A train travels a distance of 480 km at a uniform speed. If the speed had been 8 km/h less, then it would have taken 3 hours more to cover the same distance. We need to find the speed of the train.

To find the speed of the train, let's define the variables and set up the equations based on the information given.

Defining Variables

Let s be the speed of the train in km/h. The time taken to travel 480 km at speed s is:

Time = Distance / Speed = 480 / s

Setting Up the Equation

If the speed is reduced by 8 km/h, the new speed becomes (s - 8). The time taken at this reduced speed is:

Time = 480 / (s - 8)

Time Difference

According to the problem, the time taken at the reduced speed is 3 hours more than the time taken at the original speed. This gives us the equation:

480 / (s - 8) = 480 / s + 3

Solving the Equation

To eliminate the fractions, we can multiply through by s(s - 8):

• 480s = 480(s - 8) + 3s(s - 8)

Expanding both sides:

• 480s = 480s - 3840 + 3s² - 24s

Now, simplifying the equation:

• 0 = 3s² - 24s - 3840

Quadratic Equation

This simplifies to:

s² - 8s - 1280 = 0

Now we can use the quadratic formula, s = (-b ± √(b² - 4ac)) / 2a, where a = 1, b = -8, and c = -1280.

Calculating the Roots

Calculating the discriminant:

b² - 4ac = (-8)² - 4(1)(-1280) = 64 + 5120 = 5184

Now, applying the quadratic formula:

s = (8 ± √5184) / 2

Calculating the square root:

√5184 = 72

Thus, we have:

s = (8 ± 72) / 2

Finding the Speed

This gives us two potential solutions:

• s = (80) / 2 = 40
• s = (-64) / 2 = -32 (not valid)

Therefore, the speed of the train is 40 km/h.