Distance Covered Before Stopping

Oct 13, 2025 by talynt.org 5 views

Introduction

In this article, we will delve into a mathematical problem involving a train's journey between two stations. The problem requires us to calculate the train's speed and the time it takes to cover the remaining distance. We will use algebraic equations to solve the problem and provide a step-by-step solution.

The Problem

Station A and Station B are 120 miles apart. A train traveling from Station A to Station B goes exactly $\frac{1}{3}$ of the distance before it has to stop for 16 minutes. When the train continues its journey, the engineer increases the speed by 10 miles per hour. What is the total time taken by the train to travel from Station A to Station B?

Breaking Down the Problem

To solve this problem, we need to break it down into smaller parts. Let's start by calculating the distance covered by the train before it stops.

Distance Covered Before Stopping

The train travels $\frac{1}{3}$ of the total distance before it stops. Since the total distance is 120 miles, the distance covered before stopping is:

\frac{1}{3} \times 120 = 40 \text{ miles}

Time Taken to Cover the Remaining Distance

After stopping for 16 minutes, the train continues its journey. The engineer increases the speed by 10 miles per hour. Let's assume the original speed of the train is $x$ miles per hour. Then, the new speed is $x + 10$ miles per hour.

The time taken to cover the remaining distance is given by:

\text{Time} = \frac{\text{Distance}}{\text{Speed}}

In this case, the distance is 80 miles (since 40 miles were covered before stopping), and the speed is $x + 10$ miles per hour.

Setting Up the Equation

We know that the train takes a total of 16 minutes to stop and then cover the remaining distance. We can set up an equation using the formula:

\text{Time} = \frac{\text{Distance}}{\text{Speed}}

Substituting the values, we get:

16 \text{ minutes} = \frac{80 \text{ miles}}{x + 10 \text{ mph}}

Converting Time to Hours

To make the units consistent, we need to convert the time from minutes to hours. There are 60 minutes in an hour, so:

16 \text{ minutes} = \frac{16}{60} \text{ hours} = \frac{4}{15} \text{ hours}

Solving the Equation

Now we can substitute the value of time in hours into the equation:

\frac{4}{15} \text{ hours} = \frac{80 \text{ miles}}{x + 10 \text{ mph}}

Cross-multiplying, we get:

(x + 10) \times \frac{4}{15} = 80

Simplifying the equation, we get:

\frac{4x + 40}{15} = 80

Multiplying both sides by 15, we get:

4x + 40 = 1200

Subtracting 40 from both sides, we get:

4x = 1160

Dividing both sides by 4, we get:

x = 290 \text{ mph}

Finding the Total Time

Now that we have the speed of the train, we can find the total time taken to travel from Station A to Station B. The train travels 40 miles before stopping, and then covers the remaining 80 miles at a speed of 300 mph.

The time taken to cover the first 40 miles is:

\text{Time} = \frac{\text{Distance}}{\text{Speed}} = \frac{40 \text{ miles}}{290 \text{ mph}} = \frac{8}{57.8} \text{ hours} = 0.139 \text{ hours}

The time taken to cover the remaining 80 miles is:

\text{Time} = \frac{\text{Distance}}{\text{Speed}} = \frac{80 \text{ miles}}{300 \text{ mph}} = \frac{8}{30} \text{ hours} = 0.267 \text{ hours}

The total time taken is the sum of these two times:

\text{Total Time} = 0.139 \text{ hours} + 0.267 \text{ hours} = 0.406 \text{ hours}

Converting Total Time to Minutes

To convert the total time from hours to minutes, we multiply by 60:

\text{Total Time} = 0.406 \text{ hours} \times 60 = 24.36 \text{ minutes}

Conclusion

In this article, we solved a mathematical problem involving a train's journey between two stations. We used algebraic equations to find the train's speed and the time it takes to cover the remaining distance. The total time taken by the train to travel from Station A to Station B is approximately 24.36 minutes.

Key Takeaways

The train travels $\frac{1}{3}$ of the total distance before stopping.
The time taken to cover the remaining distance is given by the formula: $\text{Time} = \frac{\text{Distance}}{\text{Speed}}$ .
We can set up an equation using the formula and solve for the speed of the train.
The total time taken is the sum of the time taken to cover the first 40 miles and the time taken to cover the remaining 80 miles.

Introduction

In our previous article, we explored a mathematical problem involving a train's journey between two stations. We used algebraic equations to find the train's speed and the time it takes to cover the remaining distance. In this article, we will answer some of the most frequently asked questions related to the problem.

Q: What is the distance between Station A and Station B?

A: The distance between Station A and Station B is 120 miles.

Q: How far does the train travel before it stops?

A: The train travels $\frac{1}{3}$ of the total distance before stopping, which is 40 miles.

Q: What is the speed of the train?

A: The speed of the train is 290 mph.

Q: How long does the train take to cover the remaining distance?

A: The train takes 0.267 hours (or 16 minutes) to cover the remaining 80 miles.

Q: What is the total time taken by the train to travel from Station A to Station B?

A: The total time taken by the train to travel from Station A to Station B is approximately 24.36 minutes.

Q: How did you calculate the speed of the train?

A: We used the formula $\text{Speed} = \frac{\text{Distance}}{\text{Time}}$ to calculate the speed of the train. We first calculated the time taken to cover the remaining distance, and then used the formula to find the speed.

Q: What is the significance of the 10 mph increase in speed?

A: The 10 mph increase in speed is significant because it allows the train to cover the remaining distance in less time. Without the increase in speed, the train would have taken longer to cover the remaining distance.

Q: Can you explain the concept of algebraic equations in more detail?

A: Algebraic equations are mathematical statements that contain variables and constants. They are used to represent relationships between variables and constants. In the context of the train's journey, we used algebraic equations to represent the relationship between the distance, speed, and time.

Q: How can I apply the concepts learned in this article to real-life situations?

A: The concepts learned in this article can be applied to real-life situations in various ways. For example, you can use algebraic equations to represent relationships between variables and constants in fields such as physics, engineering, and economics.

Q: What are some common mistakes to avoid when solving algebraic equations?

A: Some common mistakes to avoid when solving algebraic equations include:

Not following the order of operations (PEMDAS)
Not simplifying the equation before solving
Not checking the solution for validity
Not considering the units of measurement

Conclusion

In this article, we answered some of the most frequently asked questions related to the train's journey between two stations. We hope that this article has been helpful in clarifying any doubts you may have had about the problem. If you have any further questions or need further clarification, please don't hesitate to ask.

Key Takeaways

The train travels $\frac{1}{3}$ of the total distance before stopping.
The time taken to cover the remaining distance is given by the formula: $\text{Time} = \frac{\text{Distance}}{\text{Speed}}$ .
We can set up an equation using the formula and solve for the speed of the train.
The total time taken is the sum of the time taken to cover the first 40 miles and the time taken to cover the remaining 80 miles.