Problem Statement

The question states:

• A train travels 132 km at a certain average speed.
• It then travels 140 km at an average speed that is 4 km/h faster than the initial speed.
• The total time for the journey is 4 hours.
• You are asked to determine:
  1. The initial average speed.
  2. The time taken to cover each part of the journey separately.

Let us solve step by step.

Let the initial average speed be $x \, \text{km/h}$.

1. Time for the first segment (132 km): The time is calculated as: $\text{Time}_1 = \frac{\text{Distance}_1}{\text{Speed}_1} = \frac{132}{x}$

2. Time for the second segment (140 km): The average speed for this segment is $x + 4 \, \text{km/h}$, so: $\text{Time}_2 = \frac{\text{Distance}_2}{\text{Speed}_2} = \frac{140}{x + 4}$

3. Total time: The total journey time is given as $4 \, \text{hours}$. Therefore: $\frac{132}{x} + \frac{140}{x + 4} = 4$

Solve for $x$

To solve: $\frac{132}{x} + \frac{140}{x + 4} = 4$

Multiply through by $x(x + 4)$ to eliminate denominators: $132(x + 4) + 140x = 4x(x + 4)$

Expand: $132x + 528 + 140x = 4x^2 + 16x$

Combine terms: $4x^2 - 16x - 132x - 140x + 528 = 0$ $4x^2 - 288x + 528 = 0$

Simplify: $x^2 - 72x + 132 = 0$

Solve using the quadratic formula: $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$ Here, $a = 1$, $b = -72$, and $c = 132$. Substitute: $x = \frac{-(-72) \pm \sqrt{(-72)^2 - 4(1)(132)}}{2(1)}$ $x = \frac{72 \pm \sqrt{5184 - 528}}{2}$ $x = \frac{72 \pm \sqrt{4656}}{2}$ $x = \frac{72 \pm 68.25}{2}$

Two solutions:

1. $x = \frac{72 + 68.25}{2} = \frac{140.25}{2} = 70.125$
2. $x = \frac{72 - 68.25}{2} = \frac{3.75}{2} = 1.875$

Since the speed must be reasonable for a train, $x = 70.125 \, \text{km/h}$ is the valid solution.

Time for each segment:

1. Time for the first segment: $\text{Time}_1 = \frac{132}{70.125} \approx 1.88 \, \text{hours}$

2. Time for the second segment: $\text{Time}_2 = \frac{140}{70.125 + 4} = \frac{140}{74.125} \approx 1.89 \, \text{hours}$

Total time: $\text{Time}_1 + \text{Time}_2 \approx 1.88 + 1.89 = 3.77 \, \text{hours} \, (\text{slightly less due to rounding errors}).$

Let me know if you'd like further clarification or another check! Here are 5 relative questions for practice:

Related Questions:

1. What is the total distance covered by the train in this journey?
2. If the total time was 5 hours instead of 4, what would the initial speed $x$ be?
3. What is the average speed for the entire journey (considering both segments)?
4. What would be the time taken if the second segment’s speed was $x + 8$ instead?
5. How does the solution change if the distance of the second segment was 160 km instead of 140 km?

Tip:

When solving equations with fractions, always eliminate denominators by multiplying through by the least common denominator!