Problem 14 - Precalculus

Question

Evaluate each expression. $$\frac{|27-5|}{|5-27|}$$

Step-by-Step Solution

Answer

The expression evaluates to 1.

Step 1: Calculate the absolute value in the numerator

First, evaluate the expression inside the absolute value in the numerator: \(27 - 5\). This simplifies to \(22\). So, the absolute value \(|27 - 5|\) is simply \(22\), since \(22\) is already positive.

Step 2: Calculate the absolute value in the denominator

Next, evaluate the expression inside the absolute value in the denominator: \(5 - 27\). This simplifies to \(-22\). The absolute value of \(-22\) is \(22\), because the absolute value function converts negative numbers to positive numbers.

Step 3: Simplify the fraction

Now you have the fraction \(\frac{22}{22}\). Divide \(22\) by \(22\) to simplify the fraction to \(1\).

Key Concepts

Understanding the Numerator

When working with fractions, the top part is called the numerator. It represents how many parts of a whole are being considered. In our exercise, the expression \( |27-5| \) represents the numerator. Here, the operation inside the absolute value, \( 27 - 5 \), simplifies to \( 22 \). Because \( 22 \) is already positive, its absolute value stays as \( 22 \).
Understanding how to handle absolute values within the numerator is crucial. Absolute values always convert a negative result into a positive one, but a positive number remains unchanged.
So always evaluate what's inside the absolute value first, then apply the absolute value, ensuring your numerator is positive.

Exploring the Denominator

The denominator is the bottom part of a fraction, indicating into how many equal parts the whole is divided. Let’s apply this to our exercise: we evaluate \( |5-27| \) as the denominator. First, calculate \( 5 - 27 \). This operation gives us \( -22 \).
When you apply the absolute value, \( |-22| \), it changes to \( 22 \). Absolute values turn negative numbers positive.
The denominator is always a positive result when absolute values are involved, as they ensure no negative division occurs. This makes calculations straightforward and consistent when dividing quantities by changing negative values to positive.

How to Simplify Fractions

Simplifying fractions means making them as simple and small as possible, reducing them to their most basic form. In our exercise, the fraction becomes \( \frac{22}{22} \) once both the numerator and denominator are computed using absolute values.

• To simplify, divide the numerator \( 22 \) by the denominator \( 22 \).
• When both the numerator and denominator are the same, as in \( \frac{22}{22} \), the fraction simplifies to \( 1 \).

Simplifying ensures any fraction is in its easiest form to understand and utilize.
Remember, if the numerator and denominator are identical, their division will always yield \( 1 \). This process not only simplifies calculations but also makes it easier to compare fractions.