Problem 5 - College Algebra

Question

If Bill was traveling \(v \mathrm{mi} / \mathrm{h}\), how would you represent Daemon's speed if he was traveling \(10 \mathrm{mi} / \mathrm{h}\) faster?

Step-by-Step Solution

Answer

Daemon's speed is \(v + 10\) mi/h.

Step 1: Understanding the Problem

We need to find a way to express Daemon's speed, given that Daemon is traveling faster than Bill. Bill's speed is represented by the variable \(v\), which is in miles per hour (mi/h). Daemon is traveling 10 miles per hour faster than Bill.

Step 2: Setting Up the Expression

Since Daemon is traveling 10 mi/h faster than Bill, we start with Bill's speed \(v\) and add 10 to it. Therefore, Daemon's speed can be represented by the expression \(v + 10\).

Step 3: Final Representation

We have established that Daemon's speed is a result of adding 10 to Bill's speed. Hence, Daemon's speed is represented as \(v + 10\) mi/h.

Key Concepts

Problem Solving

Problem solving in algebra involves understanding a situation and using mathematical techniques to find a solution. It is a systematic process where:

• We first identify the information given in the problem.
• Next, we determine what is being asked or what needs to be found out.
• We then decide on the best approach and use relevant mathematical operations.

In this exercise, the problem was to express one speed based on the difference from another. We start by pinpointing the given speeds: Bill's speed and Daemon's need to be defined. Understanding that Daemon's speed is dependent on Bill's helps us decide to use addition.
Problem solving requires clarity and a sequential approach to lead step by step to the required solution.

Variables in Algebra

In algebra, variables are symbols or letters used to represent unknown values. They serve as placeholders for numbers we do not yet know. These variables are essential for building mathematical expressions and for simplifying complex problems.
In our exercise, the variable "\(v\)" is used to represent Bill's speed. It can take any value depending on the specific situation. By using a variable, we can easily manipulate and transform the expressions to reflect changes, such as in speed.
Understanding how variables work is crucial, as it allows flexibility and abstraction in mathematical analysis, making it easier to handle a wide range of situations without specifying exact numbers right away.

Mathematical Expressions

A mathematical expression is a combination of numbers, operators, and variables grouped together to show a value. They can represent real-world situations in a concise and simplified manner. In algebra, expressions are key to forming equations and solving problems.

• Expressions can include operations like addition, subtraction, multiplication, or division.
• They can also incorporate variables to denote quantities.

In the exercise, the expression "\(v + 10\)" was formed. This represents Daemon’s speed as being 10 units faster than Bill's. Such expressions allow us to model situations and perform calculations intuitively.
Mastery of mathematical expressions empowers students to describe and solve problems efficiently, enabling them to tackle more complicated subjects with ease.