Problem 28 - Algebra A Combined Function

Question

Simplify each complex fraction. $$ \frac{3}{1-\frac{4}{3}} $$

Step-by-Step Solution

Answer

The simplified form is -9.

Step 1: Identify the Complex Fraction

The given fraction is \( \frac{3}{1-\frac{4}{3}} \). Notice that the denominator contains another fraction.

Step 2: Simplify the Denominator

To simplify the expression \( 1-\frac{4}{3} \), find a common denominator. The term 1 can be rewritten as \( \frac{3}{3} \). Thus, \( 1 - \frac{4}{3} = \frac{3}{3} - \frac{4}{3} = \frac{3-4}{3} = \frac{-1}{3} \).

Step 3: Write the Simplified Denominator

With the simplified denominator, the fraction becomes \( \frac{3}{\frac{-1}{3}} \).

Step 4: Multiply by the Reciprocal

Dividing by a fraction is equivalent to multiplying by its reciprocal. So \( \frac{3}{\frac{-1}{3}} = 3 \times \left(-3\right) = -9 \).

Step 5: Write the Simplified Expression

The simplified complex fraction is \( -9 \).

Key Concepts

Understanding Reciprocals

In mathematics, the concept of a reciprocal is related to the notion of division. The reciprocal of a fraction is created by swapping its numerator and denominator. For instance, the reciprocal of \( \frac{a}{b} \) is \( \frac{b}{a} \). When you take the reciprocal of a number, especially a fraction, you are essentially asking "What number can I multiply this by to get 1?"
For example, the reciprocal of 2 (or \( \frac{2}{1} \)) is \( \frac{1}{2} \) because \( 2 \times \frac{1}{2} = 1 \). This property is handy when dealing with division of fractions.
When simplifying complex fractions like \( \frac{3}{1 - \frac{4}{3}} \), a key step involves multiplying by the reciprocal of the simplified denominator. This simplifies the process as multiplying a fraction by its reciprocal results in elimination of the denominator component, making equations easier to visualize and solve.

Simplifying Fractions Made Easy

Simplifying fractions involves reducing them to their smallest form where the numerator and the denominator are integers with no common factors other than 1. When faced with a complex fraction, it's essential to break it down into simpler parts to make it easier to handle.
Simplifying begins by examining each part of the fraction. Take the complex fraction \( \frac{3}{1 - \frac{4}{3}} \). The denominator here \( 1 - \frac{4}{3} \) is not straightforward. The step involves converting the 1 into a fraction with the same denominator as \( \frac{4}{3} \), which is \( \frac{3}{3} \).
Thus, you perform the subtraction \( \frac{3}{3} - \frac{4}{3} = \frac{-1}{3} \). This results in a simpler expression \( \frac{3}{\frac{-1}{3}} \). Reducing this to a simpler fraction by multiplying by the reciprocal makes it easier to interpret and solve.

Finding a Common Denominator

A common denominator is crucial when you have to add or subtract fractions, as it allows the fractions to be combined in a straightforward manner. It is the common multiple of the denominators of the fractions involved.
When you see expressions like \( 1 - \frac{4}{3} \), finding a common denominator facilitates the simplification process. Here, the whole number 1 is rewritten as a fraction with the common denominator \( \frac{3}{3} \), aligning it with \( \frac{4}{3} \).
The subtraction can then be performed easily because both fractions have the same denominator, giving \( \frac{3 - 4}{3} = \frac{-1}{3} \).
This is a game-changer because it turns a potentially complex problem into simple arithmetic. This common denominator technique is a foundational skill in handling fractions proficiently.