A train travels 360 km at a uniform speed. If the speed had been 5 km/h more, it would have taken 1 hour less for the same journey. Find the speed of the train.

Answer

To solve the problem, we will follow these steps:

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

### Step 2: Write the Distance and Time Relationship
The distance traveled by the train is 360 km. The time taken to travel this distance at speed \( x \) is given by:
\[ \text{Time} = \frac{\text{Distance}}{\text{Speed}} = \frac{360}{x} \text{ hours} \]

### Step 3: Write the Time for Increased Speed
If the speed had been increased by 5 km/h, the new speed would be \( x + 5 \) km/h. The time taken at this new speed would be:
\[ \text{New Time} = \frac{360}{x + 5} \text{ hours} \]

### Step 4: Set Up the Equation
According to the problem, if the speed had been 5 km/h more, it would have taken 1 hour less for the same journey. Therefore, we can set up the equation:
\[ \frac{360}{x} - \frac{360}{x + 5} = 1 \]

### Step 5: Solve the Equation
To solve the equation, we will first find a common denominator:
\[ \frac{360(x + 5) - 360x}{x(x + 5)} = 1 \]
This simplifies to:
\[ \frac{360 \cdot 5}{x(x + 5)} = 1 \]
\[ \frac{1800}{x(x + 5)} = 1 \]

### Step 6: Cross Multiply
Cross multiplying gives us:
\[ 1800 = x(x + 5) \]
This expands to:
\[ x^2 + 5x - 1800 = 0 \]

### Step 7: Factor the Quadratic Equation
To factor the quadratic equation \( x^2 + 5x - 1800 = 0 \), we look for two numbers that multiply to -1800 and add to 5. The numbers are 45 and -40. Thus, we can factor it as:
\[ (x + 45)(x - 40) = 0 \]

### Step 8: Solve for \( x \)
Setting each factor to zero gives us:
\[ x + 45 = 0 \quad \Rightarrow \quad x = -45 \quad \text{(not valid since speed cannot be negative)} \]
\[ x - 40 = 0 \quad \Rightarrow \quad x = 40 \]

### Conclusion
The speed of the train is \( 40 \) km/h.

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