Problem 299 - Elementary Algebra

Question

In the following exercises, solve. Cassius drives his boat upstream for 45 minutes. It takes \(\begin{array}{lll}\text { him } & 30 & \text { minutes to return }\end{array}\) downstream. His speed going upstream is three miles per hour slower than his speed going downstream. Find his upstream and downstream speeds.

Step-by-Step Solution

Answer

Upstream speed is 6 mph, downstream speed is 9 mph.

Step 1 - Define Variables

Let the speed of the boat downstream be denoted as \(d\) miles per hour. Therefore, the speed of the boat upstream is \(d - 3\) miles per hour.

Step 2 - Convert Time to Hours

Convert the provided times from minutes to hours: 45 minutes equals \(\frac{45}{60} = 0.75\) hours, and 30 minutes equals \(\frac{30}{60} = 0.5\) hours.

Step 3 - Write Distance Equation

The distance traveled upstream and downstream are the same. Using the formula distance = speed × time, we have: \(d \times 0.5 = (d - 3) \times 0.75\).

Step 4 - Simplify the Equation

Expand and simplify the equation \(d \times 0.5 = (d - 3) \times 0.75\): \(0.5d = 0.75d - 2.25\). Solve for \(d\) to get: \(0.5d - 0.75d = -2.25\), which simplifies to \(-0.25d = -2.25\), and therefore \(d = 9\).

Step 5 - Solve for Upstream Speed

Since the upstream speed is \(d - 3\), substituting \(d = 9\) gives: Upstream speed = \(9 - 3 = 6\) miles per hour.

Key Concepts

Distance-Rate-Time Relationship

Understanding the relationship between distance, rate, and time is crucial for solving algebra word problems. The basic formula to remember is:

• Distance = Rate × Time

This means the distance traveled is the product of the speed (rate) and the time taken to travel. In our example, Cassius travels the same distance upstream and downstream, but the rate and time vary. By knowing two of these variables, we can easily find the third. For instance, if Cassius drives his boat for 0.75 hours upstream and 0.5 hours downstream, knowing the respective speeds allows us to solve for the distance traveled.

Linear Equations

A linear equation is an algebraic equation where each term is either a constant or the product of a constant and a single variable. In solving our problem, we set up a linear equation to equate the distances: \[d \times 0.5 = (d - 3) \times 0.75\]

• Step 1: Define variables to represent the speeds.
• Step 2: Express the distances using the distance formula.
• Step 3: Expand and simplify the equation.

After simplifying, the given problem translates to a manageable linear equation: \[0.5d = 0.75(d - 3)\] This linear equation allows us to solve for the variable representing the downstream speed, illuminating the relationship between upstream and downstream speeds.

Variable Definition

Defining variables is the first step in solving any word problem. Variables are symbols used to represent unknown values. In our exercise, we let:

• \(d\) be the speed of Cassius's boat downstream in miles per hour.
• \(d - 3\) be the speed of the boat upstream, which is three miles per hour slower than downstream.

By clearly defining these variables, we can create equations that represent the relationships described in the problem. Doing so makes it easier to substitute and solve the equations.

Unit Conversion

Unit conversion is essential for consistency in equations, especially in time-based calculations. In our problem, time is given in minutes, but we need it in hours to match the standard rate units (miles per hour). The conversion process is simple:

• 45 minutes to hours: \(\frac{45}{60} = 0.75\) hours
• 30 minutes to hours: \(\frac{30}{60} = 0.5\) hours

Ensuring all units are consistent prevents errors and makes solving the equations straightforward.