Understanding the Meeting Point Problem

This problem involves two individuals, Suman and Rekha, traveling towards each other from different locations (X and Y). They start simultaneously, meet at some point in between, and then continue their journey to the other's starting point. We are given the time each person takes to complete the remaining part of their journey after meeting, and the speed of one person. We need to find the speed of the other person.

Formula for Time After Meeting

When two people start simultaneously from two points and travel towards each other, and they meet at a point, if they take \(t_1\) and \(t_2\) time respectively after meeting to reach the destinations of the other person, and their speeds are \(v_1\) and \(v_2\) respectively, then the relationship between their speeds and times after meeting is given by:

\( \frac{v_1}{v_2} = \sqrt{\frac{t_2}{t_1}} \)

In this problem, Suman starts from X and goes towards Y, while Rekha starts from Y and goes towards X. They meet somewhere in between.

• Let \(v_S\) be the speed of Suman.
• Let \(v_R\) be the speed of Rekha.
• Let \(t_S\) be the time Suman takes to reach Y after meeting Rekha.
• Let \(t_R\) be the time Rekha takes to reach X after meeting Suman.

According to the problem:

• Suman reaches Y in 3 hours 12 minutes after meeting. So, \(t_S\) = 3 hours 12 minutes.
• Rekha reaches X in 1 hour 48 minutes after meeting. So, \(t_R\) = 1 hour 48 minutes.
• The speed of Rekha, \(v_R\), is given as 9 km/h.
• We need to find the speed of Suman, \(v_S\).

Converting Time Units

First, we need to convert the times given in hours and minutes into a single unit, either hours or minutes. It's usually easier to work with hours.

• 1 hour = 60 minutes.
• 12 minutes = \(\frac{12}{60}\) hours = \(\frac{1}{5}\) hours.
• 48 minutes = \(\frac{48}{60}\) hours = \(\frac{4}{5}\) hours.

Now, convert \(t_S\) and \(t_R\) into hours:

• \(t_S\) = 3 hours 12 minutes = \(3 + \frac{12}{60}\) hours = \(3 + \frac{1}{5}\) hours = \(\frac{15}{5} + \frac{1}{5}\) hours = \(\frac{16}{5}\) hours.
• \(t_R\) = 1 hour 48 minutes = \(1 + \frac{48}{60}\) hours = \(1 + \frac{4}{5}\) hours = \(\frac{5}{5} + \frac{4}{5}\) hours = \(\frac{9}{5}\) hours.

Calculating Suman's Speed

Now we can use the formula: \(\frac{v_S}{v_R} = \sqrt{\frac{t_R}{t_S}}\)

Substitute the known values:

\( \frac{v_S}{9} = \sqrt{\frac{\frac{9}{5}}{\frac{16}{5}}} \)

Simplify the fraction inside the square root:

\( \frac{\frac{9}{5}}{\frac{16}{5}} = \frac{9}{5} \times \frac{5}{16} = \frac{9}{16} \)

So the equation becomes:

\( \frac{v_S}{9} = \sqrt{\frac{9}{16}} \)

Calculate the square root:

\( \sqrt{\frac{9}{16}} = \frac{\sqrt{9}}{\sqrt{16}} = \frac{3}{4} \)

The equation is now:

\( \frac{v_S}{9} = \frac{3}{4} \)

Solve for \(v_S\):

\( v_S = 9 \times \frac{3}{4} \)

\( v_S = \frac{27}{4} \)

To express this as a mixed number:

\( \frac{27}{4} = 6 \text{ with a remainder of } 3 \)

So, \(v_S = 6 \frac{3}{4}\) km/h.

Conclusion

The speed of Suman is \(6\frac{3}{4}\) km/h.

Revision Table: Key Information

Additional Information on Speed, Time, and Distance

This problem is a specific case of relative speed problems. Here are some related concepts:

• Speed: The rate at which an object moves. It is calculated as distance traveled per unit of time. \( \text{Speed} = \frac{\text{Distance}}{\text{Time}} \).
• Distance: The total length of the path traveled by an object. \( \text{Distance} = \text{Speed} \times \text{Time} \).
• Time: The duration for which an object moves. \( \text{Time} = \frac{\text{Distance}}{\text{Speed}} \).
• Relative Speed: When two objects are moving, their relative speed is the speed of one object with respect to the other.
  • If they move in the same direction, relative speed is the difference between their speeds.
  • If they move in opposite directions (towards each other or away from each other), relative speed is the sum of their speeds.
• Meeting Point Problems: These often involve calculating the time or distance to the meeting point, or, as in this case, times or speeds related to the journey after meeting. The formula used above is a special case applicable when simultaneous start occurs and times *after* meeting are given.

Understanding these fundamental concepts of speed, time, and distance is crucial for solving various quantitative aptitude problems.