Problem 56 - College Algebra with Modeling and Visualization

Question

Simplify the expression. $$ \frac{x+1}{y-2} \div \frac{2 x+2}{y-2} \div \frac{x}{y} $$

Step-by-Step Solution

Answer

\(\frac{y}{2x}\)

Step 1: Rewrite Division as Multiplication

When dividing fractions, we convert the division operation to multiplication by taking the reciprocal of the divisor. So the expression \(\frac{x+1}{y-2} \div \frac{2x+2}{y-2} \div \frac{x}{y}\) becomes \(\frac{x+1}{y-2} \times \frac{y-2}{2x+2} \times \frac{y}{x}\).

Step 2: Cancel Common Factors

The fractions \(\frac{x+1}{y-2} \times \frac{y-2}{2x+2}\) allow us to cancel the \(y-2\) from the numerator of the second fraction and the denominator of the first fraction. This simplifies our expression to \((x+1) \times \frac{1}{2x+2}\). Additionally, remember that \(2x+2\) can be factored as \(2(x+1)\), allowing further simplification.

Step 3: Simplify Further

Now, consider \((x+1) \times \frac{1}{2(x+1)} \times \frac{y}{x}\). The \(x+1\) terms in the numerator of the first term and part of the denominator of the second term cancel each other, simplifying to \(\frac{1}{2} \times \frac{y}{x}\).

Step 4: Compute Final Simplified Expression

Finally, compute \(\frac{1}{2} \times \frac{y}{x}\), which yields \(\frac{y}{2x}\). This is the simplified expression.

Key Concepts

Fraction Division

To simplify expressions involving fraction division, it's essential to convert the division operation into multiplication. The idea rests on using the reciprocal of the divisor—a handy trick in mathematics! When you see a division symbol between two fractions like \(\frac{a}{b} \div \frac{c}{d}\), swap the division for multiplication and take the reciprocal of the second fraction. This transformation gives us \(\frac{a}{b} \times \frac{d}{c}\).
This approach changes the operation while maintaining the equivalence of the expression, thus making it easier to work with. Especially in algebraic fractions, this step lays the foundation for subsequent simplifications.

Multiplication of Fractions

Once you've converted division into multiplication using reciprocals, it's time to understand how multiplication of fractions works. It's straightforward: when multiplying two fractions, you multiply the numerators together and the denominators together. For instance, for two fractions \(\frac{a}{b}\) and \(\frac{c}{d}\), their product is \(\frac{a \times c}{b \times d}\).
In our specific problem, after transforming each division into multiplication, we handle the process step-by-step to simplify. This step prepares the fractions for cancelling any common factors in the next phase.

Canceling Common Factors

A crucial skill in simplifying fraction expressions is canceling common factors. This is where things start to get cleaner and neater! After multiplying the fractions, check whether there are any common factors in the numerator and the denominator.

• If you see the same term above and below the fraction line, you can cancel them. Like, if both a numerator and a denominator have \((x+1)\), you can remove each pair of matching terms.
• This is valid because any number divided by itself equals one. Mathematically, removing identical terms doesn't alter the value of the expression.

Finding and canceling these factors greatly simplifies the expression, allowing us to focus on the remaining parts of the problem.

Factoring in Algebra

Factoring in algebra involves expressing expressions as the product of their simplest algebraic components. This is an essential skill, especially when simplifying algebraic fractions.
Take expressions like \(2x+2\); they can be factored into \(2(x+1)\). Detecting such opportunities makes it easy to see common factors needing cancellation. By breaking down complex expressions into simpler factors, we can easily check for and cancel unnecessary terms, thus simplifying the fraction efficiently.
Factoring is tantamount to untangling the equation, revealing layers that might not be initially apparent, and greatly aiding in the understanding and simplification of the expression!