Problem 38 - Intermediate Algebra

Question

Comparing Travel. A bicyclist can travel 40 miles in the same time that a motorcyclist can travel 60 miles. If the bicyclist travels 12 mph slower than the motorcyclist, find the speed of the motorcyclist.

Step-by-Step Solution

Answer

The speed of the motorcyclist is 36 mph.

Step 1: Understanding the Problem

We are given that a bicyclist can travel 40 miles in the same duration that a motorcyclist can travel 60 miles, and the bicyclist travels 12 mph slower than the motorcyclist. We need to determine the speed of the motorcyclist.

Step 2: Establishing Variables

Let the speed of the motorcyclist be \( x \) mph. Therefore, the speed of the bicyclist would be \( x - 12 \) mph, since the bicyclist travels 12 mph slower.

Step 3: Expressing Time for Each Traveler

The time taken to travel a certain distance can be expressed as \( t = \frac{d}{v} \), where \( d \) is distance, and \( v \) is speed. Using this, the time for the bicyclist is \( t = \frac{40}{x-12} \) and for the motorcyclist is \( t = \frac{60}{x} \).

Step 4: Setting Up the Equation

Since the time taken is the same for both travelers, we set the equations equal: \[ \frac{40}{x-12} = \frac{60}{x} \]

Step 5: Cross-Multiplying

To eliminate the fractions, we cross-multiply: \[ 40x = 60(x - 12) \]

Step 6: Solving the Equation

Expand the right side: \[ 40x = 60x - 720 \]. Then, isolate \( x \) by subtracting \( 60x \) from both sides: \[ -20x = -720 \]. Divide by -20: \[ x = 36 \].

Step 7: Checking the Solution

The motorcyclist's speed is 36 mph. The bicyclist’s speed would then be \( 36 - 12 = 24 \) mph. The time for both should be the same, so check: \( \frac{60}{36} \) equals about 1.67 hours, and \( \frac{40}{24} \) also equals about 1.67 hours. Both times are equal, so our solution is consistent.

Key Concepts

Distance, Rate, and Time

Understanding how distance, rate, and time relate to one another is key in solving travel-related problems. These concepts are closely linked by the formula:
\[ t = \frac{d}{v} \] where

• \( t \) stands for time
• \( d \) represents distance
• \( v \) denotes speed or rate

To imagine this relation, consider a simple journey. If you travel faster, you cover more distance in the same amount of time, or you can cover a fixed distance in less time. Conversely, traveling slower means you take longer to travel a fixed distance.

In our problem, we're given specific distances and are aware that both travelers, bicyclist and motorcyclist, take the same amount of time. By expressing this time using the mentioned formula, we are equipped to set up an equation to find unknown speeds. This underscores a vital fact: understanding one part of the equation helps solve for the others.

Algebraic Equations

Algebraic equations form the core foundation of many problem-solving techniques in math. They utilize symbols to represent quantities in equations, allowing you to solve for unknown values. In our exercise, we want to find the speed of the motorcyclist, which is initially unknown. We represent this unknown speed with a variable, say \( x \).

We can set up an equation:

• The speed of the motorcyclist: \( x \)
• The speed of the bicyclist: \( x - 12 \) (since it is 12 mph slower)

The next step is expressing time using equations. Using the idea from the previous concept, we equate the time for both the bicyclist traveling 40 miles and the motorcyclist traveling 60 miles. Thus, establishing a relationship using our formula for time:\[ \frac{40}{x-12} = \frac{60}{x} \]Cross-multiplying resolves our fractions, creating an easier pathway to isolate and solve for \( x \).

This manipulation relies on understanding and transforming variables and constants, giving you powerful tools to deal with unknowns.

Problem-Solving Strategies

Problem-solving in math involves a set process that can be applied to various problems. The exercise exemplifies such a strategy, breaking down complex tasks into manageable steps.

Here's a look at effective steps:

• Understand the Problem: Grasp the details, like knowing the bicyclist is slower by 12 mph and both take the same time.
• Identify Variables: Choose what you're solving for, and express the idea with variables.
• Set Up an Equation: Use relationships between distance, rate, and time to create a solvable equation.
• Solve: Simplify and manipulate the equation to find the unknowns.
• Check Your Solution: Confirm by substituting back into the original context, ensuring consistency.

This organized breakdown turns seemingly intricate problems into a series of smaller, clearer tasks. With practice, students can apply these strategies broadly, enhancing their problem-solving skills beyond distance and rate problems.