Problem 35 - Elementary Algebra

Question

For the following problems, perform the multiplications and divisions. $$ \frac{x-5}{x-1} \div \frac{x-5}{4} $$

Step-by-Step Solution

Answer

Answer: The simplified expression after performing the division is $\frac{4}{x-1}$.

Step 1: Rewrite Division as Multiplication

We can rewrite the division of two fractions as the multiplication by the reciprocal (flip) of the second fraction: $$ \frac{x-5}{x-1} \div \frac{x-5}{4} = \frac{x-5}{x-1} \cdot \frac{4}{x-5} $$

Step 2: Simplify

We can now simplify the expression by canceling common factors from both numerator and denominator. We can see that \((x-5)\) is a common factor, which we can cancel: $$ \frac{x-5}{x-1} \cdot \frac{4}{x-5} = \frac{\cancel{(x-5)}}{x-1} \cdot \frac{4}{\cancel{(x-5)}} = \frac{4}{x-1} $$

Step 3: Final Answer

The simplified expression after performing the division is: $$ \frac{4}{x-1} $$

Key Concepts

Fraction Division

When dividing fractions, a helpful trick is to turn the division process into multiplication. Consider it like flipping the second fraction and then proceeding as if you're multiplying two fractions. This is known as multiplying by the reciprocal.
Here's how it works: if you have \( \frac{a}{b} \div \frac{c}{d} \), you can convert this to \( \frac{a}{b} \cdot \frac{d}{c} \). Essentially, you're flipping the second fraction (\( \frac{c}{d} \)) so that the numerator and denominator swap places.

• This simplifies the division process
• Allows calculations to be manageable

In our exercise, we applied this method by changing \( \frac{x-5}{x-1} \div \frac{x-5}{4} \) into \( \frac{x-5}{x-1} \cdot \frac{4}{x-5} \). This small flip makes a massive difference in easing calculations.

Fraction Multiplication

After transforming the division into a multiplication, the next step is to deal with multiplying the fractions as usual. This involves some straightforward steps.

• Multiply the numerators of both fractions together
• Multiply the denominators of both fractions together

For example, if you have \( \frac{a}{b} \cdot \frac{c}{d} \), the result will be \( \frac{a \cdot c}{b \cdot d} \). It's simple arithmetic with algebraic expressions as placeholders. As you follow this method, be cautious with operations to ensure accuracy.
In our specific case, we transformed our divided fractions \( \frac{x-5}{x-1} \cdot \frac{4}{x-5} \) into one single fraction by multiplying numerators \((x-5) \cdot 4\) and denominators \((x-1) \cdot (x-5)\). However, before you multiply, it's often beneficial tolook for opportunities to simplify the expression.

Simplifying Fractions

Simplifying fractions in algebra means reducing them to their lowest terms, making them easier to work with. You achieve this by cancelling out any common factors that appear in both the numerator and denominator.
Here's a straightforward approach to follow:

• Identify any common factors in the numerator and denominator
• Cancel these common factors
• Re-write the fraction without those factors

In the exercise problem, before proceeding with multiplication and following reciprocal conversion, we noticed that \( (x-5) \) appears both in the numerator and denominator. Once identified, we were able to cancel these factors, leading to a much simpler fraction: \( \frac{4}{x-1} \).
By simplifying early, you save time and reduce errors in your calculations. Always keep an eye out for common factors, as they provide an easy doorway to simpler expressions.