 train travels a distance of 600 km at a constant 
speed. If the speed of the train is increased by 5 
km/hr, the journey would take 4 hrs less. Find the 
speed of the train

Let the speed of the train be \( x \) km/h. The time taken to travel 600 km at this speed is:

\[
\text{Time} = \frac{600}{x} \, \text{hours}
\]

If the speed is increased by 5 km/h, the new speed is \( x + 5 \) km/h, and the time taken for the journey becomes:

\[
\text{New Time} = \frac{600}{x+5} \, \text{hours}
\]

According to the problem, the journey takes 4 hours less at the increased speed. This gives the equation:

\[
\frac{600}{x} - \frac{600}{x+5} = 4
\]

Now, solve this equation.

### Step 1: Simplify the equation
Multiply both sides by \( x(x+5) \) to eliminate the denominators:

\[
600(x+5) - 600x = 4x(x+5)
\]

\[
600x + 3000 - 600x = 4x^2 + 20x
\]

\[
3000 = 4x^2 + 20x
\]

### Step 2: Form a quadratic equation
Rearrange the terms:

\[
4x^2 + 20x - 3000 = 0
\]

### Step 3: Simplify the quadratic equation
Divide the entire equation by 4:

\[
x^2 + 5x - 750 = 0
\]

### Step 4: Solve the quadratic equation
Use the quadratic formula:

\[
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\]

Here, \( a = 1 \), \( b = 5 \), and \( c = -750 \). Substituting these values:

\[
x = \frac{-5 \pm \sqrt{5^2 - 4(1)(-750)}}{2(1)}
\]

\[
x = \frac{-5 \pm \sqrt{25 + 3000}}{2}
\]

\[
x = \frac{-5 \pm \sqrt{3025}}{2}
\]

\[
x = \frac{-5 \pm 55}{2}
\]

### Step 5: Calculate the solutions
\[
x = \frac{-5 + 55}{2} = \frac{50}{2} = 25
\]
or
\[
x = \frac{-5 - 55}{2} = \frac{-60}{2} = -30
\]

Since speed cannot be negative, the speed of the train is:

\[
x = 25 \, \text{km/h}
\]

### Final Answer:
The speed of the train is **25 km/h**.

Would you like more details or have any questions?

Here are 5 related questions:
1. How would the solution change if the distance was 700 km instead of 600 km?
2. What if the speed increase was 10 km/h instead of 5 km/h?
3. How can this problem be solved graphically?
4. What is the total time saved with the new speed?
5. How does the quadratic formula help in solving real-world problems?

**Tip:** Always check if the quadratic equation can be factored before applying the quadratic formula for faster solutions!

A train travels a distance of 600 km at a constant speed. If the speed of the train is increased by 5 km/h, the journey would take 4 hours less. Find the speed of the train.

Solve the problem of finding the speed of a train traveling 600 km using quadratic equations. Learn how to set up equations and apply the quadratic formula. This step-by-step guide is ideal for students in Grades 9-11.

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