Problem 49 - Elementary and Intermediate Algebra

Question

Simplify each complex fraction. $$ \frac{\frac{1}{p}+\frac{1}{q}}{\frac{1}{p}} $$

Step-by-Step Solution

Answer

The simplified form of the complex fraction is \( \frac{q+p}{q} \).

Step 1: Understand the Complex Fraction

A complex fraction is a fraction where the numerator or the denominator (or both) contain a fraction. In this problem, the numerator \( \frac{1}{p} + \frac{1}{q} \) and the denominator \( \frac{1}{p} \) are both fractions.

Step 2: Find a Common Denominator for the Numerator

To simplify the complex fraction, start by simplifying the numerator. The numerator is \( \frac{1}{p} + \frac{1}{q} \). To add these fractions, find a common denominator, which is \( pq \). Rewrite each term: \( \frac{1}{p} = \frac{q}{pq} \) and \( \frac{1}{q} = \frac{p}{pq} \). Thus, the numerator becomes \( \frac{q+p}{pq} \).

Step 3: Simplify the Complex Fraction

Now substitute the simplified numerator \( \frac{q+p}{pq} \) back into the complex fraction, which becomes \( \frac{\frac{q+p}{pq}}{\frac{1}{p}} \).

Step 4: Simplify by Division of Fractions

To simplify \( \frac{\frac{q+p}{pq}}{\frac{1}{p}} \), multiply by the reciprocal of the denominator: \( \frac{q+p}{pq} \cdot p \).

Step 5: Simplify the Expression

Carrying out the multiplication: \( \frac{p(q+p)}{pq} \). Cancel \( p \) in the numerator and \( p \) in the denominator to get \( \frac{q+p}{q} \).

Step 6: Final Result

The fraction \( \frac{q+p}{q} \) cannot be simplified further, so it is the final simplified form of the original complex fraction.

Key Concepts

Fraction Simplification

Simplifying fractions is all about making them easier to understand and work with. In our problem, we begin with a complex fraction like \( \frac{\frac{1}{p} + \frac{1}{q}}{\frac{1}{p}} \). Here, both the top and bottom parts are themselves fractions.
The goal is to simplify, or "reduce," these fractions into their simplest form. This means finding simpler numbers that have the same value. We often start by reorganizing or performing operations on the fractions to make them easier to manage. In this exercise, we simplified the numerator, \( \frac{1}{p} + \frac{1}{q} \), by finding a common denominator, making it easier to add the fractions together.
Once you've simplified your fraction as much as possible, check to see if you have any common factors in the numerator and denominator that cancel each other out. In our case, after simplifying, the numerator and denominator did not have common factors other than the initially divided \( p \), resulting in \( \frac{q+p}{q} \). This is the simplest form we can achieve in this problem.

Common Denominator

The concept of a common denominator is crucial when dealing with fractions. Think of the denominator as the "bottom" of the fraction, which tells us into how many parts the whole is divided. When adding or subtracting fractions, they must have the same number of parts. This is achieved through the use of common denominators.
In our case, the fraction's numerators \( \frac{1}{p} \) and \( \frac{1}{q} \) have different denominators: \( p \) and \( q \). To make these fractions comparable and easier to add, we find the common denominator, which here is \( pq \).
By rewriting \( \frac{1}{p} \) as \( \frac{q}{pq} \) and \( \frac{1}{q} \) as \( \frac{p}{pq} \), we transformed the fractions to have the same bottom number, \( pq \). This makes the fractions easier to add and paves the way for further simplification. This step is crucial because, without the same denominator, combining fractions is not possible.

Reciprocal of a Fraction

Understanding the concept of a reciprocal is fundamental to simplifying complex fractions. The reciprocal of a fraction is obtained by swapping its numerator and denominator. In essence, if you have the fraction \( \frac{a}{b} \), its reciprocal is \( \frac{b}{a} \).
When dealing with complex fractions, particularly in division, the reciprocal plays a key role. After simplifying the numerator and setting the fraction into a form where one fraction is over another, like \( \frac{\frac{q+p}{pq}}{\frac{1}{p}} \), you can apply the concept of the reciprocal. To divide by a fraction, you multiply by the reciprocal. So, \( \frac{\frac{q+p}{pq}}{\frac{1}{p}} \) becomes \( \frac{q+p}{pq} \times p \). This removes the division, allowing us to simplify further by cancelling out the common \( p \) in the numerator and denominator. The result is \( \frac{q+p}{q} \), effectively simplifying the complex fraction to its simplest form.