Question

A train travels $400 \text{ km}$ at a uniform speed. If the speed had been $10 \text{ km/h}$ more, it would have taken $2$ hours less for the same journey, then the speed of the train is _____.

• A. $30 \text{ km/h}$
• B. $40 \text{ km/h}$
• C. $50 \text{ km/h}$
• D. $80 \text{ km/h}$

Hint:

In speed-time problems: \[ \text{Time} = \frac{\text{Distance}}{\text{Speed}} \] If speed increases, time decreases. Always use this relationship carefully while forming equations.

Answer 1

The Correct Option is B

Concept: The basic relation connecting distance, speed, and time is: \[ \text{Time} = \frac{\text{Distance}}{\text{Speed}} \] When speed increases, time decreases for the same distance.

Step 1: Assume the original speed. Let the original speed of the train be: \[ x \text{ km/h} \] Distance travelled: \[ 400 \text{ km} \] Therefore, original time taken is: \[ \frac{400}{x} \text{ hours} \]

Step 2: Form the expression for increased speed. If speed increases by $10$ km/h, then new speed becomes: \[ x+10 \] New time taken becomes: \[ \frac{400}{x+10} \]

Step 3: Use the information about time reduction. According to the question: \[ \text{Original Time} - \text{New Time} = 2 \] Thus, \[ \frac{400}{x} - \frac{400}{x+10} = 2 \]

Step 4: Take LCM and simplify carefully. LCM of denominators: \[ x(x+10) \] So, \[ \frac{400(x+10)-400x}{x(x+10)} = 2 \] Expand numerator: \[ \frac{400x+4000-400x}{x(x+10)} = 2 \] \[ \frac{4000}{x(x+10)} = 2 \]

Step 5: Remove the denominator. Cross multiply: \[ 4000 = 2x(x+10) \] \[ 4000 = 2x^2 + 20x \] Divide entire equation by 2: \[ 2000 = x^2 + 10x \] Bring all terms to one side: \[ x^2 + 10x - 2000 = 0 \]

Step 6: Factorize the quadratic equation. We need two numbers whose product is: \[ 1 \times (-2000) = -2000 \] and sum is: \[ 10 \] These numbers are: \[ 50 \quad \text{and} \quad -40 \] Therefore, \[ (x+50)(x-40)=0 \]

Step 7: Find the valid value of speed. So, \[ x=-50 \] or \[ x=40 \] Since speed cannot be negative: \[ x=40 \] Hence, the speed of the train is: \[ \boxed{40 \text{ km/h}} \]