Problem 19 - Intermediate Algebra

Question

Translate each phrase to an algebraic expression. Answers may vary depending on the variables chosen. three-fourths of the pressure

Step-by-Step Solution

Answer

The algebraic expression is \( \frac{3}{4}P \).

Step 1: Identify the Key Phrase

The key phrase in the problem is "three-fourths of the pressure." This indicates a fraction and a relationship with a variable representing the pressure.

Step 2: Determine the Algebraic Representation

Choose a variable to represent the pressure. Commonly, the variable "P" is used for pressure. So, let \( P \) be the variable that represents the pressure.

Step 3: Translate Into an Algebraic Expression

The fraction "three-fourths" translates to \( \frac{3}{4} \) and "of" typically means multiplication in algebraic translations. Therefore, the algebraic expression for "three-fourths of the pressure" is \( \frac{3}{4}P \).

Key Concepts

Understanding Fractions

Fractions are a way to represent parts of a whole. They are expressed as one number (the numerator) over another number (the denominator). For instance, in the fraction \( \frac{3}{4} \), the numerator is 3, and the denominator is 4. This fraction means that we have three out of four equal parts.Fractions can also signify division. The numerator is divided by the denominator. That's why \( \frac{3}{4} \) can be understood as 3 divided by 4. This is key when translating phrases into algebraic expressions because fractions often indicate portions of a quantity.Remember:

• The denominator tells us into how many equal parts the whole is divided.
• The numerator tells us how many of those parts we are considering.

When you see phrases like "three-fourths of," you recognize that a fraction is being defined, which involves both quantities and operations like multiplication.

Variable Representation

In algebra, variables are symbols used to represent numbers or values that can change. Typically, letters of the alphabet, such as \( x \), \( y \), or \( P \), are used as variables. Choosing a variable:

• Select a letter that makes sense or is commonly associated with what it represents. For example, \( P \) for pressure.
• Once a variable is chosen, it can represent the value throughout the entire exercise or equation.

Variables help simplify expressions and equations, making it easier to manipulate them. In our context, choosing \( P \) to represent pressure allows us to conveniently express the entire problem statement algebraically.

Translating Phrases into Expressions

The skill of translating phrases into algebraic expressions allows you to convert real-world problems into a form that can be solved mathematically. It involves converting words into symbols and numbers. Understanding keywords and phrases is critical here.Key aspects to consider:

• "Three-fourths of": Here, "three-fourths" indicates the fraction \( \frac{3}{4} \). The word "of" hints at multiplication.
• If you see "the pressure," you've identified that a variable needs to stand for pressure, which we've represented as \( P \).

The phrase "three-fourths of the pressure" translates to multiplying \( \frac{3}{4} \) by \( P \), resulting in the expression \( \frac{3}{4}P \). This expression can then be used in equations to perform further calculations or analyses.