Problem 28 - Elementary and Intermediate Algebra

Question

Substitute to find the value of each expression. Work Time. Alan takes twice as long to do a job as Connor does. Suppose \(t\) represents the time it takes Connor to do the job. Then \(2 t\) represents the time it takes Alan. How long does it take Alan if Connor takes (a) \(30 \sec ?\) (b) \(35 \min ?\) (c) \(2 \frac{1}{2}\) hr?

Step-by-Step Solution

Answer

Alan takes 60 sec, 70 min, and 5 hr respectively.

Step 1: Understand the Variables

Let the time it takes Connor to do the job be represented by the variable \( t\). Therefore, the time it takes Alan to complete the same job is represented by \( 2t \).

Step 2: Substitute for (a) 30 seconds

Substitute \( t = 30 \) seconds into Alan's time equation: \[ 2(t) = 2(30) = 60 \] seconds. So, it takes Alan 60 seconds.

Step 3: Substitute for (b) 35 minutes

Substitute \( t = 35 \) minutes into Alan's time equation: \[ 2(t) = 2(35) = 70 \] minutes. So, it takes Alan 70 minutes.

Step 4: Substitute for (c) 2.5 hours

Substitute \( t = 2.5 \) hours into Alan's time equation: \[ 2(t) = 2(2.5) = 5 \] hours. So, it takes Alan 5 hours.

Key Concepts

Variable Substitution

Variable substitution is a fundamental concept in algebra where a variable is replaced with a specific value. In the given problem, we have two variables representing time: one for Connor (\t\text{t}) and one for Alan (\t\text{2t}). By substituting specific values into these variables, we can easily determine the total time it takes each to complete a task. This process involves simply replacing the variable with a given numerical value and performing the necessary arithmetic operations.For example:

• If \(\t\text{t} = 30\) seconds, then \(\t\text{2t} = 60\) seconds after substitution.
• If \(\t\text{t} = 35\) minutes, Alan’s time becomes \(\t\text{2 × 35 = 70}\) minutes.
• If \(\t\text{t} = 2.5\) hours, the substitution gives Alan’s time as \(\t\text{2 × 2.5 = 5}\) hours.

By clearly understanding how to substitute values into variables and perform simple multiplications, you can solve a variety of algebraic problems with ease.

Time Calculations

Time calculations are essential in solving real-life problems as they allow us to convert and manipulate units of time effectively. In the problem, knowing how to handle different time units is crucial. Each time unit, like seconds, minutes, and hours, needs to be handled specifically and consistently.Consider these steps when dealing with time calculations:

• Ensure the units are always the same when performing calculations. If the problem involves multiple units, convert them to a common unit before proceeding.
• If \(\t\text{Connor}\) takes 30 seconds to complete a task, Alan would take \(\t\text{60 seconds}\).
• For 35 minutes, Alan’s time calculates to \(\t\text{70 minutes}\).
• With 2.5 hours, Alan’s total time is handled as \(\t\text{5 hours}\) after substituting and performing the arithmetic.

Understanding and practicing these conversions and calculations make it easier to solve similar algebraic problems involving time.

Simple Algebra

Simple algebra involves basic operations such as addition, subtraction, multiplication, and division to solve equations. In the problem, the relationship between Alan's and Connor's time to finish a job is a straightforward multiplication relationship: \(\t\text{Alan’s time}\) is twice \(\t\text{Connor's time}\), or \(\text{2t}\).To solve these types of problems:

• Identify the variables and their relationships (e.g., \(\t\text{2t}\)).
• Substitute the known values.—Examples: \(\t\text{30 seconds}\), \(\t\text{35 minutes}\), \(\t\text{2.5 hours}\).
• Perform the calculations—multiply the known value by 2.

With each step, the same principle applies, and by handling each step systematically, you simplify algebraic expressions effectively.