Question

Five friends Puru, Qadir, Roy, Sham and Tom live on five vertices of a regular pentagonal ground in a clockwise order. One day while Puru was walking to meet Roy, met Qadir who was on the way to meet Tom. Qadir changed his mind and decided to go with Puru to meet Roy. After meeting Roy all three went to Tom’s house for a cup of tea.
If all of them took shortest routes cutting through the ground, which of the following statements is/are TRUE?

• A. Till Tom’s place, Qadir walked more distance than Puru
• B. Till Tom’s place, Puru walked more distance than Qadir
• C. Till Tom’s place, Qadir walked double the distance than Roy
• D. While returning to their respective homes from Tom’s place, Puru has to travel the least.

Answer 1

The Correct Option is C, D

To solve the problem, consider the geometry of a regular pentagon with vertices labeled as P, Q, R, S, and T representing the houses of Puru, Qadir, Roy, Sham, and Tom respectively. The friends travel across the pentagon as follows: Puru travels from P to R to meet with Roy, while Qadir travels from Q towards T but meets Puru at P and then accompanies him to R. After meeting Roy, all three travel directly to T (Tom's house).

Since it's a pentagon, the shortest paths will involve straight lines (diagonals and/or edges of the pentagon). Let's analyze the distances traveled:

• Puru travels from P to R, then R to T, equivalent to a diagonal plus a segment of the pentagon.
• Qadir initially travels from Q to P (an edge), then travels from P to R with Puru (a diagonal), finally from R to T with Puru and Roy.
• Roy travels from R to T with Puru and Qadir, which is a single diagonal.

	Distance
Puru	P to R: Diagonal (d) and R to T: Segment
Qadir	Q to P: Edge (d/2), P to R: Diagonal (d), R to T: Segment
Roy	R to T: Segment

Analyzing these paths:

1. Qadir first travels an edge QP (d/2), then a diagonal PR (d), yielding a longer travel than Puru's continuous PR route initially.
2. Roy travels RT, the shortest segment, making Qadir's total till T twice the segment since Roy takes one diagonal when considering edge and diagonal together.
3. On returning home, Puru needs to travel the least back to his home compared to Qadir or Roy since he directly returns along the shortest path whereas others may have longer edges or diagonals to traverse.

Therefore:

• Qadir walked double the distance than Roy till Tom’s place.
• While returning from Tom's place, Puru travels the least distance back to his home.

The correct statements among the options offered are:

• Till Tom’s place, Qadir walked double the distance than Roy.
• While returning to their respective homes from Tom’s place, Puru has to travel the least.