Problem 47 - Algebra 1: Concepts and Skills

Question

Steamboats carried cotton and passengers up and down the Mississippi River in the mid- 1800 s. A steamboat could travel 8 miles per hour downstream from Natchez, Mississippi, to New Orleans, Louisiana, and only 3 miles per hour upstream from New Orleans to Natchez. It was about 265 miles each way. If it took a steamboat 55 more hours to go upstream than it did to go downstream, how long did it take to complete the roundtrip? Solve \(8 t=3(t+55),\) where \(t\) is the time (in hours) it takes the steamboat to travel downstream and \((t+55)\) is the time it takes to travel upstream.

Step-by-Step Solution

Answer

The total time for the roundtrip journey is 121 hours.

Step 1: Simplify the equation

We start with the given equation \(8t = 3(t + 55)\) which represents the equal distances travelled downstream and upstream. To simplify, distribute the '3' on the right-hand side of the equation resulting in \(8t = 3t + 165\)

Step 2: Solve for t

Subtract '3t' from both sides of the equation to get the time it takes for the steamboat to travel downstream. Thus, \(8t - 3t = 165\) simplifies to \(5t = 165\). Dividing both sides by 5 gives \(t = 33\) hours.

Step 3: Calculate roundtrip time

The total roundtrip time would be the time it takes to travel downstream plus the time it takes to travel upstream. But we know from the problem that the time it takes to travel upstream is 55 hours more than the time it takes to travel downstream. Thus, the total time for the roundtrip journey is \(33 hours (downstream) + 33 hours (downstream time) + 55 hours (additional upstream time) = 121\) hours

Key Concepts

Solving Linear Equations

When approaching linear equations, the goal is to find the value of the unknown variable that makes the equation true. These equations often look simple, like the straight-forward equation from our steamboat problem: 8t = 3(t + 55). The solution starts by expanding and simplifying, which may involve operations such as distribution, combining like terms, and isolating the variable.

In this case, we start by distributing the '3' across the expression (t + 55), leading to 3t + 165. From there, we can subtract '3t' from both sides to isolate the 't' term: 8t - 3t = 165, which simplifies to 5t = 165. Finally, divide both sides by 5 to end up with t = 33, indicating that it takes the steamboat 33 hours to travel downstream.

Understanding how to manipulate these equations is a fundamental skill in algebra and will be applied in various contexts, be it straightforward or embedded in complex word problems.

Algebra Word Problems

Algebra word problems can often be intimidating, but with a structured approach, they become much more manageable. To solve these, it is essential to translate the given information into mathematical expressions or equations.

Let's dissect the steamboat problem: We are told about a steamboat's travel hours and speeds in different directions. The problem includes specific details such as speed, time, and distance, which we can correlate: A steamboat going downstream covers the same distance faster (8 miles per hour) than it does going upstream (3 miles per hour). We then create an equation reflecting the relationship: 8t = 3(t + 55).

From this concise equation, we can recognize the typical structure of word problems:

• Identify the variables.
• Formulate equations based on the relationships provided.
• Solve the equation using algebraic techniques.
• Interpret the solution in the context of the problem.

By breaking down problems into these steps, students can tackle even the most daunting word problems with confidence.

Rate Time Distance Problems

Rate time distance problems are a classic application of algebra, typically phrased like travel scenarios, as seen in the steamboat example. These problems are based on the fundamental formula Distance = Rate × Time. When two distances are equal and the rates and times vary, we can relate them through an equation.

In our steamboat scenario, the upstream and downstream trips cover the same distance. We know the rates and seek the travel times. By setting the distances equal to each other, 8t = 3(t + 55), we establish an equation that allows us to solve for the variable 't' which is the time. Once 't' is found, as shown in the step-by-step solutions, we calculate the total roundtrip time by accounting for the additional 55 hours taken upstream: 33 hours (downstream) + 33 hours (downstream time) + 55 hours (additional upstream time) = 121 hours.

This framework - understanding and applying the distance formula - is applicable to a wide variety of problems, making it a pivotal concept in algebra. Additionally, recognizing that the rates and times are inversely proportional aids in setting up the correct equations for these types of problems.