Problem 11 - Calculus Early Transcendentals: Pearson New International Edition

Question

Simplify as much as possible. Be sure to remove all parentheses and reduce all fractions. \(\frac{\frac{11}{7}-\frac{12}{21}}{\frac{11}{7}+\frac{12}{21}}\)

Step-by-Step Solution

Answer

The simplified form is \( \frac{7}{15} \).

Step 1: Simplify the fractions within the subtraction

Let's simplify the fractions \( \frac{11}{7} \) and \( \frac{12}{21} \). First, we identify the common denominator between them. The lowest common denominator of 7 and 21 is 21. Rewriting \( \frac{11}{7} \) with this common denominator, we get \( \frac{11}{7} = \frac{33}{21} \).\ Now the expression \( \frac{11}{7} - \frac{12}{21} \) becomes \( \frac{33}{21} - \frac{12}{21} = \frac{21}{21} = 1 \).

Step 2: Simplify the fractions within the addition

Using the same common denominator approach, we rewrite \( \frac{11}{7} \) as \( \frac{33}{21} \). Therefore, \( \frac{11}{7} + \frac{12}{21} = \frac{33}{21} + \frac{12}{21} = \frac{45}{21} \).

Step 3: Simplify the overall expression

Now, our expression becomes \( \frac{1}{\frac{45}{21}} \). To simplify this division of a fraction by another fraction, we multiply by the reciprocal. So, \( 1 \div \frac{45}{21} \) becomes \( 1 \times \frac{21}{45} = \frac{21}{45} \).

Step 4: Reduce the final fraction

The fraction \( \frac{21}{45} \) can be further simplified by finding the greatest common divisor of 21 and 45, which is 3. Dividing both the numerator and the denominator by 3, we get \( \frac{21}{3} = 7 \) and \( \frac{45}{3} = 15 \), resulting in the final simplified fraction: \( \frac{7}{15} \).

Key Concepts

Simplifying Fractions

When you're faced with a complex algebraic expression that includes fractions, the first step is often simplifying those fractions. Simplifying, in this context, means reducing the fractions to their simplest form where the numerator and the denominator have no common factors other than 1. This typically involves dividing both the numerator and the denominator by their greatest common divisor (GCD). For example, in the expression \( \frac{21}{45} \), the GCD is 3. By dividing the numerator and the denominator by 3, you obtain \( \frac{7}{15} \), which is the simplified form. This simplification makes expressions easier to work with and understand.
Remember that simplifying a fraction is always a useful step because it can help in evaluating algebraic expressions more effectively.

Common Denominator

Dealing with fractions often requires finding a common denominator so that the fractions can be added or subtracted. A common denominator is a shared multiple of the denominators of the fractions involved. For example, if you have the fractions \( \frac{11}{7} \) and \( \frac{12}{21} \), the lowest common denominator is 21. This means you need to convert \( \frac{11}{7} \) to \( \frac{33}{21} \), which allows you to directly add or subtract the fractions. To find a common denominator, list the multiples of each denominator and find the least value they share. Once you have it, adjust each fraction by multiplying the numerator and the denominator by the necessary factors to achieve this common denominator. This preparation step is crucial for fraction calculations in algebraic contexts.

Greatest Common Divisor

The greatest common divisor (GCD) is a key concept in simplifying fractions, helping to reduce them to the simplest form. It is the largest positive integer that divides two or more integers without leaving a remainder. For instance, when simplifying \( \frac{21}{45} \), the GCD of 21 and 45 is 3. You determine the GCD by listing the factors of each number and picking the largest one they have in common. Using this number to divide both the numerator and denominator ensures that the fraction is at its simplest. This process of reduction is not only crucial for simplicity but also necessary for solving more complex algebraic problems accurately.

Reciprocal of a Fraction

The reciprocal of a fraction is a concept that flips the roles of the numerator and the denominator. If you have a fraction like \( \frac{a}{b} \), its reciprocal is \( \frac{b}{a} \). Reciprocal operations come in handy, especially in division problems involving fractions. For instance, dividing by \( \frac{45}{21} \) is the same as multiplying by its reciprocal, \( \frac{21}{45} \). This transformation helps simplify expressions and solve equations more easily, as it converts the division into multiplication, a process that is more straightforward. Understanding the reciprocal is essential in tackling algebraic fractions efficiently.