Problem 50 - Elementary and Intermediate Algebra

Question

Simplify each complex fraction. $$ \frac{\frac{m}{n}+1}{1-\frac{m}{n}} $$

Step-by-Step Solution

Answer

\( \frac{m+n}{n-m} \)

Step 1: Convert to a Single Fraction: Numerator

The numerator of the complex fraction is \( \frac{m}{n} + 1 \). To add these two terms, convert '1' to a fraction with the same denominator \(n\). Thus, \( 1 = \frac{n}{n} \). This gives us: \[ \frac{m}{n} + \frac{n}{n} = \frac{m+n}{n} \].

Step 2: Convert to a Single Fraction: Denominator

The denominator of the complex fraction is \( 1 - \frac{m}{n} \). Convert '1' to a fraction with the same denominator \(n\), so \( 1 = \frac{n}{n} \). This gives us: \[ \frac{n}{n} - \frac{m}{n} = \frac{n-m}{n} \].

Step 3: Setup the New Fraction

The complex fraction now looks like: \[ \frac{\frac{m+n}{n}}{\frac{n-m}{n}} \].

Step 4: Simplify the Complex Fraction

When you divide by a fraction, it is equivalent to multiplying by its reciprocal. Therefore, we can write: \[ \frac{m+n}{n} \times \frac{n}{n-m} \].

Step 5: Cancel the Denominator \(n\)

Multiply the numerators together and the denominators together, then cancel similar terms before simplifying: \[ \frac{m+n}{n} \times \frac{n}{n-m} = \frac{(m+n) \cdot n}{n \cdot (n-m)} = \frac{m+n}{n-m} \].

Key Concepts

Algebra Simplification

At the heart of simplifying any mathematical expression is the concept of algebra simplification. To simplify an expression means to rewrite it in a way that is easier to understand or use in calculations. Let's consider the complex fraction, which initially seems cumbersome due to its layered numerators and denominators.

The goal of simplification here is to untangle these layers and write them as simple fractions whenever possible. We begin by making sure that each term within the expression shares a common denominator, which makes it easier to combine or subtract them. For instance, converting the whole number to a fraction with the same denominator as other fractions allows for straightforward addition or subtraction.

By performing operations step-by-step and simplifying gradually, we ensure that we achieve the simplest form of the original complex expression.

Fractions Addition

To add fractions, they must share a common denominator. This rule applies even in complex fractions where portions of the expression itself are fractions. In the given exercise, the numerator of the complex fraction is \(\frac{m}{n} + 1 \).

Since '1' is not naturally expressed as a fraction over 'n,' we need to convert it into an equivalent fraction to proceed: - Convert '1' into \(\frac{n}{n}\) because,- This makes the addition \(\frac{m}{n} + \frac{n}{n} = \frac{m+n}{n} \).Simplifying fractions and combining terms are all par for the course in addition operations involving fractions. Ensuring common denominators provides a straightforward path to combining terms and simplifying further.

Fraction Division

When dealing with complex fractions, division is inevitable. A complex fraction involves a fraction in the numerator divided by a fraction in the denominator.

For the expression \( \frac{\frac{m+n}{n}}{\frac{n-m}{n}} \), note that this is essentially a division operation between two simpler fractions. To transform this division into manageable multiplication, we use the principle of reciprocal multiplication. This step helps move away from division, simplifying the calculation by converting two layers of fractions into a single operation. It’s a must-know tactic when tackling fractions embedded within each other.

Reciprocal Multiplication

Reciprocal multiplication is a fundamental technique in simplifying complex fractions. When you divide by a fraction, you multiply by its reciprocal instead. This sounds complicated, but it's straightforward once you see it in action.

Take the complex fraction: \( \frac{\frac{m+n}{n}}{\frac{n-m}{n}} \). Here, instead of dividing, you multiply the numerator by the reciprocal of the denominator:

• The reciprocal of \(\frac{n-m}{n}\) is \(\frac{n}{n-m}\).
• Thus, \(\frac{m+n}{n} \times \frac{n}{n-m}\).

Through reciprocal multiplication, you transform the complex fraction into a form that’s easy to simplify: \(\frac{m+n}{n-m}\). Understanding and applying reciprocity makes streamlining complex math expressions both efficient and effective. This ensures fewer mistakes and a clearer path to solutions.