Problem 64 - Algebra and Trigonometry

Question

Boat Speed A boat on a river travels downstream between two points, 20 mi apart, in one hour. The return trip against the current takes 2\(\frac{1}{2}\) hours. What is the boat's speed, and how fast does the current in the river flow?

Step-by-Step Solution

Answer

The boat's speed in still water is 14 mph, and the current speed is 6 mph.

Step 1: Understand the Problem

We are asked to find the speed of the boat in still water and the current's speed in the river. Given that the boat travels 20 miles downstream in 1 hour and back upstream in 2.5 hours.

Step 2: Define Variables

Let \( b \) represent the boat's speed in still water and \( c \) represent the speed of the current. Then, downstream speed is \( b+c \) and upstream speed is \( b-c \).

Step 3: Write Equations

Using the formula \( \text{Speed} = \frac{\text{Distance}}{\text{Time}} \), create equations for downstream and upstream travel: \( b + c = \frac{20}{1} \) and \( b - c = \frac{20}{2.5} \).

Step 4: Simplify Equations

Simplify the equations: the first is \( b + c = 20 \), and the second is \( b - c = 8 \) since \( \frac{20}{2.5} = 8 \).

Step 5: Solve the System of Equations

Add the two equations to eliminate \( c \): \( 2b = 28 \), giving \( b = 14 \). Subtract the second equation from the first: \( 2c = 12 \), giving \( c = 6 \).

Step 6: Verify the Solution

Verify the calculated speeds in the original equations: Downstream: \( b+c = 14+6=20 \), and Upstream: \( b-c = 14-6=8 \). Both check out against the conditions.

Key Concepts

System of Equations

In many algebra problems, especially word problems, systems of equations come in handy. A system of equations is a collection of two or more equations with the same set of variables. In our boat speed problem, we have two equations involving two variables: the speed of the boat in still water and the current speed. Here, understanding each variable's role is key.

• First, identify the variables. Let’s define them as \( b \) for the boat's speed in still water and \( c \) for the current speed.
• Then form equations based on these definitions and the information given in the problem.
• Using the downstream and upstream scenarios, we derive the equations: \( b + c = 20 \) and \( b - c = 8 \).

By setting up these simultaneous equations, you can now solve for the unknown variables using algebraic methods such as addition or subtraction to eliminate one variable and find the other.

Speed and Distance Problems

Speed and distance problems are a practical application of algebra where you'll often need to find unknown values related to time, speed, and distance. The key formula to remember in these problems is: \[\text{Speed} = \frac{\text{Distance}}{\text{Time}}\]
Given the problem where a boat travels 20 miles downstream and takes 1 hour, it establishes that the downstream speed is \( \frac{20}{1} = 20 \) mph. Similarly, for the upstream journey taking 2.5 hours, the speed is \( \frac{20}{2.5} = 8 \) mph.

• Downstream speed combines both the speed of the boat and the current, hence \( b+c = 20 \).
• Upstream speed is reduced by the current, so it's \( b-c = 8 \).

Analyzing the distances and speeds in terms of the time taken in each direction gives a clear picture and helps set up the equations needed to find the unknown variables.

Problem Solving Steps

The process of solving word problems in algebra can often be broken down into specific steps, making them easier to tackle. For the boat speed problem, the steps would look like this:

• Understand the Problem: You need to determine what is being asked; here, it’s the speed of the boat in still water and the speed of the current.
• Define Variables: Assign symbols to unknown values. Use \( b \) and \( c \) to represent the boat speed and current speed, respectively.
• Write Equations: Formulate equations based on the relationships given by the problem, specifically the downstream and upstream data.
• Simplify Equations: Simplify the given equations to highlight the relationships between your variables accurately.
• Solve the System: Use addition or subtraction method to eliminate one variable and solve for the other.
• Verify Solution: Always check your solutions in the context of the problem to ensure they meet all the given conditions.

By following these structured steps, solving even complex word problems becomes more manageable and systematic.