Problem 5 - Algebra A Combined Function

Question

Simplify each complex fraction. $$ \frac{\frac{1+x}{6}}{\frac{1+x}{3}} $$

Step-by-Step Solution

Answer

The simplified form of the complex fraction is \( \frac{1}{2} \).

Step 1: Identify the Complex Fraction

The complex fraction is given as \[ \frac{\frac{1+x}{6}}{\frac{1+x}{3}}. \]This fraction has a numerator \( \frac{1+x}{6} \) and a denominator \( \frac{1+x}{3} \).

Step 2: Divide the Fractions

To simplify the complex fraction, divide the numerator by the denominator. Thus, the operation becomes \[ \frac{1+x}{6} \div \frac{1+x}{3}. \]

Step 3: Convert Division to Multiplication

Convert the division into a multiplication by multiplying by the reciprocal of the denominator. So, the expression becomes\[ \frac{1+x}{6} \times \frac{3}{1+x}. \]

Step 4: Cancel Common Terms

Notice that the \(1+x\) terms appear in both the numerator and the denominator. Cancel these common terms to get\[ \frac{3}{6}. \]

Step 5: Simplify the Resulting Fraction

Simplify \( \frac{3}{6} \) by dividing both the numerator and the denominator by their greatest common divisor, which is 3. Hence,\[ \frac{3 \div 3}{6 \div 3} = \frac{1}{2}. \]

Key Concepts

Understanding Simplifying Fractions

Simplifying fractions is a process that makes a fraction easier to work with by expressing it in its simplest form. The goal is to find the simplest equivalent fraction that has the smallest possible whole numbers in both the numerator and the denominator. To simplify a fraction, you need to divide both the top (numerator) and the bottom (denominator) of the fraction by the greatest common divisor (GCD). Here's how you do it:

• Identify the numerator and denominator of your fraction.
• Find the GCD of these two numbers.
• Divide both the numerator and the denominator by the GCD to get the simplified fraction.

Simplifying doesn't change the value of the fraction, it just reduces the numbers involved.

Understanding the Reciprocal

The concept of a reciprocal is crucial in simplifying complex fractions. A reciprocal is simply what you multiply a number by to get 1. In other words, if you have a fraction, say \(\frac{a}{b}\), its reciprocal is \(\frac{b}{a}\).When dealing with division of fractions, we often use the reciprocal. Here's why:

• Instead of dividing by a fraction, you multiply by its reciprocal.
• For example, dividing by \(\frac{1}{2}\) is the same as multiplying by 2, which is its reciprocal.
• This operation turns division into a more straightforward multiplication action.

Remember, the reciprocal of a fraction is essentially flipping the numerator and the denominator.

Canceling Common Terms

Once you're in the realm of multiplication after replacing division with a reciprocal, canceling terms can make calculations much simpler. This involves striking out (dividing out) terms that appear both in the numerator and the denominator. Here's how canceling terms works:

• Identify terms that are exactly the same in both the numerator and the denominator.
• Strike through these terms, which effectively simplifies the expression.
• This works because anything divided by itself equals 1.

Canceling terms allows you to bypass more complicated calculations by reducing expressions to their simplest form quickly.

Fraction Division Simplified

Fraction division might seem tricky, but with the right approach, it's straightforward. Rather than dividing directly, we rely on the reciprocity of the divisor. The steps for dividing fractions are:

• Take the reciprocal of the divisor (the fraction you are dividing by).
• Convert the division problem into a multiplication problem using this reciprocal.
• Perform the multiplication.

Working through this process turns fraction division into an operation you're more familiar with: multiplication. It’s all about flipping and multiplying to make complex calculations much simpler.