Problem 13 - Calculus Early Transcendentals: Pearson New International Edition

Question

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

Step-by-Step Solution

Answer

\(1 - \frac{1}{1+\frac{1}{2}} = \frac{1}{3}\).

Step 1: Simplify the Denominator

The expression inside the fraction is \(1 + \frac{1}{2}\). We first find a common denominator for the terms inside the parentheses. The fraction becomes \(\frac{2}{2} + \frac{1}{2} = \frac{3}{2}\). So, \(1 + \frac{1}{2} = \frac{3}{2}\).

Step 2: Simplify the Outer Fraction

Now, substitute back into the original expression to get \(1 - \frac{1}{\frac{3}{2}}\). To simplify this, we invert the denominator of the fraction, turning it into a multiplication, \(1 - \frac{2}{3}\).

Step 3: Simplify the Expression

Now, we just need to subtract \(\frac{2}{3}\) from 1. To do this, first express 1 as a fraction with a common denominator: \(1 = \frac{3}{3}\). So the expression is now \(\frac{3}{3} - \frac{2}{3}\). Perform the subtraction on the numerators: \(\frac{3-2}{3} = \frac{1}{3}\).

Key Concepts

Fractions

When dealing with fractions, understanding the basics is paramount. A fraction represents a part of a whole. It is comprised of a numerator (the top number) and a denominator (the bottom number). Here, the denominator defines the number of equal parts the whole is divided into, while the numerator specifies how many parts we have.
In solving the exercise, the fraction \(\frac{1}{1+\frac{1}{2}}\) requires us to simplify \(1+\frac{1}{2}\). This involves the operation within a larger fraction. Recognizing that \(\frac{1}{2}\) is a fraction itself, we must integrate it properly in the simplification.
Understanding fractions thoroughly helps in breaking down complex mathematics into simple, solvable steps. Mastering the arithmetic of fractions is crucial in algebra.

Common Denominators

In any fractional arithmetic, especially addition, having a common denominator is essential. It allows you to combine fractions effectively. In our exercise, to add \(1\) (which can be written as \(\frac{2}{2}\)) and \(\frac{1}{2}\), we convert \(1\) to have a denominator of 2, resulting in \(\frac{2}{2}\).
The common denominator here makes \(\frac{1 + \frac{1}{2}}{1}\) become \(\frac{\frac{2}{2} + \frac{1}{2}}{1}\), which simplifies to \(\frac{3}{2}\).

• This process is important because it ensures both fractions are speaking the same 'language' in terms of parts or pieces of the whole.
• Once you have a common denominator, you can easily work with fractions as you would with whole numbers.

This technique is incredibly useful not only for addition but also for subtraction, as evidenced in our problem when reducing \(1 - \frac{\frac{2}{3}}{1}\) to a simpler form by converting \(1\) into \(\frac{3}{3}\) first.

Arithmetic Operations

Algebra often involves operations on or with fractions. These can include addition, subtraction, multiplication, or division. In our example, we had to perform a subtraction:
\(1 - \frac{\frac{2}{3}}{1}\). This is executed by converting \(1\) into its fraction equivalent with the same denominator as \(\frac{2}{3}\); hence, \(1 = \frac{3}{3}\).

• Once both numbers are expressed as fractions with a common denominator, subtraction involves simply subtracting the numerators.
• This process brings us to our simplified result, \(\frac{1}{3}\).

Mastering these operations ensures a smooth problem-solving process. Always remember: get a common denominator, perform operations on the numerators, and simplify when needed. Simplification of fractions is about finding the greatest common factor to reduce them to their smallest form.
Being comfortable with these steps will pave the way for tackling more complicated algebraic problems with confidence.