Problem 27 - Algebra A Combined Function

Question

Simplify each complex fraction. $$ \frac{1}{2+\frac{1}{3}} $$

Step-by-Step Solution

Answer

The simplified form of the complex fraction is \( \frac{3}{7} \).

Step 1: Understand the Complex Fraction

The given expression is a complex fraction where the denominator itself is a fraction. The expression is \( \frac{1}{2 + \frac{1}{3}} \). Our goal is to simplify this by converting it into a simpler fraction.

Step 2: Combine Denominator Terms

To simplify, first address the terms in the denominator, \( 2 + \frac{1}{3} \). Convert \( 2 \) into a fraction with a denominator of 3. Thus, \( 2 = \frac{6}{3} \). Now combine the fractions: \( 2 + \frac{1}{3} = \frac{6}{3} + \frac{1}{3} = \frac{7}{3} \).

Step 3: Rewrite Complex Fraction

Now, rewrite the original expression using the combined denominator. The complex fraction becomes \( \frac{1}{\frac{7}{3}} \).

Step 4: Simplify by Inverting and Multiplying

Simplify \( \frac{1}{\frac{7}{3}} \) by realizing this is equivalent to multiplying by the reciprocal: \( 1 \times \frac{3}{7} = \frac{3}{7} \). This gives us the simplified expression.

Key Concepts

Simplifying Fractions

When dealing with fractions, simplification means rewriting them in their simplest form. For complex fractions, like \( \frac{1}{2 + \frac{1}{3}} \), the task involves simplifying both the numerator and the denominator whenever applicable.

• Start by focusing on any fractional components within the overall fraction. Here, the fraction is inside the denominator.
• Each whole number in the fraction should be expressed with a common denominator to combine it easily. In our example, the number 2 is written as \( \frac{6}{3} \).
• Add or subtract fractions as necessary. By finding a common denominator, we can combine \( \frac{6}{3} + \frac{1}{3} \) to get \( \frac{7}{3} \).

Once any smaller fractions within the fraction are combined, the complex fraction can often be simplified further.

Reciprocal of a Fraction

The reciprocal of a fraction is a key concept in simplifying complex fractions. It is simply flipping the numerator and denominator of the fraction.

• For a fraction \( \frac{a}{b} \), its reciprocal is \( \frac{b}{a} \).
• Reciprocals are especially useful when dividing fractions, as dividing by a fraction is the same as multiplying by its reciprocal.
• In our exercise, the reciprocal of \( \frac{7}{3} \) helps us simplify the expression \( \frac{1}{\frac{7}{3}} \). By taking the reciprocal of \( \frac{7}{3} \), we invert it to get \( \frac{3}{7} \).

After flipping the fraction, you multiply as usual to get the simplified result.

Combining Fractions

Combining fractions involves performing operations like addition or subtraction of fractions with different denominators. It's important to ensure all fractions are described using a common denominator.

• Identify the common denominator for the fractions you are working with.
• Convert each fraction to equivalent fractions with the common denominator. For example, convert 2 into \( \frac{6}{3} \) so that it can be combined with \( \frac{1}{3} \).
• Perform the addition or subtraction: \( \frac{6}{3} + \frac{1}{3} = \frac{7}{3} \).

By achieving a single fraction where several existed before, you make further operations, such as simplification, much easier to carry out. This technique is fundamental and used frequently when simplifying more complex expressions.