Problem 11 - Algebra A Combined Function

Question

Simplify each complex fraction. $$ \frac{\frac{1}{3}}{\frac{1}{2}-\frac{1}{4}} $$

Step-by-Step Solution

Answer

The simplified form is \(\frac{4}{3}\).

Step 1: Simplify the Denominator

First, we need to simplify the expression in the denominator, which is \(\frac{1}{2} - \frac{1}{4}\). To do this, find a common denominator. The common denominator of 2 and 4 is 4. Rewrite \(\frac{1}{2}\) as \(\frac{2}{4}\):\[\frac{1}{2} = \frac{2}{4}\]Now, subtract:\[\frac{2}{4} - \frac{1}{4} = \frac{1}{4}\]

Step 2: Rewrite the Complex Fraction

Now that we have simplified the denominator to \(\frac{1}{4}\), rewrite the entire complex fraction:\[\frac{\frac{1}{3}}{\frac{1}{4}}\]This fraction can be expressed as the multiplication of the numerator by the reciprocal of the denominator:\[\frac{1}{3} \times \frac{4}{1}\]

Step 3: Perform the Multiplication

Multiply the numerators and the denominators:\[\frac{1 \times 4}{3 \times 1} = \frac{4}{3}\]

Step 4: Simplified Result

The final simplified form of the complex fraction is \(\frac{4}{3}\).

Key Concepts

Simplifying Fractions

Simplifying fractions is a basic and important skill in mathematics that helps in uncovering the clearest possible expression of a fraction. A fraction, generally in the form \( \frac{a}{b} \), can often be simplified when the numerator (a) and denominator (b) have common factors. By dividing both by their greatest common divisor, you get the simplest form of the fraction.
Consider a fraction \( \frac{2}{4} \). Both 2 and 4 have a common factor of 2. So, dividing both the numerator and the denominator by 2 results in \( \frac{1}{2} \). This is the simplified form of \( \frac{2}{4} \).
When you simplify fractions, it makes calculations easier and results more comprehensible. In the exercise, the denominator \( \frac{2}{4} - \frac{1}{4} \) simplifies to \( \frac{1}{4} \), leading us to a more manageable expression: \( \frac{\frac{1}{3}}{\frac{1}{4}} \).

Common Denominator

Finding a common denominator is essential when you need to add or subtract fractions with different denominators. The least common denominator (LCD) is the smallest number that is a common multiple of the denominators involved. This step ensures that each fraction part refers to segments of equal size.
Let's examine the fractions \( \frac{1}{2} \) and \( \frac{1}{4} \). The denominators here are 2 and 4. The smallest number that is a multiple of both 2 and 4 is 4. Hence, 4 is their common denominator.
To adjust the fraction \( \frac{1}{2} \) to have this common denominator, it is rewritten as \( \frac{2}{4} \). Now, subtracting \( \frac{1}{4} \) from \( \frac{2}{4} \) is straightforward, giving you \( \frac{1}{4} \). This simplification is critical in the exercise as it allows for easier handling of the compound fraction.

Reciprocals

Reciprocals play a crucial role in simplifying complex fractions, especially during division. The reciprocal of a fraction \( \frac{a}{b} \) is simply the fraction flipped, turning it into \( \frac{b}{a} \). This transformation is key when dividing by a fraction.
In the exercise, to simplify \( \frac{\frac{1}{3}}{\frac{1}{4}} \), we use the reciprocal of \( \frac{1}{4} \), which is \( \frac{4}{1} \). By multiplying \( \frac{1}{3} \) with \( \frac{4}{1} \), you essentially convert the division into a multiplication, simplifying the entire expression.
Understanding reciprocals is especially useful because it allows one to turn division problems into simpler multiplication tasks, making calculations more straightforward and intuitive.

Fraction Multiplication

Multiplying fractions is a simpler process compared to adding or subtracting them. When multiplying two fractions, all you need to do is multiply the numerators together and the denominators together.
Given two fractions \( \frac{a}{b} \) and \( \frac{c}{d} \), their product is \( \frac{a \times c}{b \times d} \). This straightforward method makes multiplication of fractions easier to manage.
In the exercise, after rewriting \( \frac{\frac{1}{3}}{\frac{1}{4}} \) as \( \frac{1}{3} \times \frac{4}{1} \), you multiply across to get \( \frac{1 \times 4}{3 \times 1} \), resulting in \( \frac{4}{3} \).
This multiplication approach helps achieve the final simplified form of the fraction conveniently and highlights the effectiveness of turning division tasks into multiplication.