Question

A train covers a distance of 180 km at a uniform speed. If the speed had been 5 km/hour more, then it would have taken $\dfrac{1}{2}$ hour less for the same journey. Find the speed of the train.

Hint:

When dealing with speed–distance–time problems, use \( \text{Time} = \frac{\text{Distance}}{\text{Speed}} \), and form an equation comparing the two time conditions.

Answer 1

Step 1: Let the speed of the train be \( x \) km/h.
Distance \( = 180 \) km. Time taken \( = \frac{180}{x} \) hours.
Step 2: Write the equation for the second condition.
If the speed is 5 km/h more, then the time taken is \( \frac{180}{x + 5} \) hours. According to the question: \[ \frac{180}{x} - \frac{180}{x + 5} = \frac{1}{2} \]
Step 3: Simplify.
\[ 180 \left(\frac{(x + 5) - x}{x(x + 5)}\right) = \frac{1}{2} \] \[ 180 \times \frac{5}{x(x + 5)} = \frac{1}{2} \] \[ \frac{900}{x(x + 5)} = \frac{1}{2} \] \[ x(x + 5) = 1800 \] \[ x^2 + 5x - 1800 = 0 \]
Step 4: Solve the quadratic equation.
\[ x = \frac{-5 \pm \sqrt{5^2 - 4(1)(-1800)}}{2} \] \[ x = \frac{-5 \pm \sqrt{25 + 7200}}{2} = \frac{-5 \pm \sqrt{7225}}{2} \] \[ x = \frac{-5 \pm 85}{2} \] Taking the positive value, \[ x = \frac{80}{2} = 40 \] Step 5: Conclusion.
The speed of the train is \( \boxed{40\ \text{km/h}} \).