Problem 57 - Algebra and Trigonometry

Question

Distance, Speed, and Time Wendy took a trip from Davenport to Omaha, a distance of 300 \(\mathrm{mi}\) . She traveled part of the way by bus, which arrived at the train station just in time for Wendy to complete her journey by train. The bus averaged \(40 \mathrm{mi} / \mathrm{h},\) and the train averaged 60 \(\mathrm{mi} / \mathrm{h}\) . The entire trip took \(5 \frac{1}{2} \mathrm{h} .\) How long did Wendy spend on the train?

Step-by-Step Solution

Answer

Wendy spent 4 hours on the train.

Step 1: Understand the Problem

Wendy travels a total distance of 300 miles using a bus and a train. The total time for the trip is 5.5 hours. We need to find out how much time she spent on the train, given the bus speed is 40 mph and the train speed is 60 mph.

Step 2: Define Variables

Let \( x \) represent the time spent on the bus, which implies \( 5.5 - x \) is the time spent on the train. We will use these variables to form equations based on the distances covered.

Step 3: Calculate Distance by Each Mode

From the formula \( \text{Distance} = \text{Speed} \times \text{Time} \), the distance traveled by bus is \( 40x \) and the distance by train is \( 60(5.5 - x) \).

Step 4: Set Up the Equation for Total Distance

The total distance traveled is given as 300 miles. Therefore, the equation is: \[ 40x + 60(5.5 - x) = 300 \]

Step 5: Solve the Equation

Simplify the equation: \[ 40x + 330 - 60x = 300 \] \[ -20x + 330 = 300 \] \[ -20x = -30 \] \[ x = 1.5 \] This means Wendy spent 1.5 hours on the bus.

Step 6: Calculate Time Spent on Train

Since total time is 5.5 hours, time on train is \( 5.5 - 1.5 = 4 \text{ hours} \).

Step 7: Verify the Solution

Check the total distances: - Distance by bus: \( 40 \times 1.5 = 60 \) miles.- Distance by train: \( 60 \times 4 = 240 \) miles.- Total distance: \( 60 + 240 = 300 \) miles, which matches the problem statement.

Key Concepts

Algebraic Equations

Algebraic equations are essential tools in solving real-world problems, especially when multiple variables are in play. In Wendy's trip scenario, we set up an algebraic equation by expressing different segments of the journey using variables. This process allows us to create a relationship between the time spent on each mode of transport and their respective speeds. By letting \( x \) represent the time on the bus, we could then use \( 5.5 - x \) to express the time spent on the train. By applying these variables in the context of the given speeds, we could formulate an algebraic equation to help solve the problem effectively.

Distance Formula

The distance formula, \( \text{Distance} = \text{Speed} \times \text{Time} \), is the backbone for solving problems involving movement. It is important to identify the correct elements for speed and time. In Wendy's journey, the distance formula was applied twice, once for each segment of her travel. For the bus segment, we calculated the distance as \( 40x \) since the bus speed was 40 mph and the time was \( x \) hours. Similarly, for the train, the distance was \( 60(5.5 - x) \) since the train traveled for \( 5.5 - x \) hours at 60 mph. This format makes it easy to see how the total distance was composed of two interconnected parts.

Word Problems

Word problems translate real-life scenarios into mathematics, reinforcing the need to comprehend and structure them into equations. When dealing with word problems like Wendy's trip, it’s essential to identify what is being asked and what information is provided. The problem provided the total distance, total time, and speeds of each mode of transport. From this, we needed to determine how much time was spent traveling by each mode. Breaking down the scenario, defining variables, and recognizing what the problem requires are steps that bring word problems from a narrative context to a mathematical solution.

Rate and Time

Understanding the concept of rate and time is crucial, especially in problems involving travel. Rate refers to speed, seen as distance per unit of time. Knowing the rates, 40 mph for the bus and 60 mph for the train, allowed us to understand how fast each mode was and set up meaningful equations. Time, as a variable, is often the sought-after quantity in these problems. In this exercise, the total time was given, and the subtraction of time spent on the bus provided the time on the train. Recognizing the interplay between rate and time allows the effective use of equations to solve for unknowns.