Question

Given below are two statements:
Statement I: Ravi walks from his house at a speed of 5 km per hour and reaches the college 10 minutes late. If he increases the speed by 1 km per hour next day, he reaches the college 4 minutes earlier than the scheduled time. If the college is P km far from his house, then P = 7.5 km.
Statement II: Amit runs \(2\frac{1}{3}\) times as fast as Babita. If Amit gives Babita a start of 80 meters, then the winning post must be 140 meters far so that Amit and Babita might reach it at the same time.
In the light of the above statements, choose the correct answer from the options given below.

• A. Both Statement I and Statement II are true
• B. Both Statement I and Statement II are false
• C. Statement I is true but Statement II is false
• D. Statement I is false but Statement II is true

Answer 1

The Correct Option is D

Let's analyze each statement step by step to determine their validity. 

1. Statement I Analysis:
   • Ravi walks from his house at a speed of 5 km per hour and reaches the college 10 minutes late.
   • If he increases the speed by 1 km per hour, he reaches the college 4 minutes earlier.

• Let \( T \) be the scheduled time to reach college in hours.
• Equation 1 (initial day):
  Ravi's speed = 5 km/h.
  Time taken = \( T + \frac{10}{60} \text{ hours} \).
  Distance \( P = 5 \times (T + \frac{10}{60}) \).
• Equation 2 (next day):
  Ravi's speed = 6 km/h.
  Time taken = \( T - \frac{4}{60} \text{ hours} \).
  Distance \( P = 6 \times (T - \frac{4}{60}) \).
• \(5T + \frac{5}{6} = 6T - \frac{6}{15}\)
• \(\frac{5}{6} + \frac{6}{15} = 6T - 5T\)
• \(T = \frac{5}{2} = 2.5\) hours
• \(P = 5 \times (2.5 + \frac{1}{6}) = 13.75 \text{ km}\)

1. Statement II Analysis:
   • Amit runs \(2\frac{1}{3}\) times as fast as Babita.
   • If Amit gives Babita a start of 80 meters, the post must be 140 meters away for them to meet simultaneously.

• Time taken by Babita to cover 140 meters is \( \frac{140}{b} \).
• Time taken by Amit to cover \(140 - 80 = 60\) meters is \( \frac{60}{\frac{7}{3}b} = \frac{60 \times 3}{7b} \).

Therefore, the correct answer is: Statement I is false but Statement II is true.