Problem 229 - Elementary Algebra

Question

A motor boat traveled 18 miles down a river in two hours but going back upstream, it took 4.5 hours due to the current. Find the rate of the motor boat in still water and the rate of the current.

Step-by-Step Solution

Answer

The rate of the motorboat in still water is 6.5 mph, and the rate of the current is 2.5 mph.

Step 1 - Define Variables

Let the rate of the motorboat in still water be denoted by \( b \) (in miles per hour) and the rate of the current be denoted by \( c \) (in miles per hour).

Step 2 - Write Down Equations for Downstream

When traveling downstream, the effective speed of the boat is \( b + c \). Since it traveled 18 miles in 2 hours, we can write the equation: \[ 18 = (b + c) \times 2 \]

Step 3 - Simplify Downstream Equation

Simplify the equation from Step 2: \[ b + c = \frac{18}{2} \]\[ b + c = 9 \]

Step 4 - Write Down Equations for Upstream

When traveling upstream, the effective speed of the boat is \( b - c \). Since it traveled 18 miles in 4.5 hours, we can write the equation: \[ 18 = (b - c) \times 4.5 \]

Step 5 - Simplify Upstream Equation

Simplify the equation from Step 4: \[ b - c = \frac{18}{4.5} \]\[ b - c = 4 \]

Step 6 - Solve the System of Equations

We now have the system of equations: \[ b + c = 9 \]\[ b - c = 4 \]Add these equations to eliminate \( c \): \[ (b + c) + (b - c) = 9 + 4 \]\[ 2b = 13 \]\[ b = 6.5 \]

Step 7 - Find the Rate of the Current

Substitute \( b = 6.5 \) into the equation \( b + c = 9 \): \[ 6.5 + c = 9 \]\[ c = 9 - 6.5 \]\[ c = 2.5 \]

Key Concepts

Algebraic Equations

When working with word problems that involve rates or speeds, we often need to set up algebraic equations. This helps us to represent the given information mathematically. For instance, if a motorboat's speed in still water is denoted by 'b' and the current speed is 'c', traveling downstream can be modeled by the equation: \[ 18 = (b + c) \times 2 \]This form helps to clearly show the relationship between the distance, speed, and time. When done correctly, these equations become powerful tools to solve the problem step-by-step. The key to mastering algebra in rate problems is understanding how to extract and convert the word problem into accurate algebraic equations.

Problem-Solving

Solving rate problems can be easy once you understand the structure. Follow these steps:1. **Define your variables**: Clearly note what each variable stands for. In our example, 'b' stands for the boat's speed in still water and 'c' for the current speed.2. **Set up equations**: Use the given information to form equations. For instance, downstream speed is represented by \[ 18 = (b + c) \times 2 \]and upstream speed by \[ 18 = (b - c) \times 4.5 \].3. **Solve the equations**: Simplify the equations step-by-step and solve for the variables. Combine equations if needed to eliminate one variable and find the other.Breaking down the problem into smaller, manageable steps makes it much more approachable.

System of Equations

In many rate problems, particularly those involving multiple variables, we end up with a system of equations. A system of equations is a set of two or more equations with the same variables. They're solved simultaneously to find a common solution. In our exercise:1. From downstream: \[ b + c = 9 \]2. From upstream: \[ b - c = 4 \]To solve, we can add these equations to eliminate one variable. In this case, adding both equations:\[ (b + c) + (b - c) = 9 + 4 \]This simplifies to \[ 2b = 13 \]Then divide by 2 to find 'b'. Once 'b' is found, substitute back into one of the original equations to find 'c'. Working with systems of equations is made easier through consistent practice.

Current Speed

Understanding current speed is crucial in rate problems involving rivers or moving bodies of water. The current can either aid or hinder travel based on the direction. In our exercise, the motorboat's effective speed changes as it travels downstream or upstream:- **Downstream**: Boat speed + Current speed = Effective speed- **Upstream**: Boat speed - Current speed = Effective speedIn our exercise, when traveling downstream: \[ b + c = 9 \]When traveling upstream: \[ b - c = 4 \]By finding these effective speeds and solving the system of equations, we determine the actual boat speed in still water and the speed of the current. Identifying and using current speed appropriately can simplify and solve many rate problems effectively.