Problem 57 - Algebra for College Students

Question

Solve each of Problems \(47-62\) by setting up. Bret started on a 70-mile bicycle ride at 20 miles per hour. After a time he became a little tired and slowed down to 12 miles per hour for the rest of the trip. The entire trip of 70 miles took \(4 \frac{1}{2}\) hours. How far had Bret ridden when he reduced his speed to 12 miles per hour?

Step-by-Step Solution

Answer

Bret had ridden about 26.67 miles when he reduced his speed.

Step 1: Define Variables

Let the time Bret traveled at 20 miles per hour be \( t_1 \) hours and the distance traveled during this time be \( d_1 \) miles. Let the time Bret traveled at 12 miles per hour be \( t_2 \) hours and the distance traveled during this time be \( d_2 \) miles.

Step 2: Establish Time Equation

The total time for the trip is given as \(4 \frac{1}{2}\) hours, which is equivalent to 4.5 hours. Thus, we have the equation \( t_1 + t_2 = 4.5 \).

Step 3: Establish Distance Equation

The total distance covered during the trip is 70 miles. So, we have the equation \( d_1 + d_2 = 70 \).

Step 4: Express Distances in Terms of Time

Using the formula \( \text{distance} = \text{speed} \times \text{time} \), we can write \( d_1 = 20t_1 \) and \( d_2 = 12t_2 \).

Step 5: Substitute Distances into Total Distance Equation

Substitute the expressions for \( d_1 \) and \( d_2 \) into the total distance equation: \( 20t_1 + 12t_2 = 70 \).

Step 6: Solve the System of Equations

We have two equations: \( t_1 + t_2 = 4.5 \) and \( 20t_1 + 12t_2 = 70 \). Solve the first equation for \( t_2: t_2 = 4.5 - t_1 \). Substitute \( t_2 \) into the second equation and solve for \( t_1 \):\[20t_1 + 12(4.5 - t_1) = 70 \12t_1 + 54 = 70 \12t_1 = 16 \t_1 = \frac{4}{3} \text{ hours (or 1.33 hours)}\]

Step 7: Calculate Distance Traveling at 20 mph

Using \( d_1 = 20t_1 \), substitute \( t_1 = \frac{4}{3} \):\[d_1 = 20 \times \frac{4}{3} = \frac{80}{3} \text{ miles}\]This simplifies to approximately 26.67 miles.

Key Concepts

Algebraic Equations

Algebraic equations are mathematical statements that show the relationship between different variables. They are fundamental in solving various real-life problems, like finding distances or time taken for a journey. In the problem with Bret's bicycle ride, algebraic equations help us analyze the situation by relating the time and distance covered at different speeds.
Using the formula:

• \( \text{distance} = \text{speed} \times \text{time} \)

we can create equations for Bret's trip. Bret rode at two different speeds, 20 mph and 12 mph. Therefore, we need to define variables for the time spent at each speed, like \( t_1 \) and \( t_2 \). We then set up the equations to match the known total time and total journey distance. This approach helps identify unknowns, leading us towards the solution.
By expressing distance in terms of time and speed, we derive the equations \( d_1 = 20t_1 \) and \( d_2 = 12t_2 \). These equations are crucial as they allow the substitution needed to solve the problem.

Rate Time Distance Problems

Rate time distance problems are a category of math problems focused on calculating either the rate, time, or distance of moving objects. Solving these typically involves understanding the relationship expressed as:

• \( \text{distance} = \text{rate} \times \text{time} \)

This type of problem is often encountered in travel scenarios as we're asked to determine unknown values given certain conditions, such as speeds and travel time.
In the scenario with Bret, we have two segments of his journey with two different rates. The challenge is to find out how long he traveled at each rate given the total distance and total time. By setting up an equation for each segment of the journey, we capture his speed during each segment and multiply by the time spent at that speed to find the distances.* This structured approach highlights how variables interact and allows us to find the missing information efficiently.

Systems of Equations

When handling complex problems involving multiple variables and conditions, systems of equations become a valuable tool. They consist of multiple equations that are solved together to find the values of unknown variables.
For Bret's journey, we formed a system of equations to incorporate all the known and unknown factors. The equations include:

• \( t_1 + t_2 = 4.5 \) \( \text{hours} \)
• \( 20t_1 + 12t_2 = 70 \) \( \text{miles} \)

The first equation represents the total time spent biking, while the second accounts for the total distance.
To solve these equations, we use substitution. We express one variable in terms of the other from the first equation (e.g., \( t_2 = 4.5 - t_1 \)) and substitute it into the second equation. This process simplifies the system into a single equation that we solve to find the value of one variable. After determining \( t_1 \), we can plug it back to find \( t_2 \), giving us the complete solution. Systems of equations, thus, provide a systematic method for tackling multi-variable problems, ensuring all aspects of a scenario are considered in the solution.