

Key Concepts

Finding a Common Denominator

When you're dealing with fractions, especially when adding or subtracting them, it's crucial to work with a common denominator. This means ensuring that each fraction involved in the operation is expressed in terms of a similar base. To do this, identify the least common multiple (LCM) of the denominators involved. For instance, if you're working with fractions like \( \frac{1}{2} \), \( \frac{3}{4} \), and \( \frac{7}{8} \), you need to find the smallest number that all these denominators can divide into, which is 8.
Here's how you convert:

• For \( \frac{1}{2} \), multiply the numerator and the denominator by 4 to get \( \frac{4}{8} \).
• For \( \frac{3}{4} \), multiply by 2 to get \( \frac{6}{8} \).
• The fraction \( \frac{7}{8} \) already has the denominator of 8, so it remains the same.

Using a common denominator not only makes it simpler to perform operations like addition or subtraction but also clears the way for further simplification.

Fraction Addition and Subtraction

After ensuring each fraction has the same denominator, you can proceed with the actual addition or subtraction. This process involves simply adding or subtracting the numerators, while the denominators stay unchanged. For example, if you're starting with the fractions \( \frac{4}{8} \), \( \frac{6}{8} \), and \( \frac{7}{8} \), you deal only with the numerators:

• First, subtract: \( 4 - 6 = -2 \).
• Then, add (or continue the operation): \( -2 + 7 = 5 \).

Thus, the numerator is 5 while the denominator is 8, leading to the fraction \( \frac{5}{8} \). This is how we simplify using addition or subtraction once a common denominator is established.
This simplification method can help clarify mathematical expressions involving fractions, as it effectively breaks down the process into manageable steps.

Fraction Division

Dividing fractions might initially seem tricky, but the process is made simple by the rule: multiply by the reciprocal. Let's take the fraction division example \( \frac{\frac{5}{8}}{\frac{3}{8}} \). The steps are straightforward:

• Flip the fraction in the denominator (reciprocal) so \( \frac{3}{8} \) becomes \( \frac{8}{3} \).
• Multiply the numerators: \( 5 \times 8 = 40 \).
• Multiply the denominators: \( 8 \times 3 = 24 \).

This results in \( \frac{40}{24} \). Further simplification would yield \( \frac{5}{3} \), after dividing both the numerator and the denominator by their greatest common divisor, which is 8.
Fraction division becomes less daunting once you translate it into multiplication with a reciprocal. This technique underlines a deep connection between division and multiplication in mathematics.

Mathematical Expressions

Mathematical expressions can involve various operations, including addition, subtraction, multiplication, division, and more. Simplifying a mathematical expression with fractions means breaking down each term using fundamental arithmetic operations and ensuring every part is as reduced as possible. Let's recap how we simplified the given complex expression:

• First, simplify both the numerator and denominator to bring clarity to the expression.
• Use fraction addition and subtraction to handle numerators and denominators separately.
• Apply division rules, like multiplying by the reciprocal.
• Always check if the final fraction can be further reduced for simplicity.

Breaking complex expressions into simpler components not only helps in understanding but also in efficient problem-solving.
In essence, simplifying mathematical expressions involves a systematic approach to dealing with operations, leading to clearer and reduced forms.