A train covers a distance of 480 km at a uniform speed. If the speed had been 8 km/hr less then it would have taken 3 hours more to cover the same distance. Find the usual speed of the train.

Explanation

To solve the problem, we need to find the usual speed of the train given that it covers a distance of 480 km. If the speed were reduced by 8 km/hr, it would take 3 hours longer to cover the same distance.

Let's break down the solution step by step.

### Step 1: Define the Variables
Let the usual speed of the train be \( x \) km/hr.

### Step 2: Write the Time Equation
The time taken to cover 480 km at speed \( x \) is given by:
\[ t = \frac{480}{x} \]

### Step 3: Write the Time Equation for Reduced Speed
If the speed is reduced by 8 km/hr, the new speed becomes \( x - 8 \) km/hr. The time taken at this reduced speed is:
\[ t + 3 = \frac{480}{x - 8} \]

### Step 4: Set Up the Equation
From the above, we can set up the equation:
\[ \frac{480}{x - 8} = \frac{480}{x} + 3 \]

### Step 5: Clear the Denominators
Multiply through by \( x(x - 8) \) to eliminate the fractions:
\[ 480x = 480(x - 8) + 3x(x - 8) \]

### Step 6: Expand and Simplify
Expanding both sides:
\[ 480x = 480x - 3840 + 3x^2 - 24x \]
Now, simplify:
\[ 0 = 3x^2 - 24x - 3840 \]

### Step 7: Rearrange the Equation
Rearranging gives us:
\[ 3x^2 - 24x - 3840 = 0 \]

### Step 8: Divide by 3
To simplify, divide the entire equation by 3:
\[ x^2 - 8x - 1280 = 0 \]

### Step 9: Factor the Quadratic Equation
We need to factor the quadratic equation \( x^2 - 8x - 1280 = 0 \). We look for two numbers that multiply to -1280 and add to -8. These numbers are -40 and 32.

Thus, we can factor it as:
\[ (x - 40)(x + 32) = 0 \]

### Step 10: Solve for \( x \)
Setting each factor to zero gives:
1. \( x - 40 = 0 \) → \( x = 40 \)
2. \( x + 32 = 0 \) → \( x = -32 \) (not valid since speed cannot be negative)

### Conclusion
The usual speed of the train is:
\[ \boxed{40 \text{ km/hr}} \]