Problem 83 - Algebra A Combined Function

Question

In ice hockey, penalty killing percentage is a statistic calculated as \(1-\frac{G}{P},\) where \(G=\) opponent's power play goals and \(P=\) opponent's power play opportunities. Simplify this expression.

Step-by-Step Solution

Answer

The simplified expression for the penalty killing percentage is \(\frac{P - G}{P}\).

Step 1: Understanding the Given Expression

The expression given is the penalty killing percentage, which is calculated as \(1 - \frac{G}{P}\). Here, \(G\) represents the number of goals scored by the opponents during a power play, and \(P\) is the total number of times the opponents have a power play opportunity.

Step 2: Combine Terms Under a Common Denominator

To simplify the expression, we need to combine the terms under a single denominator. Rewrite \(1\) as \(\frac{P}{P}\) to have a common denominator with \(\frac{G}{P}\): \[1 - \frac{G}{P} = \frac{P}{P} - \frac{G}{P}\]

Step 3: Subtract Fractions

With a common denominator, subtract the numerators:\[\frac{P}{P} - \frac{G}{P} = \frac{P - G}{P}\]

Step 4: Simplified Expression

The simplified expression for the penalty killing percentage is \(\frac{P - G}{P}\). This represents the fraction of power play opportunities in which the opponents did not score.

Key Concepts

Fractions

A fraction represents a part of a whole. It is composed of two numbers separated by a horizontal line.

• The top number is called the numerator, indicating how many parts are taken or considered.
• The bottom number is the denominator, showing the total number of equal parts into which the whole is divided.

For example, in the fraction \( \frac{1}{2} \), the numerator is 1, and the denominator is 2, expressing that one part out of two equal parts is considered. Understanding fractions is crucial in simplifying algebraic expressions, as it involves operations such as addition, subtraction, and finding a common denominator.

Numerators and Denominators

In any given fraction, identifying the numerator and denominator is key to performing operations. The **numerator**, located above the line, tells you how many parts you have, while the **denominator** determines the size of each part by indicating the total number of parts into which the whole is divided. In our example, simplifying the algebraic expression involves fractions like \( \frac{G}{P} \). To simplify such expressions:

• Ensure the fractions have a common denominator, which allows us to easily perform arithmetic operations like addition or subtraction.
• When the denominators are the same, arithmetic operations only impact numerators.
• In the case of subtraction, subtract the numerator of the second fraction from the numerator of the first fraction.

Understanding how numerators and denominators interact in arithmetic operations is vital, particularly when simplifying expressions such as the penalty killing percentage in ice hockey.

Arithmetic Operations

Arithmetic operations are the foundation of working with fractions and include addition, subtraction, multiplication, and division. These basic operations can be applied to both numbers and algebraic expressions.In the context of the problem, we focused on **subtraction** of fractions. Here’s how it was done:

• To subtract two fractions, align them to have a common denominator.
• Convert whole numbers into fractions if necessary by expressing them over 1 or as fractions with a common denominator.
• Subtract the numerators and retain the common denominator.

For instance, transforming \(1\) into \( \frac{P}{P} \) allowed the expression \(1 - \frac{G}{P}\) to transform to \(\frac{P}{P} - \frac{G}{P}\). This facilitated simple subtraction of the numerators, leading to the final simplified expression \(\frac{P-G}{P}\). Arithmetic operations thus enable us to manipulate and simplify expressions effectively by following systematic steps.