Problem 28 - Prealgebra

Question

Translate each phrase into an algebraic expression. two seconds slower than Joseph's time

Step-by-Step Solution

Answer

\( t - 2 \)

Step 1: Identify the Variable

First, identify the variable for Joseph's time in seconds. Let's denote Joseph's time as \( t \).

Step 2: Translate the Phrase

The phrase 'two seconds slower than Joseph's time' indicates that something is 2 seconds less than \( t \).

Step 3: Formulate the Expression

To express 'two seconds slower than Joseph's time' algebraically, we subtract 2 from \( t \). This can be written as \( t - 2 \).

Key Concepts

Identifying Variables

In algebra, identifying variables is a crucial first step in solving any problem. A variable is a symbol, often a letter, that represents a number we don't yet know. In our exercise, we're dealing with Joseph's time, and this is an unknown quantity that we need to express algebraically. To do this, we assign a variable to the unknown.

Here, Joseph's time is represented by the variable \( t \). Think of \( t \) as a container that holds the value of Joseph's time in seconds. This makes it easier to manipulate and translate the given phrase into an algebraic expression. Identifying variables simplifies complex scenarios and helps us construct meaningful expressions and equations. Always remember, choosing a succinct and relevant variable is the first step in translating any real-world problem into algebra.

Subtraction in Algebra

Subtraction is a fundamental operation in algebra that involves taking away a certain value from another. It is essential to understand subtraction both in numerical terms and when working with variables. When we translate phrases into algebraic expressions, subtraction often signifies a decrease or reduction from a given quantity.

In our example, the phrase 'two seconds slower than Joseph’s time' suggests a reduction from Joseph's time. This is mathematically represented by subtracting 2 from the variable \( t \) which symbolizes Joseph's time. Hence, the expression becomes \( t - 2 \).

Always keep in mind the order: in expressions like 'less than' or 'slower than,' subtraction typically follows from the given quantity, which in this case is \( t \). This understanding of subtraction will support the creation of correct and clear algebraic expressions.

Algebraic Translation Steps

Translating phrases into algebraic expressions involves a few deliberate steps. Let's break them down step-by-step for clarity and comprehension.

**1. Understand the phrase:** Start by reading the problem carefully. Identify what needs to be represented in algebraic form.

**2. Identify the variables:** As we discussed earlier, determine the unknown quantities and assign them variables. For Joseph’s time, we used \( t \).

**3. Recognize key phrases:** Look for words like 'more than', 'less than', 'increased by', or 'decreased by', as they indicate mathematical operations. In our case, 'slower than' means subtraction.

**4. Perform the operation:** Apply the recognized mathematical operation to the variable, forming the complete expression. For 'two seconds slower than Joseph's time', we subtract 2 from \( t \), resulting in \( t - 2 \).

Following these structured steps ensures accurate and effective translation of words into algebra, simplifying the process of solving mathematical problems.