Problem 43 - Algebra 1: Concepts and Skills

Question

In Exercises 43–45, use the following information. You are riding your bike to a pond that is 8 miles away. You have a choice to ride in the woods, on the road, or both. In the woods, you can ride at a speed of 10 mi/h. On the road, you can ride at a speed of 20 mi/h. Write an expression for your total time.

Step-by-Step Solution

Answer

The total time expression is \( t = d_w / 10 + d_r / 20 \), for \( d_w \) the distance ridden in the woods and \( d_r \) on the road. The sum of both distances equals 8 miles.

Step 1: Define the Variables

Let's define \( d_w \) as the distance ridden in the woods and \( d_r \) as the distance ridden on the road. Both distances are parts of the total 8 miles, so \( d_w + d_r = 8 \).

Step 2: Write the Individual Time Expressions

Using the formula Time = Distance / Speed, the time (\( t \)) you will need to ride in the woods is \( t_w = d_w / 10 \) and the time needed to ride on the road is \( t_r = d_r / 20 \).

Step 3: Write the Total Time Expression

The total time, \( t \), will be the sum of the individual times: \( t = t_w + t_r = d_w / 10 + d_r / 20 \).

Key Concepts

Understanding Algebraic Expressions

Algebraic expressions are the cornerstone of algebra, providing a way to represent mathematical situations using variables and numbers. When dealing with a problem like calculating the total time for a bike ride, we use algebraic expressions to succinctly describe the relationships among distance, speed, and time.

An algebraic expression is made up of terms, which are the individual parts of the expression separated by addition or subtraction signs. Each term can include numbers (coefficients), variables (which represent unknown quantities), and exponents. It's like a recipe; each ingredient (term) contributes to the overall outcome - in this case, the total time of the bike ride. By translating the problem into algebra, we create a powerful tool that allows us to solve for unknowns and make predictions.

Solving Speed and Distance Problems

Speed and distance problems are a practical application of algebra, particularly when variables are involved. The key to solving these problems is understanding the fundamental relationship given by the formula: Time = Distance / Speed.

If you're biking to a pond and want to determine how long the trip will take, knowing the speed at which you travel through different terrains (woods versus road) and the distance you need to cover becomes vital. Once these pieces of the puzzle are identified, you can use algebraic expressions to determine the total time spent traveling. By writing separate expressions for each leg of the journey and then combining them, you get a clear picture of the total time. It's like connecting dots on a map: once all the segments are connected, you can see the complete route and its duration.

Defining Variables in Algebra

Defining variables is essential when translating real-world situations into algebraic language. A variable is a symbol, typically a letter, that stands in for an unknown value that we are either trying to determine or that can vary in the problem context.

In our biking scenario, we define variables for the distances on different terrains: \( d_w \) for the distance in the woods and \( d_r \) for the distance on the road. These variables help us break down the problem into smaller parts, making it more manageable. By setting up a relationship between these variables, such as \( d_w + d_r = 8 \), we create an equation that captures the total distance in a way that lets us solve for each part separately. Think of variables as placeholders; they hold a spot for the actual numbers and allow us to focus on the structure of the problem, which remains consistent regardless of specific values.